M“ . (.‘nl f‘,’ 1 .1. air}! . I ‘ l '0’ (:1 (Suit. 1.1., kin; III .I '1. I) If... . sun! 5‘. r. . l..9v.0-vo¢ DU... ff...“ i..la\‘flff I <01." n’.‘ O. 1.3%....» .35 fit}. .3! '33. .. Ir] ..”‘9r f I In. 1!. (.f‘lr‘ .' O . 0": .7 .f To .J.rg.u.ne‘zv r tn...- . “‘. .V‘,‘ III- "’L-oo." 9"“ . up)". an.“ . . .Iutvfrvr)‘ x. o t. t, P, ' .$uoc.r$ cl. . . n o . f Y... 11.! I .1. .-ul. . . _ . .?.f- : . . .. 1.7.. 2.... . . . fr?!) ‘8» . a... . -10.. 1.Juh§.wo . .no av)?.:-..¢o\lt il.}\. _ plu— o.‘”ll O‘lfil. Ir r fl‘IIlv’len-IBJ . .IA'V S... “3‘. . .l. . 'I vailu.....i: 0 . 1|.Aptrlt?a‘s.’lo. leak-.53! ..u¢v.v.-..|fy.u.pi! :r.I,O; i$.4:ll.l . . . : Iii“: 3.1.“!- . .. 4 . ..::...L... ..v£ .uL, .. . \r u 1 v... .. Ln. «Inn 0A."- f....‘.-.. C. .0». 1%...“ ..\..I 1.-.}...1 a. {31.}. 1...... . ....F.. .. I. it... .. . S: ..l§.1ar.ovz (.51... ‘....¢: 1‘ 1.. REL, v . a...) i \l,.. .r 3.). . . .1. ms? . . ..l 7. . . n It. ‘tbffi . . l E s ‘ . e. 11.1.21.) n y 0-. It’ll), at..- I o . .‘ 1'! {sgl,'¢.iil fry": *l‘g‘u’ . . in pk}: : .1». .. L.I...v:.clra .shmphoufl. or 5% . (.31.; ll Cull! It!) ; 99)"?! ‘1..1\.HIDJ$ . ii} cl: E.J.ofv‘1rlI-lo OI. Iclvt. Suri‘cfi flint gL‘CigtaZUo r. .V 5335» (0% 13.1““; ‘15:!” Car .’ . .{II} I... .t. v ..... .2 .. f. .o- I)! fit. list-31E?! Ifhrr L. lu.|.1n."l§lllll.l:t.in .‘r. Elli: (Olhgla p [A 0. lit/Pulifil i!!! (It!) . :W‘...‘ I‘ll . I .Kuuordnrlalu... ‘2’! (’0‘ .3593“. . n ‘ y’): v (pg...) 1.. o.v ; I In; . .13 an); . 611-! 52.1.5I‘uiilll Quill: . .I- \II'H'a.I(rv fiat-I). E'ig "mind-"n ‘m :‘m‘ v‘m » O Cu. .3- . . n . , n. 2...} . ,. .12.. 191' . . (7:2... -5141... yoly‘l'uktfl‘l'.) run}: .5 . . n. . .. lad... . . L3»! J! .1. -l . . , I... . Limit. .\ 3.1 It: . . .1 E r. y .3 ..!w0\. O I tKIY.a\.v".£!QI11x.. u-‘ l.llI\V1| I.“ 0 .D:...h '1}! I.’ I -Ot00.. t... .teO. :v\\ . r'. :1 b. \lb'\l I’I‘"II‘ I! III 4 v. . n. . Q: l.1u..|.....|!.r -«wrnvcnruz .. . r. I I 9 1 I II- .> Q. . .N .. .C .. '1. .2 v .‘..p.rl...ab... i; . fi ..§2I:Fu;e..llr .. 3.525! . {I ‘ . I n . I A! 1"}. \ . , v In) L;al!pll|t).§k.lk l!!§.?tp.r¢ o .. I. . (lo. 19...! . .u :2. . z .. d¢.rl.’..¢|r¢oil spy}... . .;I‘.';. r tcrgg . . - y i. . I . Int . ... . 40.16..-.551153. . 5 $1 . 2...-4537.hflwnfget.£.)é£.5 , . . . 1.. D t a t . . ; . ‘0lllvo Jlr I‘lO..1.llsv$(‘-vl. .l: 1'. (I! 1.).l I‘Sé“! A. .{I‘it n éé.§.él . . , ..l . . D ., . Ax . .l . . II. 0;... {hr (nil. 1.4!... vulfll~| 5.! I!|.\‘A\l>r|.1lv\tel f. :Il‘v‘t'i150..ul§1¢e=.(§/t;l I!!! .éfl , Illif‘ll’l v .. .YQK. . 0. I. v I If 1".”- -( A .. 1A. I Z. If .1 it giving 2? xi .r?1!flr‘a§§ 5:0"; ‘II, :Ig‘Ir‘ f? g‘iv‘gc‘lig . «r .[ \..Io.vo\- . l!- I It kits-IMF... . ‘bflflflu; 7.x. tot U).llur:.-f0.lfl(oM2.h..-..:'§llot it?! nll .33., pt? I435. (I .035"? - .CI... 4 :(‘rl 2%;‘J11fl-7’ I. ttslx I‘.A..1r\‘lc|l).l| In I! GOA A t I. . ¢- 1 ‘5 . c .- . .\. . . .c V II . . Vu’v. Iv. .il. . I) .115. |-.|.u 0 . til. .(v’ :I \J - .31.! 6‘. 5%. MI. zit-5.... cpl. I {IS (’1'. .13.- ii‘!‘ II 1) ,..lb‘.i.fihkiifie-€.fi§ffifid.3u..-..n..um..u......eu...n.d.uv.rwfln.mi$$..fi inf.lo..r.rmv:.h?. .fiu 4.5.3.1. .5a+...$z..:.....u......uwnn.:. . {.31. .. Wu: .rwco. mm)". in .II. 1|! 9:: . 2“- 2|..- 11 ‘1' A1 Aalx a. .1. . . .i . I .5.-. or. 1 Wing). . s . . t .4. r u: .n. . . . .Ilqt' .. v: 3- il.vt\|.l I. ..\ .L .. SI.) 0 km. .|.. .. .11. c! L‘: 4... Mr! .1. .1 3..-... gilt}. 3’ .aiiu;5.lvo 11)... .IV 14.14.35 .u: t . (Ir . "A; . n) p r . . , . x n .. . . . .1 v.-- L...x.\ ...... 1.1.. u ... . J. . .v. 2 § 2-, $5.2}... 4 13%,...“ . 3.75.213?.u...¢10...1......f.l. -9? glfisfiu. :n b I .. I A ll '1 f.’ -ll'irl 01!- o r . . ...||uur.....n 1.1.1... ...cl|..- I}. . . 4. ..| , I... . . :4}... .53!!!ch uh . . ....O&.Bllu. (til (ml. - . n| . . .31.»! I“ 0.0 . .o I... .1“... up. I. fluid-"HIV. U . . ..,.z.v’.!.ll .r.. ‘.v .7 A I\. ! MELIMJ MICHIGAN SIATE UNI IVFRSITYII I I II III II III IIIIIIIIIIIIII 31293 00605 8444 I .7; II“. “III IJBRARY Michigan State University This is to certify that the dissertation entitled CHARACTERIZING THE REGION OF ATTRACTION OF A POWER SYSTEM TRANSIENT STABILITY MODEL presented by Jichang Lo has been accepted towards fulfillment of the requirements for Ph. D. degreein Systems Science C/Muz‘a 0% Major professor Date Feb. 15, 1990 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to romovo this chockout from your record. TO AVOID FINES roturn on or Moro duo duo. DATE DUE DATE DUE DATE DUE MSU Io An Affirmdivo ActloNEquol Opportunity Institution CHARACTERIZING THE REGION OF ATTRACTION OF A POWER SYSTEM TRANSIENT STABILITY MODEL By Jichang Lo A DISSERTATION Submitted to Michigan State University in pardal fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and System Science 1990 0054/64 ABSTRACT CHARACTERIZING THE REGION OF ATTRACTION OF A POWER SYSTEM TRANSIENT STABILITY MODEL By Jichang Lo A fast accurate method for assessing the retention or loss of stability is developed. The method does not require transient stability simulation of the fault, and does not require computation associated with selecting and determining a specific u.e.p.. This method is the first that is fast and accurate enough to be used on-line for dynamic security assessment and emergency control in an energy management system. A closed form expression is hypothesized and then proven to be a portion of the stable manifold associated with each type 1 u.e.p.. The closure of these stable mani- folds for the type 1 u.e.p., that lie on the boundary of the region of attraction, is the boundary of the region of attraction if certain assumptions hold. This boundary of the region of attraction is the boundary between the region of