STEADY-STATE AND/TRANSffiNT STABILITY ANALYSES OF MULTIMACHINE POWER SYSTEMS USING FIVE DIFFERENT CATASTROPHE MODELS B Y KIN MING SUM B.A.Sc, UNIVERSITY OF BRITISH COLUMBIA, 1988 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR T H E MASTER DEGREE OF APPLIED SCIENCE IN THE F A C U L T Y OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as confirming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA May 1998 ©Kin Ming Sum, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) 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 steady-state 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 are derived for steady-state stability assessment. 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. i i i TABLE OF CONTENTS Page 1. ABSTRACT ii 2. T A B L E OF CONTENTS *v 3. LIST OF TABLES vl 4. LIST OF FIGURES vi? 5. A C K N O W L E D G E M E N T vii; 6. CHAPTER 1 Introduction.. 1 7. CHAPTER 2 Application of Catastrophe Theory to Steady-State Analysis of Multimachine Power Systems 4 8. CHAPTER 3 Application of Catastrophe Theory to Transient Stability Analysis of Multimachine Power Systems 33 9. CHAPTER 4 Discussion And Conclusions 51 10. REFERENCES : 54 11. APPENDIX A Derivation of One-Machine Infinite Bus Dynamic Equivalents of Multimachine Power Systems .. 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 m 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 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 steady-state 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. Changes in power flows and bus voltages are also of concern. 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 system when subjected to disturbances. Hence, power system stability becomes an increasingly important consideration in system planning and operation. Extensive stability studies are l 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. Also, these fast direct methods will help reduce the number of cases to be studied off-line. 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. Limitations and accuracy of these catastrophes will be discussed. 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. Much work has been done in the area of direct and fast assessment of transient stability [1]. 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]. Results and observations of the two test systems will be discussed in Section 2.5. Finally, conclusions will be stated in Section 2.6. 5 Catastrophe Theory Catastrophe theory was originally presented by Professor Rene Thorn and published in his book "Structural Stability and Morphogenesis" [8]. Thorn used differential topology to explain sudden changes in morphogenesis. 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. Given such a system, catastrophe theory tells us the following: 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. Specifically, if the number of the control parameters is not greater than four, 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 Rnand the control space R r respectively. 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 n X R r defined by . V Y V C ( Y ) = 0 , . (2.2) where V C (Y) = V ( Y , C ) and W i s the partial derivative with respect to Y . 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. These are the points at which V Y V C ( Y ) = 0 and V 2 Y V C ( Y ) =0 (2.3) The singularity set, S, is then projected down onto the control space R rby eliminating the 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 Y V C ( Y ) = y 3 + wy + x = 0 (2.4) where y is the only state variable and w, x are the control parameters and therefore V 2 y Vc(Y) = 3 y 2 + w = 0 (2 5) By algebraic manipulation of Equations (2.4) and (2.5), the cusp bifurcation set may be put in the form of 4w 3 + 27x 2 = 0 (2.6) 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 State Space Function Catastrophe Manifold Fold 1 1 y3/3+xy=0 y2+x=0 Cusp 2 1 y4/4+wy2/2+xy=0 y3+wy+x=0 Swallowtail 3 1 y5/5+vy3/3+wy2/2+xy=0 y4-r-vy2+wy-r-x=0 Butterfly 4 1 y6/6+Uy4/4+vy3/3+wy2/2+xy=0 y5+uy3+vy2-rwy+x=0 Wigwam 5 1 y7/7+ty5/5+uy4/4+vy3/3+wy2/2+xy=0 y6+ty4-ruy3+vy2+wy+x=0 y is state variable and t, u, v, w, x are control parameters . Table 2.2.1: Single State Space Dimension Catastrophes 8 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 Pe; E [E; Ej (gy cos 8ij + bij sin Sy)] (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 m . = mechanical power input of generator i ' p e. = electrical power output of generator i p a. = accelerating power of generator i 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. These machines are called critical machines; the other machines are non-critical. 9 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: k M k = £ mj (2.9) j=i I k 6k = — £ mj8j (2.10) M k j = 1 . 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: n Mo - Z nij (2.H) j=k+l V ' I n 80 = — £ mj5j (2.12) • Mo J k-l By suitable algebraic manipulation, the swing equation for the group of critical machines can be put in the form of: M ¥ k = P m - P c - T k S i n ( y k + ak) (2.13) where Mo M k M = (2.14) M 0 + M k Mo-Mk V = (2.15) Mo + M k ^k = 5 k - 5 0 (2.16) 10 Mo k M k „ Pm = — • E.Pmj " EPnij (217) M 0 +M k j 1 Mo+Mk j = k + 1 M„ k k M k n n Pc= [ E E EiEj (gy cosSy + by sin5,j)]-[- E £ EEj (gy cosSy + by sinSy)] (2.18) Mo+Mk i = l j = 1 '.. ' M 0 + M k i = k + l j = k + 1 T k = VAk 2 + B k 2 (2.19) k n Ak E E [ Ei Ej (u g i j cos (8U - Te) + b y sin (8y - T k )) ] (2.20) i-lj-k+l k n B k • = E E [ E ; Ej (by cos (8ij - T k ) - u g i j sin (8;j - Tk)) ] (2.21) i=lj=k+l a k = tan -1 (Ak /Bk) (2.22) or by defining Mi = 8;-8 k i= 1,2,3,.'. , k critical machines (2.23) • (p; = 8i - 80 i = k+l,k+2,...., n non-critical machines (2.24) P c, Ak and B k may be represented in the form of: Mo k k M k n n p c =[ — £ EEiEj (gijCOSTiij+bijSinriijJl-f £ £ EEjfejCos co ij+bySin (p ;j)] (2.25) Mo+M k , = , J = 1 . Mo+Mk , = k + l j = k + 1 k n Ak = EE [ E i E j ( p g i j C O s ( r i i - ( / ) j ) + bijSin(rii-(p j))] (2.26) i=lj=k+l k n Bk =' E E [ E ; Ej (by cos fa - 5y+E>6 (2.50) = y7+7fiy6+21B2y5+35l5y+35B4y3+21B5y2+7B6y+ft7 (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 6 , Equation (2.44) may be put in the following form: K2 K.2 K2 K1+K3 AT k 6 + 7 A ^-42 A ¥^ -210 AxPk3+840AxFk2+2520 A ^-5040 - = 0 (2.52) K.3 K3 K3 K3 Substitute A ^ = y + P, expand (y+P)n where n = 1, 2,..,6, and make the coefficient of y 5 zero so that: K 2 6p + 7 = 0, or (2.53) K 3 7K 2 P = (2-54) 6K 3 Hence we get the wigwam manifold equation y 6 + ty4 + uy3 + vy2 + wy + x = 0 ' ' (2.55) and it may be shown that (see Appendix B) t = -42- 15p2 (2.56) u = 4p (3 -10p 2 ) (2.57) v = 840 + P2 (288 - 45p 2) (2.58) w - - 12p(40-31p 2 + 2p 4 ) . (2.59) K, + K 3 x = - p 2 ( 1320 - 138p2 + 5 p 4 ) - 5040 (2.60) Note that K i is the only parameter which varies as A P m varies; K 2 and K 3 remain constant. Hence, P remains constant and x is the only control parameter which varies with A Pm. 15 2.4.3Butterfly Catastrophe Selecting terms of A T k to the fifth degree and normalizing A T k 5 , Equation (2.44) may be put in the following form: K3 K3 K1+K3 A T k 5 - 6- A T k 4 -30 A T k 3 +120 A T k 2 +360 A T k -720 = 0 (2.61) K 2 K 2 K2 Substitute A T k = y + p, expand (y+P)n where n = 1, 2,..,5, and make the coefficient of y 4 zero so that: (2.62) 6K 3 P -5K 2 Hence we get the butterfly manifold equation y5 + uy3 + vy2 + wy + x = 0 (2.63) where u = -30-10p 2 (2.64) v • = 1 0 P ( l - 2 p 2 ) (2.65) w = 5p2 (22 - 3p 2 ) + 360 (2.66) K, x = -p ( 240 - 70p2 + 4p 4 ) - 720 (2.67) K 2 . . Note again that Ki is the only parameter which varies as A P m varies; K 2 and K 3 remain constant. 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 4, Equation (2.44) may be put in the following form: K 2 K 2 K1+K3 A ^ k 4 + 5 : A ^ k 3 - 20 A ^ k 2 - 60 A Y k + 120—: = 0 (2:68) K3 K3 K3 Substitute A ^ k = y + P, expand (y+P)n where n = 1, 2,..,4, and make the coefficient of y 3 zero so that: 5K 2 P = (2,69) 4K 3 ' Hence we get the swallowtail manifold equation y4 + vy 2 + wy + x = 0 (2.70) where v - - 20 - 6p 2 (2.71) w = 8P ( 1 - p 2 ) (2.72) K, + K 3 x = P2(28-3P2)+ 120- (2.73) K 3 Note again that K\ is the only parameter which varies as A P m varies; K 2 and K 3 remain constant. Hence, p remains constant and x is the only control parameter which varies with A P m . 17 2.4.5Cusp Catastrophe Selecting terms of A T k to the third degree and normalizing A T k 3 , Equation (2.44) may be put in the following form: K3 K1+K3 - — A T k 2 - 1 2 A T k + 24-— K? K 9 A T k 3 - 4 - k 2 k =0 (2.74) Substitute A T k = y + P, expand (y+P)n where n = 1, 2,3, and make the coefficient of y2 zero so that: . 4K 3 (2.75) 3K 2 . Hence we get the cusp manifold equation y 3 + wy + x = 0 (2.76) where w = -12 - 3 p 2 (2.77) K i . x = 2P (3 - p2) + 24 (2.78) K 2 Note again that K i is the only parameter which varies as A P m varies; K 2 and K 3 remain constant. Hence, P remains constant and x is the only control parameter which varies with A P m . 18 2.4.6FoId Catastrophe Selecting terms of A^Pk to the second degree, normalizing A ^ k 2 , Equation (2.44) may be put in the following form: K 2 K1+K3 A T k - 6 — K3 K 3 A Y k 2 + 3 Vk-6— =0 (2.79) Substitute A Y k = y + P, expand (y+P)n where n = 1, 2, and make the coefficient of y zero, we find P as follows: 3K 3 P = (2.80) 2K 2 -Hence we get the fold manifold equation y 2 + x = 0 (2.81) where K, + K 3 . x = - P2 - 6 — . (2.82) K 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 m , of power systems can be determined when area A l 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 m ° + A P m - P c - T k sin(Tk+ak)]dTk T k ° AreaA2 = - $ [ P m ° + A P m - P c - T k s i n ( v P k + a k ) ] d T k After algebraic manipulation, Area (A2 - A l ) can be determined by : Area (A2-A1) = T k [cos(W + a k ) - c o s ( T k ° + a k ) ] - ( P m ° + A P m - P c ) ( T k u - T k p ) (2.85) 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. (2.83) (2.84) 20 r o + ' © o 1 ^—\ CO. + CN ' CQ. 0 0 r o © CN r o i—i ' *-—' c o . o CS r o /-—s m CO r o r-H CN Cd. 1 :ATASTROPHE CONTROL PARAMETERS /-^ o a + CN o a o t -i o CS c a i o CS CN c a • r o 1 0 0 CS CN' c a CS + CN c a • CO c a / CS CU +•> he Models: Manifold and Control Pan :ATASTROPHE CONTROL PARAMETERS > CN . c a m t 0 0 0 0 he Models: Manifold and Control Pan a /—\ c a © i—i • r o c a • * CN c a o t-H 1 o CO . 1 O l . •«-> V) « « u cu * J CN c a • >ri 1 CS CS 1 « cu ca i4 CN • 14 . r -• >n ND i4 • CN CS CN CO 1 N - < CZ5 fN CU TROPHE MODELS c -o H "II o II X i + > > o II X + + > + o II X • + > © II X + o II X + « CATAS " w PH l ' : cu s pa "3 :S . o t/3 a. S • CJ • 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 m , to power systems before the 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. System data are shown in Table 2.5.1 and 2.5.2. 18kV 230kV © Load A LoadC 230kV 13.8kV © ® ® LoadB • © Fig. 2.5.1 - Three-Machine WSCC System Configuration (three machines, nine buses) 22 Bus ' Generator Data No. Volt. Mag. /p.u. . Volt. Ang Real Gen. Power / p.u. Imag.Gen Pwr /p.u. RealLoad Power /p.u. Imag. Load Power /p.u. R+X'd /p.u. 1 1.04 0.00 0.71 0.27 0.00 • 0.00 0.000 0.060 2 1.02 9.30 1.63 • 0.06 0.00 0.00 0.000 0.119 3 1.02 4.70' 0.85 -0.109 0.00 0.00 0.000 • 0.181 4 . 