attraction and the unstable region is state space rather than a boundary (a) that is dependent on the Lypapunov function chosen and (b) that can not properly be used to indicate loss of stability if crossed by transient trajectory for a specific fault. The acceleration manifold description of the stable manifold associated with each generator is used along with the peak predictor to develop an algorithm for testing for retention and loss of stability for a particular fault, fault clearing time, and fault clear- ing action. The algorithm is also used to estimate critical clearing time. The algo- rithm indicates that the system is stable if the peak angle predictor lies on the stable side of each generator’s acceleration manifold. A computer program (see Appendix A) was developed to implement this algo- rithm for three phase faults. The algorithm was tested on nine fault cases on the New England 39 bus system. These results suggest that an accurate identification of the critical machine that loses stability is achieved based on comparison with the real transient simulation run, The results indicate that critical clearing time can be accurately assessed if the linear- ized peak angle predictor that estimates post fault angle excursion is evaluated at a point along the post fault trajectory and not at the prefault or post fault s.e.p. Copyright by JICHANG L0 1990 Dedicated to my parents and my sisters ACKNOWLEDGEMENTS I would like to thank my parents for encouraging me in pursuing higher educa- tion, my sisters for their understanding and support while studying oversea. I sincerely wish to express my thanks and appreciation to my advisor, Dr. Robert A. Schlueter, for his valuable assistance in the guidance, support, professional criti- cism, and constant encouragement made during the last four years of my Ph.D. studies at MSU. I am also indebted to Professors: Dr. Richard 0. Hill, Dr. Hassan K. Khalil, and Dr. Gerald L. Park for their advise and comments in the guidance committee. TABLE OF CONTENTS LIST OF TABLES .................................................................................................... viii LIST OF FIGURES ...................................................................................................... x CHAPTER 1. INTRODUCTION ............................................................................... l 1.1 TRANSIENT STABILITY PROBLEM ...................................... 1 1.2 OBJECTIVES .............................................................................. 3 1.3 LITERATURE REVIEWS .......................................................... 4 1.3.1 Characterization of the Boundary of the Region of Attraction ......................................................... 5 1.3.1.1 Differential Topology Approach ......................... 5 1.3.1.2 Lyapunov Approach ............................................ 8 1.3.2 Approximations to the Region of Stability ........................................................................... 10 1.3.2.1 Lyapunov Based Approximation of the Region of Stability .............................................. 11 1.3.2.2 Cutset Approximation of the Region of Stability ............................................................... 15 1.3.2.3 Approximation of the Differential Topology Description of the Region of Stability ............................................................... 16 1.3.3 Diagram of the Region of Stability by Various Approaches ........................................................ 18 CHAPTER 2. MODEL AND APPROACH ............................................................. 23 2.1 POWER SYSTEM MODEL ..................................................... 23 2.1.1 Definition ......................................................................... 25 2.1.2 Loads ................................................................................ 25 2.1.3 The Network ................... 26 2.1.4 Generator Model .............................................................. 26 2.2 METHOD OF APPROACH ...................................................... 29 2.2.1 Mathematical Derivation ................................................. 30 2.3 THEORETICAL RESULTS ...................................................... 33 2.4 BOUNDARY CHARACTERIZATION .................................... 42 2.5 JUSTIFICATION OF A CSG REFERENCE BASED CHAPTER 3. CHARACTERIZATION OF THE BOUNDARY OF THE REGION OF STABILITY FOR M MACHINE U.E.P. ..................................... 58 3.1 INTRODUCTION ...................................................................... 5 8 3.2 MATHEMATICAL MODEL .................................................... 59 3.3 ENERGY FUNCTION .............................................................. 61 3.4 THEORETICAL RESULTS ...................................................... 65 3.5 BOUNDARY CHARACTERIZATION .................................... 71 4.2 LINEARIZED POWER SYSTEM MODEL ........................... 82 4.3 DISTURBANCE MODEL ....................................................... 82 4.3.1 Pulse Disturbance ..................................................... 83 4.3.2 Impulse Disturbance ................................................. 83 4.4 A PEAK ANGLE PREDICTOR FOR FAULTS WITH SHORT CLEARING TIME ................................................... 84 4.4.1 Peak Angle Predictor for Impulse Disturbance ................................................................... 85 4.4.2 Peak Angle Predictor for Pulse Disturbance ................................................................... 