1.02 -2.200 0.00 0.00 0.00 0.00 _ 5 0.99 -4.000 0.00 0.00 1.25 0.50 _ _ 6 ' 1.01 -3.700 • 0.00 0.00 0.90 0.30 _ • 7 1.02 3.70 • 0.00 0.00 0.00 0.00 _ _ 8 1.01 0.70 0.00 0.00 1.00 0.35 • _ 9 1.03 2.00 0.00 0.00 0.00 0.00 _ _ Transimission Line Bu Admittance Shunt/2 From To 1 4 . 0.00 - - _ 2 •7 0.00 - - _ • 3 9 0.00 . • - _ . • 4 5 1.365 - 0.00 0.08 4 6 1.942 - 0.00 0.07 5 7 1.187 - 0.00 0.15 6 9 1.28 - - 0.00 0.17 7 8 1.617 _ 0.00 0.074 8 9 1.155 - 0.00 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) ( % ) ( 1 > (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 •1.11 3.9 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 ,9.5 31.8 200 8 9 • 6 39.5 300 9 11 24.7 97 200 Note: ( 1 ) These values include the transformer's reactances and are expressed on a 100 MVA 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 / \ Cusp Catastrophe Bifurcation \ Envelope / ^ P M increase •% / — \ ^ S Motor Stable Region jBenerator Stable Reg ion / >. X Fig 2.5.3: The Cusp Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameters) 25 w 0 CtXneratorNR segion ^ X ^ x ^ ^ ^ V ^ Pm increase \ ( ^ ' ^ ^ > S s > 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) . / Generator Regions ^ ^ ^ * s « ^ / 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-Pm increase \ Generator \ o ^ Ji- Wigwam Catastrophe Bifurcation / . Envelope Fig 2.5.6: The Wigwam Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameter) - Generator w o 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 on right half 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. 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 E E A C (AP m ) 1.387184 p.u 1.460789 p.u. 1.484128 p.u. CATASTROPHE MODELS A P m . % Error A P m . % Error A P m . % 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% 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) E E A C Value ) X 100% c. 30 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. This is unsafe if the butterfly or the wigwam is used for stability assessment. 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 MACHINE 7 E E A C Solution (AP m ) 2.80473 p.u. CATASTROPHE MODELS A P m % Error Cusp 2.57149 p.u. .8,316% Swallowtail 2.62972213 p.u. 6.240% Butterfly 2.84079 p.u. -1.286%. Wigwam 2.82713 p.u. -0.799% 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) E E A C Value ) X 100% 31 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 THREE APPLICATION OF CATASTROPHE THEORY MODELTO 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 by finding the 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). Conclusions will be stated in Section 3.5. 33 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 c f l t - T k f l t s i n ( Y k + c O ] d ¥ k + S [Pm p r e-Pcp o s-Tk p o ssin(Yk+ak p o s)] d ¥ k = 0 (3.1) Y k ° ^ k c 34 For the limiting case of the system to be stable, i.e. (PnT - P c f l t) ( T k c - T k ° ) + T k f l t cos ( T k c + 4! 6! + K 3 A ¥ k 3 A ¥ k 5 A T k + +. 3! 5! + BC, = 0 (3.12) (3.13) (A^k) 2 = (y + ii) 2 = y2+2fly+fl2 (3.14) ( A Y , ) 3 = (y + fi)3 = y3+3fiy2+3fl2y+fj3 (3.15) (A*F k) 4 = (y + Ii)4 •= y4+4i5y3+6fl2y2+4fi3y+fi4 (3.16) ( A ¥ k ) 5 = (y + fi)5 = y5+5fiy4+10B2y3+l OBy+SIiV+fi5 Q.IT) ( A ¥ k ) 6 = (y + fi)6 = y6+6fiy5+l 5fi2y4+20I53y3+l 5By+6fi 5 y+B 6 (3.18) By selecting as many terms of (AYk) 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. 3 6 3.2.2Wigwam Catastrophe Selecting terms of A T k to the sixth degree and normalizing A T k 6 , Equation (3.12) may be put in the following form: K3 K3 Ki+ K3 K2+K4 A»Pk6 -6 A^k 5 -30ATk 4 +120 A T k 3 +360A*Fk2 -720 A f k -720 r- "='0 (3.19) K2 K-2 K.2 K.2 Substitute A T k = y + p\ expand (y+P)n where n = 1, 2,..,6, and make the coefficient of y5 zero so that: K 3 6 0-6 - = 0 , or . , • (3.20) K 2 K 3 K 2 Hence, we get the wigwam manifold equation (3.21) y 6 + ty4 + uy3 + vy2 + wy + x = 0 (3.22) and it may be shown that t = -15 (2+p 2) (3.23) u = - 40 p 3 (3.24) v = 45(8 + 4 p 2 - p 4 ) , (3.25) K, w = 24p3 ( 10 - p 2 ) - 720 (3.26) K 2 K 4 + K,P x = -5P 2 (72-18p 2 + p 4 ) -720 ( 1 + —-)• (3.27) K 2 Note that P is a function of K 2 and K 3 ; K 2 , K 3 ,and K 4 are functions of clearing angle (T k c ); Therefore, the wigwam control parameters t, u, v, w, and x varies with T k c . 37 3.2.3Butterfly Catastrophe Selecting terms of A^Fk to the fifth degree and normalizing A ^ k 5 , Equation (3.12) may be put in the following form: K.2 K2 K.1+K3 K2+K4 A T k 5 + 5 A T ^ - 2 0 A 4 ^ - 6 0 A ¥ ^ + 1 2 0 A ¥ k + 1 2 0 — = 0 (3.28) K 3 K3 K3 K 3 Substitute A ¥ k = y + p\ expand (y+P)n where n = 1, 2,..,5, and make the coefficient of y 4 zero so that: K 2 P (3.29) K 3 Hence, we get the butterfly manifold equation y 5 + uy3 + vy 2 +wy + x = 0 (3.30) where u = - 10(2 + p 2 ) (3.31) v -20P3 (3.32) K, w = 1 5 p 2 ( 4 - p 2 ) + 1 2 0 ( 1 + ^ — ) (3.33) K 3 ; K4+K1P x = 4P3( 1 0 - p 2 ) + 120 (3.34) K3 • Note that P is a function of K 2 and K 3 ; K 2 ,K 3 ,and K 4 are functions of clearing angle (^k0). Therefore, the butterfly control parameters u, v, w, and x varies with % 38 3.2.4SwalIowtaiICatastrophe Selecting terms of A T k to the fourth degree and normalizing A Tk 4 , Equation (3.12) may be put in the following form: K 3 K1+K3 K2+K4 A T k 4 - 4 — A T k 3 - 1 2 A T k 2 + 24 —— - A T k + 24 • = 0 (3.35) K 2 K 2 K 2 Substitute A T k = y + P, expand (y+P)n where n = 1, 2,..,4, and make the coefficient of y 3 zero so that: K 3 p = (3.36) K 2 Hence we get the swallowtail