92 4.4.3 Accuracy of the Peak Angle Predictor ........................................................................ 94 4.5 COMPUTATIONAL ALGORITHM ....................................... 96 CHAPTER 5. SIMULATION RESULTS OF THE DIRECT METHOD .............. 98 5.1 INTRODUCTION ...................................................................... 98 5.2 SIMULATIONS RESULTS .................................................... 104 5.2.1 F07-06* case .................................................................. 105 5.3 DISCUSSION OF INACCURACY ........................................ 111 CHAPTER 6. CONCLUSIONS AND TOPICS FOR FUTURE RESEARCH 114 5.1 CONCLUSIONS ...................................................................... 114 5.2 TOPICS FOR FUTURE RESEARCH .................................... 118 APPENDIX A - PEAK PREDICTOR PROGRAM ........... '. .................................... 120 APPENDIX B - CONTINGENCY LOAD FLOWS ............................................... 129 APPENDIX C - SIMULATION RESULTS ............................................................ 138 LIST OF REFERENCES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO LIST OF TABLES Table 5.1.121 - 39 New England system base case (bus data) ................................ 102 Table 5.1.1b - 39 New England system base case (line data) ................................ 103 Table 5.1.2a - Computational output for F06-07 case (prefault) ............................ 106 Table 5.1.2b - Computational output for F06-07 case (postfault) .......................... 107 Table 5.1.2c - Computational output for FO6-07 case (at Tclear) ........ . ................. 108 Table 5.3.1 - Critical time clearing obtained by simulation and predictor ............ 112 Table 5.3.2 - Critical time clearing obtained by tuning method for F28-29 .......... 113 Table 3.1 - Load flow solution when line 06—07 is removed ................................. 129 Table 8.2 - Load flow solution when line 10-11 is removed ................................. 130 Table 3.3 - Load flow solution when line 10-13 is removed ................................. 131 Table B.4 - Load flow solution when line 23-24 is removed ................................. 132 Table 8.5 - Load flow solution when line 22-23 is removed ................................. 133 Table B.6 - Load flow solution when line 02-25 is removed ................................. 134 Table 3.7 - Load flow solution when line 25-26 is removed ................................. 135 Table B.8 - Load flow solution when line 28-29 is removed ................................. 136 Table 3.9 - Load flow solution when line 01-02 is removed ................................. 137 Table C.1a - Computational output for FlO-ll case (prefault) .............................. 138 Table C.1b - Computational output for F10—11 case (postfault) ............................ 139 Table C.1c - Computational output for FlO-ll case (at Tclear) ............................ 140 Table C.2a - Computational output for Flo-13 case (prefault) .............................. 141 Table C.2b - Computational output for Flo-13 case (postfault) ............................ 142 Table C.2c - Computational output for F10—l3 case (at Tclear) ............................ 143 Table C.3a - Computational output for F23-24 case (prefault) .............................. 144 Table C.3b - Computational output for F23-24 case (postfault) ............................ 145 Table C.3c - Computational output for F23-24 case (at Tclear) ............................ 146 Table C.4a - Computational output for F22-23 case (prefault) .............................. 147 Table C.4b - Computational output for F22-23 case (postfault) ............................ 148 Table C.4c - Computational output for F22-23 case (at Tclear) ................. 149 Table C.5a - Computational output for F02-25 case (prefault) .............................. 150 Table C.5b - Computational output for F02-25 case (postfault) ............................ 151 Table C.5c - Computational output for F02-25 case (at Tclear) ............................ 152 Table C.6a - Computational output for F25-26 case (prefault) .............................. 153 Table C.6b - Computational output for F25-26 case (postfault) ............................ 154 Table C.6c - Computational output for F25-26 case (at Tclear) ............................ 155 Table C.7a - Computational output for F01-02 case (prefault) .............................. 156 Table C.7b - Computational output for F01-02 case (postfault) ............................ 157 Table C.7c - Computational output for POI-02 case (at Tclear) ............................ 158 LIST OF FIGURES Figure 1.3.1 - Manifold diagram of different dimension ............................................ 7 Figure 1.3.2 - Energy contour in angle space ........................................................... 12 Figure 1.3.3 - Pictorial diagram for region of stability ............................................. 19 Figure 2.5.1a - Acceleration vs time curve (CSG model) ........................................ 49 Figure 2.5.1b - Acceleration vs time curve (COA model) ................... 50 Figure 2.5.2a - Angular speed vs time curve (CSG model) ..................................... 51 Figure 2.5.2b - Angular speed vs time curve (COA model) .................................... 52 Figure 2.5.3a - Angle vs time curve (CSG model) ................................................... 53 Figure 2.5.3b - Angle vs time curve (COA model) .................................................. 54 Figure 5.1.1 - The 39 bus New England System .................................................... 101 CHAPTER 1 INTRODUCTION 1.1 TRANSIENT STABILITY PROBLEM In a very complex power system, the primary concern of power engineers is to provide reliable and uninterrupted service to the demands when the power system is subjected to a disturbance. If the disturbance does not cause any appreciable power unbalance, the system will undergo a small deviation from its nominal operating state, and return to its original state; or to another state, this is called steady state stability. If a power system experiences a large disturbance such as a fault, loss of generation or change in network configuration, the system makes a transition from the original equilibrium state to a new equilibrium state; this is called transient stability. The terms steady state and transient stability do not indicate whether stability is retained or lost but rather whether linear or nonlinear system analysis is needed to study whether stability is retained or lost. A power system is said to retain transient stability if it does not lose synchronism under sudden large disturbances. A major disturbance causes an imbalance between the mechanical input power from the steam turbine and the real power injected into the electrical netwOrk. Some of these generators have net accelerating power while others have net decelerating power. Consequently, the rotor angles of the machines accelerate above or decelerate below the synchronous speed for t > 0. If the rotor angles are plotted as a function of time, there exist two possibilities. 1. The rotor angles increase together and swing against one another and ultimately settle to new equilibrium angles. Since the relative rotor angles do not continue to increase indefinitely, the system is said to be stable and synchronism is preserved. 2. One or more of the machine angles accelerate faster than the rest of them, so that ultimately the rotor angles of the accelerated group continues to increase indefinitely and thus diverge with respect to the stationary group. Such a mode of behavior is classified as unstable and a loss of synchronism is said to have occurred; If the fault is cleared quickly, the tendency to separate may be arrested through the post-fault network by a redistribution of the acceleration energy gained by the sys- tem during the faulted period. This acceleration energy is redistributed through the post fault network during the post fault period. The maximum time a fault can be allowed to remain on the system without observing subsequent loss of synchronism subse- quently is called the critical clearing time ta. Mathematically, the dynamic behavior of a power system is modeled by a non- linear vector differential equation i(t)=f(x(t)) (1.1) where x (t) is the state vector of the system. The stable equilibrium point (s.e.p.), x‘ 1 of equation (1.1) is defined as a state that satisfies the equation 0 = f (x‘ 1(t )) In transient stability analysis, the power system undergoes two stages: (1) the fault-on period to) =ff(x(t)) o < t < tc (1.2) (2) the postfault period 12(2) =fPf(x(t)) t > tc (1.3) where (C is called the clearing time, superscript f denotes fault-on system, and superscript pf denotes the postfault system. If the stability could be restored after the fault is cleared, then the system trajectory will converge to s.e.p., x”, such that 0 -- 1’” (x32) If the synchronism of the system is lost, then the trajectory will move toward an unstable equilibrium point (u.e.p.), x“, such that o = f” 1x“) The most widely used method of assessing retention or loss of transient stability is through step by step integration of the machine’s rotor angles. This numerical integra- tion method inherits some difficulties: (1) Several guesses of fault duration is necessary before one could determine the crit— ical clearing time that the particular system could sustain for that operating condi- tion, fault, and fault clearing action; (2) The computational burden increases rapidly as the model used to describe the sys- tem increases. 1.2 OBJECTIVES A fast, computationally efficient algorithm to perform transient stability assess- ments for a particular system fault and fault clearing action is an urgent need of the power industry, because (a) a VAX 780 computer or equivalent and the associated workforce is kept busy around the clock year round to set transfer and operating limits. These operating limits assure the system will not lose transient stability for all anticipated operat- ing conditions and credible contingencies; (b) the computation of one transient stability simulation on a large system data base for a reasonably complex model can take 15 cpu minutes. Cutting the computa- tion by a factor of one hundred or more could be required if on-line determina- tion of retention of stability is to be feasible on an EMS control center; (c) system protection studies require searching for the worst fault, fault location, and clearing combination. This search often requires screening a very large number of fault contingencies. The direct stability assessments are claimed to be accurate enough for screening (10% error in critical clearing time). The accuracy could potentially also be improved greatly. The objective of this thesis is to produce an extremely fast accurate assessment of retention or loss of stability for a particular prefault system, fault, and fault clearing action by (a) a precise closed form differential topology based characterization of the boundary of the region of stability; (b) a peak angle predictor that specifies the peak angle excursion in terms of prefault load flow conditions, fault, fault clearing time, and post fault load flow. The predictor does not require trajectory simulation but directly predicts the peak angle excursion of the transient stability model; (c) a computer program that can establish retention or loss of stability for a particular clearing time by testing whether the predicted peak angle excursion lies inside or outside the boundary of the region of stability. The program can determine reten- tion or loss of stability far more accurately than any existing method. The com- putational requirements are also less than any other reasonably accurate direct sta- bility assessment program. 