manifold equation y 4 + vy2 + wy + x = 0 (3.37) where v = - 6 ( 2 + p 2 ) (3.38) w = - 8p2 + 24 K, / K 2 (3.39) K4 + K,p x = 3p 2 ( 4 - p2) + 24( 1 +—• ) . (3.40) K 2 Note that P is a function of K 2 and K 3 ; K 2 , K 3 ,and K 4 are functions of clearing angle (TkC). Therefore, the swallowtail control parameters v, w, and x varies with Tk c. 39 3.2.5Cusp Catastrophe Selecting terms of A T k to the third degree and normalizing A T k 3 , Equation (3.12) may be put in the following form: K.2 K 1 + K 3 K 2 + K 4 A T k 3 + 3 A T k 2 - 6 A T k - 6 — — =0 (3.41) K 3 K 3 K 3 Substitute A T k = y + P, expand (y+P)n where n = 1, 2,.3, and make the coefficient of y 2 zero, we find P as follows: K 2 p = (3.42) K 3 Hence we get the cusp manifold equation y 3 +wy + x = 0 (3.43) where w = -3p2 + 6 ( l + - — ) (3.44) K 3 K4+K1P x = 2p 3 -6 (3.45) K 3 Note that P is a function of K 2 and K 3 ; K 2 , K 3 ,and K 4 are functions of clearing angle (Tk°). Therefore, the cusp control parameters w and x varies with TkC. 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 3.2.6Fold Catastrophe Selecting terms of A*F k to the second degree and normalizing A^k, Equation (3.12) may be put in the following form: K1+K3 K.2+K4 A^Fk 2 -2 • A 4 V 2 - =0 (3.46) K2 K2 Substitute A ^ k = y + P, expand (y+P)n where n = 1, 2, and make the coefficient of y zero sot that: K,+K3 p = (3.47) K2 Hence we get the fold manifold equation y 2 + x = 0 (3.48) where x - p 2 - 2( 1 + - — ) (3.49) : K2 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 c f l t - T k f l t s i n ^ + a ^ d ^ (3.50) 4V Area 2 = - $ [PnT + Pc"05 - T ^ 8 sin(^k+ak)]dTk (3.51) Refer to Section 3.2 for symbol T k ^ c o s W + c O - T k ^ r o ^ + Tk f l tcos(4V + a k f l V T ^ ^ (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 H H O UJ Q O. w en O H H ca CO. +V o C N ca + ca oo C N CQ. © C N r -CO. "* C N ca + 00 ca o + . If + + /- N c a o C N ca • o ca + CQ. 4 ca o CN ca + C N / \ CQ. + CN + ca CO C N + ca oo ca + C N + • + ti C s ea *4 o + + > •a o CO + ca C N + ca CO i4 + + . •v. s U Ui + x + o u —< E [ E E; Ej (gy cos 5y + by sin 8y)] Mo+Mk i=k+l j=l M„ P - P Mo+Mk k n E [ E E; Ej (gy cos 5y + by sin 8y)] | i=l j=k+l M k I k n + ^ ,E [ E E; Ej (gy cos 5y - by sin 6y)] Mo+Mk li=l j=k+l Mo P -P A m A c Mo+Mk k n E [ E Ei Ej ( gy COS( gy-Tk+Tk ) + by sin( 6y-T k +T k ))] i=l j=k+l ' M k + — -< Mo+Mk k n E [ E E; Ej (gy cos(8y-T k+T k) - by sin( 8y -T k +T k ) ) ] i=l j=k+l p - p A m -1- c Mo k n E E i i=l j=k+l Mo+Mk M k k n + E E Mo+Mk i=l j=k+l Ei Ej gy [cos(8ij - Tk) cosTk - sin(8y-Tk) sinT k ] + Ei Ej by [sin(8y - T k ) cosT k + cos(8y - T k ) sinT k ] Ej Ej gy [cos(8y - T k ) cosT k - sin(8y - T k ) sinT k ] - Ei Ej by [sin(8y - T k ) cosT k + cos(8y - T k ) sinT k ] 57 + Pm-P c _J k „ Mo+Mk i=1J=?+i ^ Mo Ej Ej [gy cos(8ij-Yk) + ,bij sinCSij-^k) ] -Mk E; Ej [gy cos(8ij-¥k) - by sinCSy-^ fc) ] Mo Ei Ej [-gy sinCSij-^ k) + by cos(5y-¥k) ] - M k Ei Ej [-gy sinCSy-^k) - by cos^y-^) ] > COS^k sin^k P - P E E E; Ej [ .11 gy cosCSy-Yk) + by sin(8s-,Pk) ] COS T k i=lj=k+l k n + E E E; Ej [ by cos(8y-Tk) - u gij sinCSy-^k) ] sin Wk i=lj=k+l P m - P c - [ T k sin ak cos *Pk + T k cos a k sin Y k ] Pm-Pc-[T k s inCPk + a k ) ] 58 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 2 + 7(K 2 /K 3 ) (5p)-42 = 15p 2 + (-6P)(5p)-42 = - 42 - 15 p 2 b. Parameter u (coefficient of y 3 ) = 20 p 3 + 7 (K 2 / K 3 ) (10p2) - 42 (4P)-210(K2 / K 3 ) . = 20 P3 + (-63) (10P2) - 168P - 30(-6P) = 20p 3-60p 3-168p+180p = -40p3+12p = 4p(3-10p 2) c. Parameter v (coefficient of y 2 ) = 15 p 4 + 7 (K 2 / K 3 ) (10p3) - 42 (6p2) -210(K2 / K 3 ) (3p) + 840 = 15 P4 + (-6p)(10p3) - 42 (6P2) - 30(-6p) (3p) + 840 = -45 P4 - 252 P2 + 540 p 2 +840 = -45 p 4 + 288 p 2 +840 = 840 + p2(288 -45p2) 59 Parameter w ( coefficient of y) = 6 p5 + 7 (K2 / K3) (5p4) - 42 (4p3) -210(K2 / K3) (3p2) + 840(2P) +2520(K2 / K 3) = 6 p5 + (-6p)(5p4) - 42 (4P3) - 30(-6p) (3p2) + 840(2p) + 360 (-6P) --24p 5 - 168p3 + 540p3 + 1680P - 2160P =-24P5 + 372p3 - 480p = -12P(40-31p2 + 2p4) Parameter x (constant term) = P6+7(X2/K3)(p5)-42(P4)-210(X^ = p6+ (-6p)(p5) - 42(p4) - 30(-6p)(p3) + 840(P2) + 360 (-6p)P - 5040(K,+K3)/K3 = -5p6 - 42(p4) + 180p4 + 840(p2) - 2160p2 - 5040(K1+K3)/K3 = -5p6 + 138P4 - 1320p2 - 5040(Ki+K3)/K3 =-p2 (1320 - 138P2 + 5p4) - 5040(K1+K3)/K3 P = 6 K 3 / 5 K 2 (By setting coefficient of y5 to zero) 60 A P P E N D I X C T H R E E - M A C H I N E W S C C T E S T S Y S T E M C A T A S T R O P H E M O D E L S S I M U L A T I O N R E S U L T S ( S T E A D Y - S T A T E S T A B I L I T Y A N A L Y S I S ^ ( C R I T I C A L M A C H I N E = 2) C A S E 00: S Y S T E M D A T A A N D R E S U L T S U M M A R Y C A S E C I : C U S P C A T A S T R O P H E C A S E C2: S W A L L O W T A I L C A T A S T R O P H E C A S E C3: B U T T E R F L Y C A T A S T R O P H E C A S E C4: W I G W A M C A T A S T R O P H E 61 CASE 00: WSCC System'Data and Result Summary Table 00: WSCC System Data of One Machine Infinite Bus (3 Machines, 9 Buses) NofCm 1 1 1 2 2 2 Cm 1 2 3 1 &2 1&3 2 & 3 Mk 0.1254 0.034 0.016 0.1594 0.1414 0.0499 Mo 0.0499 0.1414 0.1594 0.016 0.034 0.1254 Pm° -1.5711 1.0113 0.5598 -0.5598 -1.0113 1.5711 PC -0.5911 0.0497 0.0769 -0.0769 -0.0497 0.5911 Tk 2.9998 2.8487 2.4813 2.4813 2.8487 2.9998 Alpha -0.0643 0.0608 0.0701 -0.0701 -0.0608 0.0643 Delk 0.0396 0.3447 0.2304 0.1046 0.0612 0.3082 Delo 0.3082 0.0612 0.1046 0.2304 0.3447 0.0396 Table CI: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Cusp Catastrophe Method # of Critical Machine(s) 1 1 2 Critical Machine(s) 2 3 2 & 3 PM Limit (1) 1.38718 