1.3 LITERATURE REVIEWS Efforts at characterizing the boundary of the region of attraction for the classical power system model have produced considerable progress. However, despite this pro- gress, there are no methods that can compute every point on the boundary of the region of attraction for systems with a large number of generators. Results that permit one to characterize mathematically the boundary of the region of attraction are a recent development [1234]. Although in theory one can characterize the stability boundary using such methods [123,4], one can not at present determine closed form expressions for the stability boundary. Taylor series approximations [25] are possibly the most precise characterization. These methods require computing all of the type 1 u.e.p.’s [1] and then evaluating series approximations to the stability boundary around each u.e.p. These approximations are only accurate at points on the stability boundary that are close to a u.e.p. This thesis will develop closed form expressions for portions of the stability boundary characterized using the differential topology based description given in [1234] The closed form expression will neither need to compute a u.e.p. or compute Taylor series approximations to the stability boundary. In this section, the theory behind the transient stability assessment and methods of approximating the region of attraction are reviewed. Differential topology, Lyapunov and cutset approximation based methods will be discussed since all three methods will be utilized in obtaining the closed form expression for the stability boundary. 1.3.1 Characterization of the Boundary of the Region of Attraction 1.3.1.1 Differential Tapology Approach The differential topology approach has been applied to characterizing the region of stability of the power system transient stability model in several recent papers [23,4675]. The differential topology approach is applied to a nonlinear autonomous dynamical system described by i=fda» man where f(x) is a C1 map R" —> R". A review of the terminology and theory used to characterize the boundary of the region of attraction using differential topology is now reviewed. A point x,- is called an equilibrium point of (1.3.1) if f (xi)=0. The derivative of f at x,- is called the Jacobian matrix at x,- and is denoted by J (xi). We say that an equilibrium point is hyperbolic if the Jacobian has no eigenvalues with zero real part. A hyperbolic equilibrium pOint is a stable equilibrium point (s.e.p.) if all the eigen- values of the Jacobian have negative real part; otherwise, it is an unstable equilibrium point (u.e.p.). The type of the equilibrium. point x,- is defined to be the number of eigenvalues of J (xi) with positive real part. The solution curve of (1.3.1) starting from x at t=0 is called a trajectory, denotedby no. The difficulties with Lyapunov stability methods are: 1. that the region of stability depends on the particular Lyapunov function chosen. 2. if the state at clearing time tc exceeds the region of stability 0, no statement can be made about retention or loss of stability because there is no general necessary condition for stability. The region of stability and the boundary of the region of stability suffer no such ambiguity. If x(tc) lies outside the boundary (1.3.5), the trajectory is unstable and when it lies inside (1.3.5), the trajectory is stable. 3. the most common method of describing the region of stability for power systems characterizes the region of stability in terms of a critical potential energy. The total energy evaluated at clearing time tc must be less than this critical potential energy for the system to be stable. The method for characterizing the critical potential energy is slightly different in each of the methods [7-18]. Thus, each of these methods produce a different approximation of the boundary of the region stability [1,3,6] 4. the potential energy boundary surface is not the boundary of the region of stabil- ity (1.3.5). The PEBS can lie inside or outside the true boundary of the region of attraction. 5. the connected component of the set {x :V(x )=V(x,-)} used to characterize the sta- bility boundary in [9] does not contain any points on the stable manifold of x,- where x,- is an u.e.p. and V(x) is the energy function. Thus u.e.p. method which 10 assume that V(x) is constant near a u.e.p. or that energy at the u.e.p. can charac- terize the boundary of the region of attraction are in error. The error possibly depends on how close to the u.e.p. the stable manifold is crossed. However, since the energy function that properly describes the stable manifold is path dependent, all the methods that utilize the non path dependent total system energy V(x) to characterize the stable manifold would appear even more susceptible to error. 6. the potential energy boundary surface and controlling u.e.p.’s provide some bound. on angle difference along the trajectory but not a precise constraint. 7. These standard energy/Lyapunov functions based on aggregated network entails non-negligible transfer conductance even if the actual transmission network is assumed lossless. Consequently, significant error is introduced when a trajectory approximation is made or when transfer conductance is neglected. 1.3.2 Approximations to the Region of Stability Several methods have been used to approximate the region of stability since the exact boundary of the region of attraction (1.3.5) was not able to be expressed in closed form. A discussion of the various methods of approximating the boundary of the region of attraction are discussed here because they will be used in the derivation of a precise closed form description of the boundary of the region of attraction based on definitions of stable manifolds of equilibria (1.3.3) and the differential topology based description of the boundary of the region of attraction (1.3.5). 1.3.2.1 Lyapunov Based Approximation of the Region of Stability Among various approaches to the transient stability problem, the energy type Lyapunov function provides some promising results in evaluating transient energy. 1. Closest u.e.p. method [1011] 11 2. Controlling u.e.p. method [12] 3. PEBS method [1314] 4. Individual machine PEBS method [15161718] 5. Critical transient energy method [919] The Lyapunov approach is better understood through Figure 1.3.2 which represents the energy contour on the angular space, s.e.p. is the stable equilibrium point, the dotted curves joining the u.e.p.’s are referred to as the potential energy boundary surface. The basic idea in using direct methods is described as follows. At the instant of fault clearing, a faulted system possesses a transient energy in a form of kinetic energy. Since this kinetic energy causes the machine to swing away from the equilibriunr, the system transmission network must absorb this excess energy and con- vert it into potential energy in order for the system to remain stable after the fault clearing action is taken. I Two key iSsues play an important role in applying any one of these direct methods; 1. Finding an appropriate special energy function to evaluate the energy at a post- fault equilibrium point. 2. determining a criterion for deciding retention and/or loss of stability given tran- sient energy at the fault clearing time. In the earliest application of the Lyapunov theorem to power systems [810], 12 Figure 1.3.2 - Energy contour in angle space 13 the primary concern was to find the minimal potential energy Vmin such that the sys- tem is asymptotically stable for x such that V(x) S Vmin. In these early works, researchers [810] focused on energy functions obtained by the first integral principle. The value of Vmin used was that for the u.e.p. which had the minimum potential energy among all u.e.p.’s. This closest u.e.p. method has been proven to be too con- servative because it incorrectly assumes the trajectory will cross the stability boundary at the point where its potential energy is minimum. Investigator [12] reduced the conservativeness of the closest u.e.p. method by identifying the u.e.p. closest to the fault-trajectory as controlling u.e.p. (or relevant u.e.p.) for the particular fault under consideration. The potential energy associated with this controlling u.e.p. is called the critical transient energy. Approximately one decade ago, Fouad er al [15,16] initially made corrections to the controlling u.e.p. method by noting that part of the kinetic energy obtained during fault-on period does not contri- bute to system separation. Only the kinetic energy between the inertial centers of the "accelerated group" and "stationary group" of machines contribute to determining retention or .1053 of stability. The kinetic energy of machines in each group with respect to the inertial center were determined to have no effect on retention or loss of stability. This result motivated the development of individual and group machine energy function methods [ll-14, 20] which will be shown in this thesis to provide the means of establishing the form of the stable manifold. More recently, Fouad et al have developed an improved method of identifying the u.e.p. which defines the critical energy. This "mode of instability" method attempts to approximate best the state of the accelerated group which exists as the trajectory approaches the boundary of the region of stability. The "mode of instability" method is inconsistent with the differential topology based description of the region of stability. Each u.e.p. and the associated portion of the boundary of the region of stability, represented by the stable manifold for that u.e.p., is associated with the loss of stability by a specific machine or 14 group of machines. The accuracy of the differential topology based description is associated with correctly identifying the group of machines and thus the stable mani- fold associated with a specific u.e.p. crossed by the transient trajectory. The most attractive method for approximating the boundary of the region of sta- bility is the PEBS (Potential Energy Boundary Surface) method. In this method, the PEBS is defined as the surface connecting the unstable equilibrium points (saddle points) and orthogonal to the equipotential (constant VPE) surfaces. A Taylor series approximation of the fault on trajectory is used to determine the PEBS crOssing point where the system potential energy V”; is maximized. This energy is compared with the total system energy at the clearing time to establish whether the system is stable or unstable for this clearing time. In this approach, two major weak points