1.460789 1.484128 PM Limit (2) 1.27761 1.3396 1.36613 % Error 7.90% 8.30% 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 1.460789 1.484128 PM Limit (2) 1.3342 1.3811 1.428 % Error 3.82% 5.46% 3.78% Table C3: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Butterfly Catastrophe Method # of Critical Machirie(s) 1 1 2 Critical Machine(s) 2 3 2&3 PM Limit (1) 1.38718 1.460789 1.484128 PM Limit (2) 1.3993 1.4779 1.4974 % Error -0.87% -1.17% -0.89% Table C4: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Wigwam Catastrophe Method # of Critical Machine(s) 1 1 2 Critical Machine(s) 2 3 2 & 3 PM Limit (1) 1.38718 1.460789 L484128 PM Limit (2) 1.3914 1.4696 1.494 % Error -0.30% -0.60% -0.67% 62 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 CASE CI: Cusp Catastrophe Table CIA: WSCC System Steady-State Stab lity Analysis Cusp Catastrophe Numerical Result U of CM 1 2 CMs 2 3 2&3 > W -12.6859 -12.21 -12.6373 X Limit . 17.3913 16.4217 17.2913 PM Chg. x , cm = 2 cm = 3 cm = 2,3 0.000 5.9563 3.212 5.7267 0.100 6.8514 4.1981 6.5733 0.200 7.7464 5.1842 7.4198 0.300 8.6414 6.1703 8.2663 0.400 9.5364 .7.1564 9.1128 0.500 10.4315 8.1425 9.9593 0.600 11.3265 9.1286 10.8059 0.700 12.2215 10.1146 11.6524 0.800 13.1166 11.1007 12.4989 0.900 14.0116 12.0868 13.3454 1.000 14.9066 13.0729 14.1919 1.040 15.2646 13.4673 14.5305 1.050 15.3541 13.5659 14.6152 1.060 15.4436 13.6646 14.6998 1.070 . 15.5331 . 13.7632 14.7845 1.080 15.6226 13.8618 14.8691 1.090 15.7121 13.9604 14.9538 1.100 15.8016 14.059 15.0384 1.110 15.8912 14.1576 15.1231 . 1.120 15.9807 14.2562 .15.2078 1.130 16.0702 14.3548 15.2924 1.140 16.1597 14.4534 . 15.3771 1.150 16.2492 < 14.552 15.4617 1.160 16.3387 14.6506 15.5464 1.170 16.4282 14.7493 15.631 1.180 16.5177 14.8479 15.7157 1.190 16.6072 14.9465 15.8003 1.200 16.6967 15.0451 . 15.885 1.210 16.7862 15.1437 15.9696 1.220. 16.8757 15.2423 16.0543 1.230 . 16.9652 15.3409 16.1389 1.240 17.0547 15.4395 16.2236 1.250 17.1442 • 15.5381 16.3082 . 1.260 ' 17.2337 - 15.6367 16.3929 1 270 17 3232 15.7353 ' 16.4775 1 2S0 17.4127 15.8339 16.5622 1.290 17.5022 • 15.9326 16.6468 1.300 17.5917 16.0312 16.7315 1.310 17.6812 16.1298 16.8161 1.320 17.7707 16.2284 16.9008 1 330 17.8602 te 327 16.9854 1 340 . 17:9497 16 4256 17.0701 1.350 18.0392 16.5242 17.1547 1 360 18.1287 16.6228 17.2394 I 370 18.2182 16.7214 17.3241 1.380 18.3077 16.82 17.4087 1.390 18.3972 16.9186 17.4934 64 JOJDIUBJBJ AV CASE C2 : Swallowtail Catastrophe Table C2A:WSCC System Steady-State Stablity Analysis Swallowtail Catastrophe Numerical Result #ofCM 2 CMs •2 3 2&3 V -92.9012 -258.0824 -98.4546 W -310.9311 -1949.2561 -349.3318 X Limit 1 -269.1737 -3141.8005 -321.3642 X Limit 2 190.5849 -3955.6934 154.5994 X Limit 3 4393.9056 40400.7569 5013.4189 PM Chg. X cm = 2 cm = 3 cm = 2,3 0.000 -102.6763 -3612.5506 -146.8055 0.100 -115.1555 -3637.397 . -159.0497 0.200 -127.6348 -3662.2433 -171.2939 0.300 -140.114 -3687.0897 -183.5381 0.400 -152.5933 -3711.9361 -195.7823 0.500 -165.0725 -3736.7825 -208.0265 0.600 -177.5518 -3761.6289 -220.2707 0.700 -190.031 -3786.4753 -232.5149 0.800 -202.5103 -3811.3217 -244.7591 0.900 -214.9896 -3836.1681 -257.0033 0.910 -216.2375 -3838.6527 -258.2277 0.920 -217.4854. -3841.1374 -259.4521 0.930 -218.7333 -3843.622 -260.6765 0.940 -219.9813 -3846.1066 -261.9009 0.950 -221.2292 -3848.5913 -263.1254 0.960 -222.4771 -3851.0759 -264.3498 0.970 -223.725 -3853.5606 -265.5742 0.980 -224.973 -3856.0452 -266.7986 0.990 -226.2209 -3858.5298 -268.023 1.000 -227.4688 -3861.0145 -269.2475 1.100 -239.9481 -3885.8609 -281.4917 1.200 -252.4273 -3910.7073 -293.7359 1.210 -253.6752 -3913.1919 -294.9603 1.220 -254.9232 -3915.6765 -296.1847 1.230 -256.1711 -3918.1612 -297.4091 1.240 -257.419 -3920.6458 -298.6335 1.250 -258.6669 -3923.1304 -299.858 1.260 -259.9149 . -3925.6151 -301.0824 1:270 -261.1628 -3928.0997 -302.3068 1.280 -262.4107 -3930.5844 -303.5312 1.290 -263.6586 -3933.069 -304.7556 1.300 -264.9066 -3935.5536 -305.9801 1.310 -266.1545 . . -3938.0383 -307.2045 1.320 -267.4024 -3940.5229 -308.4289 1.130 -268 6503 -3943.0076 -309.6533 1.340 -269 898^ -3945.4922 -310.8777 1.350 -271.1462 -3947.9768 -312.1022 1.360 -272.3941 -3950.4615 -313.3266 1.370. -273.642 -3952.9461 -314.551 1.380 -274.89 -3955 4308 -315.7754 L390 -276.1379 M^m'i -316.9998 1.400 -277.3858 -3960.4 -318.2243 1.410 -278.6337 -3962.8847 -319.4487 1.420 -279.8817 -3965.3693 -320 6731 1,430 -281.1296 -3967.8539 '4mwi 1.440 -282.3775 -3970.3386 -323.1219 66 - 1 m l_ J3)3UIGJBJ M CASE C3: Swallowtail Catastrophe Table C3A:WSCC System Steady-State Stablity Analysis Butterfly Catastrophe Numerical Result #ofCM 1 2 CMs 2 3 2 & 3 u -31.8518 -30.567 -31.7207 v -2.7095 -2.1112 -2.7206 w 379.8557 366.1891 378.484 X Limit 1 521.1786 522.0345 522.4602 X Limit 2 566.7675 548.7188 565.4026 X Limit 3 -536.1656 -523.8765 -534.5544 X Limit 4 -448.2173 -469.4369 -449.7495 PM Chg. X cm = 2 cm = 3 cm = 2,3 0.000 -160.4373 -86.6668 -154.282 0.100 ,. -187.2882 -116.2494 -179.6775 0.200 -214.139 -145.832 -205.0731 0.300 -240.9899 -175.4146 -230.4687 0.400 -267.8407 -204.9972 -255.8643 0.500 -294.6916 -234.5798 -281.2599 0.600 -321.5424 -264.1624 -306.6554 0.700 -348.3933 -293.7451 -332.051 0.800 -375.2442 -323.3277 -357.4466 0.900 -402.095 -352.9103 -382.8422 1.000 -428.9459 -382.4929 -408.2378 1.150 -469.2222 -426.8668 -446.3311 1.200 -482.6476 -441.6581 -459.0289 1.210 -485.3327 -444.6164 -461.5685 1.220 -488.0178 -447.5747 -464.108 1.230 -490.7029. -450.5329 -466.6476 1.240 -493.3879 -453.4912 -469.1872 1.250 , -496.073 -456.4494 •471.7267 1.260 -498.7581 -459.4077 -474.2663 1.270 -501.4432 -462.366 -476.8058 1.280 -504.1283 -465.3242 -479.3454 1.290 -506.8134 -468.2825 •481.885 1.300 -509.4985 •471.2408 -484.4245 1.310 -512.1835 -474.199 -486.9641 1.320 -514.8686 -477.1573 -489.5036 1.330 -517.5537 480.1155 -492.0432 . 1.340 -520.2388 -483.0738 -494.5827 1.350. -522.9239 -486.0321 .-497.1223 1.360 -525.609 -488.9903 -499.6619 1.370 -528.2941 -491.9486. -502.2014 1.380 -530.9791 -494.9068 -504.741 I 390 -533 6642 497.8651 r507.2805 1 400 -536.3493 -500.8234 -509.8201 1.410 -539.0344 . -503.7816 -512.3596 1.420 -541.7195 . -506.7399 -514.8992 1.430 -544.4046 -509.6982 -517.4388 1.440 -547.0897 -512.6564 -519.9783 1.450 -549.7747 -515.6.147 -522.5179 1.460 -552.4598 -518.5729 -525.0574 1.470 -555.1449 -521.5312 -527.597 1.480 -557.83 -524.4895 -530.1366 ' . 1.490 -560.5151 -527.4477 -532.6761 1 500 . -563.2002 -530.406 -535.2157 68 008- ' — J » ) D 1 U R J B J M CASE C4: Wigwam Catastrophe Table C4A:WSCC System Steady-State Stablity Analysis Wigwam Catastrophe Numerical Result #ofCM 1 2 CMs 2 3 2 & 3 t -200.7626 -560.4905 -212.8567 u -1338.3129 -8058.3963 -1497.1980 V -1152.8688 -42971.4657 -1717.9517 w 2500.8998 -95814.1378 2171.5658 x Limit 1 -786.6426 -71906.9151 -492.5274 x Limit 2 2852.3826 -75600.0695 3136.1450 x Limit 3 4086299.7255 133355984.4960 5063398.1860 PM Chg. X cm = 2 cm = 3 cm = 2,3 0.000 -4440.142 -87242.6306 -4520.0676 0.100 -3916.0135 -86199.0822 -4005.8112 0.200 -3391.8849 -85155.5338 -3491.5547 0.300 -2867.7563 -84111.9854 -2977.2983 0.400 -2343.6278 -83068.4371 -2463.0419 0.500 -1819.4992 -82024.8887 -1948.7854 0.600 -1295.3706 -80981.3403 -1434.529 0.700 -771.242 . -79937.7919 -920.2726 0.800 -247.1135 -78894.2435 -406.0161 0.900 277.0151 -77850.6951 108.2403 1.000 801.1437 -76807.1468 622.4967 1.050 . 1063.208 -76285.3726 879.625 1.100 1325.2723 -75763.5984 1136.7532 1.150 1587.3365 -75241.8242 1393.8814 1.200 1849.4008 -74720.05 1651.0096 1.250 2111.4651 -74198.2758 1908.1378 1.270 2216.2908 -73989.5661 2010.9891 1.280 2268.7037 -73885.2113 2062.4148 1.290 2321.1165 -73780.8564 2113.8404 1.300 2373.5294 -73676.5016 2165.266 1.310 2425.9423 -73572.1468 2216.6917 1.320 2478.3551 -73467.7919 2268.1173 1.330 2530.768 -73363.4371 ,2319.543 1.340 2583.1808 -73259.0823 2370.9686 1.350 2635.5937 -73154.7274 2422.3943 1.360 2688.0065 -73050.3726 2473.8199 . 1.370 2740.4194 -72946.0177 2525.2455 1.380 2792.8323 -72841.6629 2576.6712 1390 2845 2451 -72737.3081 2628.0968 I 400 2897*65$ -72632.9532 2679.5225 1.410 2950.0708 -72528.5984 2730.9481 1.420 3002.4837 . -72424.2436 2782.3738 1.430 3054.8965 -72319.8887. 2833.7994 1.440 3107.3094 -72215.5339 2885.2251 1.450 3159.7223 -72111.179 2936.6507 1 4o0 3212.1351 -72006 8242 2988.0763 1.47(1 3264.548 -71902.4694 3039.502 1.480 3316.9608 -71798.1145 3090.9276 1 490 3369.3737 -71693.7597 3 1.500 3421.7865 -71589.4048 3193.77*9 70 APPENDIX D SEVEN-MACHINE CIGRE TEST S Y S T E M CATASTROPHE MODELS SIMULATION RESULTS (STEADY-STATE STABILITY ANALYSIS^) (CRITICAL MACHINE = 7) CASE DO: SYSTEM D A T A AND RESULT S U M M A R Y CASED1: CUSP CATASTROPHE CASE D2: SWALLOWTAIL CATASTROPHE CASE D3: B U T T E R F L Y CATASTROPHE CASE D4: WIGWAM CATASTROPHE 72 CASE DO: CIGRE System Data and Result Summary Table DO: CIGRE System Data of One Machine Infinite Bus (7 Machines, 17 Buses) No of Critical Machine = 1, Critical Machine = 7 Mk Mo Pmo PC Tk Alpha - Delk Delo 0.0568 0.4005 -0.0684 . -0.6525 4.4313 0.0869 0.1304 0.0851 Table Dl : 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 Table DO A: EEAC Solution of Steady-State Stablity forCIGRE System Critical Machine: machine 7 Chg,PM Thgc Thgu Area 1(A1) Area 2(A2) A2-A1 0 0.045297 2.922496 0.000000 7.104699 7.104699 0.1 0.068099 2.899694 0.001139 6.819258 6.818119 0.2 0.090982 2.876810 0.004563 6.538386 6.533823 0.3 0.113960 2.853832 0.010280 6.262099 6.251820 0.4 0.137047 2.830746 0.018299 5.990419 5.972119 0.5 0.160255 2.807538 . 0.028634 5.723367 5.694734 0.6 0.183599 2.784193 0.041295 5.460971 5.419676 0.7 0.207096 2.760696 0.056299 5.203259 5.146960 0.8 , 0.230762 2.737031 0.073661 4.950262 4.876601 0.9 0.254613 2.713180 0.093398 4.702017 4.608619 1 0.278668 2.689125 0.115531 4.458563 4.343032 1.1 0.302947 2.664846 0.140080 4.219941 4.079861 1.2 0.327470 2.640323 0.167069 3.986199 3.819130 1.3 0.352261 2.615532 0.196523 3.757388 . 3.560865 1.4 0.377344 2.590449 0.228471 3.533564 3.305093 1.5 0.402746 2.565047 0.262943 3.314788 3.051845 1.6 0.428497 2.539296 0.299973 3.101127 2.801155 .1.7 0.454629 2.513164 0.339596 2.892654 2.553058 1.8 0.481179 2.486614 0.381853 2.689448 2.307595 1.9 . 0.508187 2.459606 0.426788 2.491598 2,064810 2 0.535697 2.432095 0.474448 2.299198 1.824750 2.1 0.563763 2.404030 0.524886 2.112355 1.587469 2.2 0.592441 2.375351 0.578161 1.931185 1.353024 2.3 0.621800 2.345993 0.634338 1.755818 1.121480 2.4 0.651917 2.315875 0.693487 1.586397 0.892910 2.5 0.682885 2.284908 0.755690 1.423083 0.667393 2.6 0.714812 2.252981 0.821037 1.266057 0.445020 2.7 0.747829 2.219964 0.889629 1.115522 0.225893 2.71 0.751196 2.216597 0.896672 1.100834 0.2Q4163 2.72 0.754576 2.213216 0.903747 1.086214 0.182467 2.73 0.757969 2.209823 ' 0.910857 1.071662 0.160805 2.74 0.761375 2.206417 0.918001 1.057177 0.139176 • 2.75 0.764795 2.202998 0.925179 1.042761 0.117582 2.76 0.768227 2.199565 0.932391 1.028413 0.096022 2.77 0.771673 2.196119 0.939637 1.014134 0.074497 ' 2.78 0.775134 2.192659 0.946919 0.999924 0.053006 . 