occur: 1. If the potential energy function has very high gradients near the PEBS crossing, then the fault on trajectory may pass through a potential energy peak which is different from the correct one. 2. the system may have some kinetic energy Va, while passing through the PEBS. Thus, there is some kinetic energy that does not contribute to instability. An iterative PEBS method [14] is derived to correct those two weak points. The fault is simulated for t > ’c using the post fault system and the kinetic energy term is corrected so that only the kinetic energy between the accelerated and stationary groups is used. The ldnetic energy within each group is ignored. This discussion shows a nearly consistent trend toward using only the energy of accelerated individual machines or groups of machines with respect to some stationary group in characterizing the region of stability. This trend is apparent in both PEBS and in u.e.p. based energy methods. Furthermore, the individual machine or group energy function methods [12-14] were shown to provide the most accurate characteri- zation of stability. New individual machine and group energy functions will be developed in this thesis in order to prove the proposed form of the stable manifold for 15 a particular u.e.p. is a portion of the stable manifold for that u.e.p. Previous energy functions did not clearly separate the energy on the stable and unstable manifolds. Thus, they could not perfectly characterize the true region of stability when the trajec- tory crosses the stable manifold associated with that u.e.p. These new individual machine and group energy functions can characterize the energy associated with the stable and unstable manifolds for the specific u.e.p. Differential topology concepts must be used to prove that the proposed form of the stable manifold is a portion of the stable manifold for that u.e.p. Lyapunov stability theory will be used as part of the differential topology argument but could not be used alone to establish the boundary of the region of attraction. 1.3.2.2 Cutset Approximation of the Region of Stability In using Lyapunov’s second method, the constant impedance load is absorbed into the bus admittance matrix for a reduced network, and hence transfer conductance occurs, even if the actual transmission networks is assumed to be lossless originally. As a consequence, a path-dependent Lyapunov function is constructed. Hence, some approximations are needed to replace the path dependent term by a non-path dependent approximation. Bergen and Hill [21] developed a tOpological Lyapunov function where the energy is associated with the magnetic fields in the actual transmission net- work as well as with the kinetic and potential energy of synchronous generators and frequency dependent loads. This topological approach has the advantages that it avoids the transfer conductance term that appears in the reduced model. The characterization of the region of stability for a topological Lyapunov function is associated with the identifying critical cut set where loss of synchronism occurs. The cutset based method of characterization region of stability was also first initiated by Bergen and Hill [21]. They developed an expression for the maximum energy absorption on any cutset. This maximum energy absorption on a cutset is the sum of the energy of all branches on a cutset evaluated over 6;} to n—oij- , where 6;} is the 16 branch angle at prefault base case. The critical cutset, where loss of synchronism will supposedly occur, is identified by finding the cutset with the minimum energy. Sigari [22] developed a topological Lyapunov function for a power system model that includes both mechanical and flux decay dynamics of generation, constant power real load model, constant current and impedance reactive load model, and the magnetic energy of all branches in the network. Sigari proposed a Popov-like criterion for this power system model applied to every cutset to describe the region of stability. A very restrictive set of necessary and sufficient conditions for retention of stability across a cutset was derived. This description of the region of stability was derived based on assuming the Popov criterion was satisfied across this cutset. Rastgoufard proposed a critical energy method for multi-machine power system. His method derives the max- imum potential energy on the cutsets that encircles a particular generator or group based on the form of the individual or group energy function derived in [7- 18]. The cutset methods recognizes that the boundary of the region of attraction (1) depends on a constraint on the angle changes across the cutset encircling the gen- erator or generator group that loses stability. This fact is verified in his thesis by proving that a particular constraint on the system state is indeed the stable mani- fold for a particular u.e.p. (2) can not be accurately characterized by the critical energy at a point on the stabil- ity boundary. It is known that potential energy of the u.e.p. only characterizes the energy at One point on the boundary of the region of attraction. The energy function that will be derived to establish the stable manifold for any u.e.p. has mostly path dependent terms. Thus, the energy absorbed to reach each point on the stable manifold is different and thus energy based characterizations of the boundary of the region of attraction are inherently inaccurate. (3) The closed form expression for a portion of the stable manifold for any u.e.p., which is derived in this thesis, is a function of the power flows across the cutset 17 that surrounds a generator or a group of generators. This generator or group of generators would lose synchronism if this stable manifold were crossed by the transient trajectory. Thus, the cutset methods have led to the derivation of this closed form expression for portions of the stability boundary. These closed form expressions are related to the ability of a cutset that surrounds a generator group to decelerate this group of generators. 