2.79 0.778608 2.189185 0.954234 0.985784 0.031550 2.8 0.782096 2.185697 0.961585 0.971713 0.010128 2.802 0.782795 2.184997 0.963059 0.968907 0.005848 2.803 0.783145 2.184647 0.963797 0.967505 0.003709 2.804731 0.783751 2.184041 0.965074 0.965080 0.000006 2.X047336 0.783752 2.1X4040 0.965076 0 965077 0.000000 2.8047346 0.783753' 2.184040 0.965077 0.965075 -0.000002 74 CASE DI: Cusp Catastrophe Table DI A: CIGRE System Steady-State Stablity Analysis Cusp Catastrophe Numerical Result #ofCM CMs w x Limit 1 x Limit 2 , 1 . • 7 ' . • • . ' ' -12,0943 16.18897003 -16.18897003 PM Chg. X PMChg X PMChg X 0.000 2.1389 1.000 • 7.6026 2.000 13.0662 0.050 2.4121 1.050 .7.8758 2.050 13.3394 0.100 2.6853 1.100 8.1489 2.100 13.6126 . 0.150 2.9585 1.150 8.4221 2.150 13.8858 0.200 3.2317 1.200 8.6953 2.200 14.159 0.250 3.5048 1.250 8.9685 2.250 14.4321 0.300 3.778 1.300 9.2417 2.300 14.7053 0.350 . 4.0512 1.350 9.5149 2.350 14.9785 0.360 4.1058 1.360 9.5695 2.360 15.0331 0.370 4.1605 1.370 9.6241 2.370 15.0878 0.380 4.2151 1.380 9.6788 2.380 15.1424 0.390 4.2697 1.390 9.7334 2.390 15.1971 0.400 4.3244 1.400 9.788 2.400 15.2517 0.410 4.379 1.410 9.8427 2.410. 15.3063 0.420 4.4337 1.420 9.8973 2.420 15.361 0.430 4.4883 1.430 . 9.9519 2.430 15.4156 0.440 4.5429 . 1.440 10.0066 2.440 15.4702 0.450 4.5976 1.450 10.0612 2.450 - 15.5249 0.460 4.6522 1.460 10.1159 2.460 15.5795 0.470 4.7068 1.470 10.1705 2.470 15.6342 0.480 4.7615 1.480 10.2251 2.480 15.6888 0.490 4.8161 - 1.490 10.2798 2.490 15.7434 0.500 4.8707 1.500 10.3344 2.500 15.7981 0.510 4.9254 1.5.10 10.389 2.510 15.8527 0.520 4.98 1.520 10.4437 2.520 15.9073 0.530 • 5.0347 1.530 10.4983 2.530 15.962 . 0.540 5.0893 1.540 10.553 2.540 16.0166 0.550 5.1439 1.550 10.6076 2.550 16.0712 0.560 .5.1986 1.560 10.6622 2.560 16.1259 0.570 5.2532 1.570 10.7169 2 570 164805 0.580 5.3078 1.580 10.7715 2 580 16,2352 0.590 5.3625 1.590 10.8261 2.590 16.2898 0.600 5.4171 1.600 10.8808 2.600 ' 16.3444 0.650 5.6903 1.650 11.154 2.650 16.6176 0.700 5.9635 1.700 11.4271 2.700 16.8908 0.750 6.2367 1.750 11.7003 2.750 17.164 0.800 6.5098 1.800 11.9735 2.800 17.4372 0.850 6.783 1.850 12.2467 2.850 17.7103 0.900 . . 7.0562 1.900 12.5199 2.900 17.9835 0.950 7.3294 1.950 12.793 2.950 18.2567 75 < II S 3 1 f S ta "« 5 l i s CO O. II ' s S | 3 31 S 3 1 •= u S ° " M « I U II J 3 J J I U B J F J . « CASE D2: Swallowtail Catastrophe Table D2: CIGRE System Steady-State Stablity Analysis Swallowtail Catastrophe Numerical Result #ofCM . CMs •. y w X Limit 1 X Limit 2 1 7 -550.1042 -6568.4261 -20,275.68197 -21483.91942 PM Chg. X PMChg X PMChg X 0.000 -20943.715 1.000 -21149.1377 2.000 -21354.5603 0.100 -21)^ 04.2^ 73 1.100 -21169.6799 2.100 -21375.1025 0.200 -20984.7996 1.200 -21190.2222 2.200 -21395.6448 0.210 -20986.8538 1.210 -21192.2764 2.210 -21397.699 0.220 1.220 -21194.3306 2.220 • -21399.7533 0.230 -20990.9622 1.230 -21196.3849 2.230 -21401.8075 0.240 -2(jyy3.U165 1.240 -21198.4391 2.240 -21403.8617 0.250 -20995.0707 1.250 -21200.4933 2.250 -21405.9159 0.260 -20997.1249 1.260 -21202.5475 2.260 -21407.9702 0.270 -20999.1791 1.270 \ -21204.6018 2.270 -21410.0244 0.280 -21001.2334 1.280 • -21206.656 2.280 -21412.0786 0.290 -21003.2876 1.290 -21208.7102. 2.290 -21414.1328 • 0.300 -21005.3418 1.300 -21210.7644 2.300 -21416.1871 0.310 -21007.396 1.310 -21212.8187 2.310 -21418.2413 0.320 -21UUy.45U3 1.320 -21214.8729 2.320 -21420.2955 0.330 -21U11.5U45 •• 1.330 -21216.9271 2.330 -21422.3497 0.340 -21U13.5587 1.340 -21218.9814 2.340 -21424.404 0.350 -21U15.bl3 1.350 -21221.0356 2.350 -21426.4582 0.360 -21017.6672 1.360 -21223.0898 2.360 -21428.5124 0.370 ,-21Uiy.7214 1.370 -21225.144 2.370 -21430.5667 0.380 -21021.7756 1.380 -21227.1983 2.380 -21432.6209 0.390 -21U23.82yy 1.390 -21229.2525 2.390 -21434.6751 0.400 -21U25.SS41 1.400 -21231.3067 2.400 -21436.7293 0.410 -21U27.y3«3 1.410 -21233.3609 2.410 -21438.7836 0.420 -2iu2y.yy25 1.420 -21235.4152 2.420 -21440.8378 0.430 -21U32.U46S 1.430 -21237.4694 2.430 -21442.892 0.440 -21U34.1U1 1.440 -21239.5236 2.440 -21444.9462 0.450 -21U36.1552 1.450 -21241.5778 2.450- -21447.0005 0.460 -21U38.2Uy4 1.460 -21243.6321 2.460 -21449.0547 0.470 -21U4U.2637 1.470 -21245.6863 2.470 -21451.1089 0.480 -21U42.31/y 1.480 -21247.7405 2.480 -21453.1631 0.490 -21U44.3721 1.490 -21249.7947 2.490 -21455.2174 0.500 -21U4t>.4263 1.500 -21251.849 2.500 -21457.2716 0.510 -21U48.48U6 •1.510 , -21253.9032 2.510 -21459.3258 0.520 -21Um^34« 1.520 -21255.9574 2.520 -21461.38 0.530 -21U52.!)«y 1.530 -21258.0116 2.530 • -21463.4343 0.540 -21UM.6433 1.540 • -21260.0659 2.540 -21465.4885 0.550 1.550 -21262.1201 2.550 -21467.5427 0.560 -211D8.7M7 1.560 • -21264.1743 2.560 -21469.597 0.570 -21U6U.8U5y 1.570 -21266.2286 2.570 -21471.6512 0.580 -21Ub2.8b(J2 1.580 . -21268.2828 2.580 -21473.7054 0.590 -21Ub4.yi44 . 1.590 -21270.337 2.590 -21475.7596 0.600 -21U66.y686 1.600 -21272.3912 2.600 -21477.8139 0.610 -21Uby.U22« 1.610 -21274.4455 2.610 -21479.8681 0.620 . -21U71.U771 1.620 -21276.4997 2.620 -2148L9223 0.630 -2111/3.1313 1.630 -21278.5539 2.630 -21483.9765 77 CASE D3: Swallowtail Catastrophe Table D3: CIGRE System Steady-State Stablity Analysis Butterfly Catastrophe Numerical Result #ofCM CMs U V w X Limit 1 X Limit 2 X Limit 3 X Limit 4 1 . 7 -30.2547 ' -1.5145 • 362.79160 '516.045393 541.380387 -523.36753 ' -479.07337 PMChg. x PM Chg. . . X • PM Chg. X 0.000 -57.7339 1.000 -221.6436 2.000 -385.5532 0.050 -65.9294 1.050 -229.8391 2.050 -393.7487 0.100 -74.1249 1.100 -238.0345 2.100 . 401.9442 0.150 -82.3204 1.150 -246.23 2.150 410.1397 0.200 -90.5158 1.200 -254.4255 . 2.200 418.3352 0.250 -98.7113 1.250 -262.621 2-250. 426.5306 0.300 -106.9068 1.300 -270.8165 2.300 434.7261 0.350 -115.1023 1.350 -279.0119 2.350 442.9216 0.400 -123.2978 1.400 " -287.2074 2.400 451.1171 0.450 -131.4933 1.450 -295.4029 2.450 459.3126 0.500 -139.6887 1.500 -303.5984 2.500 467.5081 0.550 -147.8842 1.550 -311.7939 2.550 475.7035 0.600 -156.0797 1.600 -319.9894 2.600 483.899 0.610 -157.7188 1.610 -321.6285 2.610 485.5381 0.620 -159.3579 1.620 -323.2676 2.620 487.1772 0.630 -160.997 1.630 -324.9067 2.630 488.8163 0.640 -162.6361 1.640 -326.5458 2.640 490.4554 0.650 -164.2752 1.650 -328.1848 . 2.650 492.0945 0.660 : -165.9143 1.660 -329.8239 2.660 493.7336 . 0.670 -167.5534 1.670 ' -331.463 2.670 495.3727 0.680 -169.1925 1.680 -333.1021 2.680 497.0118 0.690 -170.8316 1.690 -334.7412 2.690 498.6509 0.700 -172.4707 1.700 -336.3803 2.700 -500.29 0.710 -174.1098 1.710 -338.0194 2.710 -501.9291 0.720 -175.7489 1.720 -339.6585 2.720 -503.5682 0.730 -177.388 1.730 -341.2976 2.730 -505.2073 0.740 -179.0271 1.740 -342.9367 2.740 -506.8464 • 0.750 -180.6662 1.750 -344.5758 2.750 -508.4855 0.760 -182.3052 1.760 -346.2149 2.760 -510.1246 0.770 -183.9443 1.770 -347.854 2.770 -511.7637 0.780 -185.5834 .. 1.780 -349.4931 2.780 -513.4028 0.790 -187.2225 1.790 -351.1322 2.790 , -515.0419 0.800 -188.8616 1.800 -352.7713 2.800 -516.681 0.810 • -190.5007 1.810 -354.4104 2.810 -518.3201 0.820 -192.1398 1.820. -356.0495 2.820 -519.9592 0.830 -193.7789 1.830 -357.6886 2.830 -521.5983 0.840 -195.418 1.840 -359.3277 2 8-10 -523.2373 0.850 -197.0571 1.850 -360.9668 2.85!) -524 876-t 0.900 -205.2526 1.900 -369.1623 2.900 -533.0719 0.950 -213.4481 1.950 -377.3577 2.950 -541.2674 79 CASE D4: Swallowtail Catastrophe Table D4: CIGRE System Steady-State Stablity Analysis Wigwam Catastrophe Numerical Result : #ofCM CMs t u V w x Limit 1 x Limit 2 x Limit 3 x Limit 4 x Limit 5 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 PM Chg. X PM Chg. X PM Chg. X 0.000 -1563569.865 1.000 -1554942.115 2.000 -1546314.365 0.050 -1563138.478 1.050 -1554510.727 2.050 . -1545882.977 0.100 -1562707.09 1.100 -1554079.34 2.100 -1545451.59 0.150 -1562275.703 1.150 -1553647.952 2.150 -1545020.202 0.200 -1561844.315 1.200 -1553216.565 2.200 -1544588.815 0.250 -1561412.927 1.250 -1552785.177 2.250 -1544157.427 0.300 -1560981.54 1.300 -1552353.79 2.300 -1543726.04 0.350 -1560550.152 1.350 -1551922.402 2.350 -1543294.652 0.400 . -1560118.765 . 1.400 -1551491.015 2.400 • -1542863.265 0.450 -1559687.377 1.450 -1551059.627 2.450 -1542431.877 0.500 -1559255.99 1.500 -1550628.24 2.500 -1542000.49 0.550 -1558824.602 1.550 -1550196.852 2.550 -1541569.102 0.600 -1558393.215 1.600 -1549765.465 2.600 -1541137.715 0.650. -1557961.827 1.650 -1549334.077 2.650 -1540706.327 0.660 -1557875.55 1.660 -1549247.8 2.660. -1540620.05 0.670 -1557789.272 1.670 -1549161.522 2.670 -1540533.772 0.680 -1557702.995 1.680 -1549075.245 2.680 -1540447.495 0.690 -1557616.717 1.690 -1548988.967 2.690 -1540361.217 0.700 -1557530.44 1.700 -1548902.69 2.700 -1540274.94 0.710 -1557444.162 1.710 -1548816.412 2.710 -1540188.662 0.720 -1557357.885 1.720 -1548730.135 2.720 -1540102.384 0.730 -1557271.607 1.730 -1548643.857 2.730 -1540016.107 0.740 -1557185.33 1.740 -1548557.58 2.740 - - -1539929.829 0.750 -1557099.052 1.750 -1548471.302 2.750 -1539843.552 0.760 -1557012.775 1.760 -1548385.025 2.760 -1539757.274 0.770 -1556926.497 1.770 -1548298.747 2.770 . -1539670.997 0.780 -1556840.22 1.780 -1548212.47 2.780 -1539584.719 \. •. 0.790 -1556753.942 1.790 -1548126.192 2.790 -1539498.442 0.800 -1556667.665 1.800 -1548039.915 2.800 -1539412.164 0.810 -1556581.387 1.810 -1547953.637 2.810 -1539325.887 0.820 -1556495.11 1.820 -1547867.36 2.820 -I53Y2 39 6(^ 1 0.830 -1556408.832 1.830 -1547781.082 2 830 -I5VJ1S3 332 0.840 -1556322.555 1.840 -1547694.805 2.840 -1539067.054 0.850 -1556236.277 1.850 -1547608.527 2.850 -1538980.777 0.900 -1555804.89 1.900 -1547177.14 2.900 -1538549.389 . 0.950 -1555373.502 1.950 -1546745.752 2.950 -1538118.002 81 00O009I-0000S91-A P P E N D I X E T H R E E - M A C H I N E W S C C T E S T S Y S T E M C A T A S T R O P H E M O D E L S S I M U L A T I O N R E S U L T S ( T R A N S I E N T S T A B I L I T Y A N A L Y S I S ) ( C R I T I C A L M A C H I N E = 2) A T F A U L T B U S = 5 A N D L I N E O P E N B E T W E E N N O D E 4 A N D 5 83 cs © SO O N O N O NO NO © I—H ON O N o co ON NO OS OS C S NO NO © • o 00 ro r» O N O N O N CN " O f N CO C S CO cs o o VO C S 00 s o w >> 00 •/"> | O ' •O 00 3 3 P Q m C S 2 II 8 o I' Q E •8 3 O CO w> O N ro ro O © Ti 00 NO ON r—( IS' I C S n ON ro I co © •8 21 cs o NO 1—I C S C S 00 OS •n •/-> 00 IT) C S C S 00 OS a 60 ft e P 79 8 o ca •S o OS SO 00 SO o SO OS ft •a •4—> 1 a & o o 00 oo. 00 ro C S 00 o CS' p cs 00 •a •a t~>l 1*1 •c. 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Tf >Ti rt WO Tf wo WO v i Vl v i V N v i *r\ wo CO CO CO CO CO CO CO CO CO 00 00 00 00 00 00 00 00 CO cs CO Tf v i NO r~ 00 ON 0 d d d d d d d O ^ vo oo VO 00 NO 00 NO 00 NO 00 NO 00 NO 00 VO 00, VO ON ON ON ON ON ON ON ON ON vo VO NO *o NO NO NO VO VO 00 00 00 00 00 00 00 00 00 r~ C~ c~ t-; r -cs cs CS cs cs cs cs' cs cs o cs CO Tf VI NO WO d d d d d d d d VO 00 w-i CO cs C S wo Tf r- 0 ON 00 VO C S ON r -vo Tf WO *r\ 00 CO CO ON. T ? : ON a UJ m Tf J 3 ) 3 U 1 B J BJ M APPENDIX F SEVEN-MACHINE CIGRE TEST SYSTEM CATASTROPHE MODELS SIMULATION RESULTS (TRANSIENT STABILITY ANALYSIS') (CRITICAL MACHINE = 7) AT F A U L T B U S = 9 AND LINE OPEN B E T W E E N NODE 9 AND 10 87 3 a a O Q 3 a m GO PL, M Xi U S • • OTJ « .a oo s o ja 2 2 60 t> , 3 ! * O II 60 H " g I E e 6 3 w CJ 5 o e 6 o a p S &l « » 2 •f ^ O O « CO SO X CQ 3 II ON 00 x