1.3.2.3 Approximation of the Differential topology Description of the Region of Stability Researchers [7,5] identified those u.e.p.’s lying on the boundary by checking whether the unstable direction will converge to the s.e.p. using a numerical integration method, and then integrate a trajectory in the stable direction to construct the boun- dary. For higher dimensional systems, the numerical integration can only provide a set of trajectories on the stable manifold. A second method of characterizing the boundary is to approximate each stable manifold W‘ (xi) by a series approximation [1,2,21] around an u.e.p. x,- . This method requires computing each of the n-l type 1 u.e.p. and then requires determining the hyperplane that is tangent to the stable manifold around the u.e.p. The accuracy of the approximation of the hyperplane to the stable manifold will only be assured if the tra- jectory crosses the stable manifold sufficiently close to the u.e.p. This method of characterizing the true boundary of the region of attraction is potentially the most accurate. A third method for characterizing the boundary of the region of attraction is given in [7]. A controlling u.e.p. is chosen based on closeness to the point that the fault on trajectory crosses the PEBS. The potential energy at the controlling u.e.p. is evaluated. The potential energy at the fault on trajectory is integrated backward along the fault trajectory until changes become small. This method does not assure that the critical energy obtained is that which one could obtain at the boundary. Furthermore, the method does not assure a boundary is produced that consistently lies within the 18 true boundary of the region of attraction. If the boundary produced by the method lies outside the region of attraction, the method would indicate the system is stable when it actually is not stable. This method is proposed because the energy at a u.e.p. is the only point along the true boundary of the region of attraction with that energy. Thus using the energy at a u.e.p. as the critical energy could estimate a boundary that lies inside or outside the boundary of the region of attraction. 1.3.3 Diagram of the Region of Stability by Various Approaches A pictorial diagram that illustrates the region of attraction characterized by several approaches is shown in Figure 1.3.3. The solid line represents the true boun- dary characterized by differential topology; u.e.p.’s lie on this manifold. Since the boundary is the union of the stable manifold of the u.e.p.’s, it covers the largest domain of attraction. The dotted line represents the PEBS which is the surface con- necting all the u.e.p.’s and the potential energy is maximized on this surface. This PEBS surface could be inside or outside the true boundary. The ellipse represents a constant energy surface through a particular u.e.p. closest to the fault trajectory charac- terized by the controlling u.e.p. method The circle represents constant energy surface through the lowest u.e.p. that is closest to the s.e.p.. The conservative nature of the closest u.e.p. method is clearly shown in this diagram, The controlling u.e.p. method provides a larger stability boundary, which is the constant energy surface associated with the u.e.p., which the fault-on trajectory is moving toward. 19 fault trajectory @w ”W“ uep 3 / / controlling uep uep 1 closest uep SCP uep 4 uep 7 uep 5 uep 6 \ PEBS Figure 1.3.3 - Pictorial diagram for region Of stability 20 1.4 CHAPTER OUTLINE Chapter 2 derives closed form expressions for portions of the stable manifolds of u.e.p. that lie on the boundary of the region of attraction of a s.e.p. The form of the portion of the stable manifold is hypothesized to be a constraint on the power flows on the cutset that surrounds the single generator that loses stability if the transient trajec- tory crosses this stable manifold A center of stationary group (CSG) power system model is derived. The reference for this model is the inertial center Of the machines that are assumed to be stationary or not significantly accelerated during the fault. An individual machine energy function is derived based on this CSG power system model. An unique individual machine energy function is thus produced depending on the sin- gle generator that is accelerated. The CSG reference for a specific individual machine energy function is the inertial center of all machines other than the one for _which the specific individual machine energy- function is written. The form of the stable mani- fold for a u.e.p. where only one generator loses stability is hypothesized. The hypothesized form of the stable manifold for a specific 1 machine u.e.p. is shown to be a portion of the stable manifold for‘that u.e.p. The proof requires showing the u.e.p. lies on the stable manifold and that all trajectories on the stable manifold converge asymptotically to the u.e.p. The individual machine energy function written in the CSG reference frame is used to prove this result. The "1 machine u.e.p." is then shown to be a type 1 u.e.p. Chapter 3 repeats chapter 2 results for the case where m machines can lose stabil- ity. A CSG model and group energy function are derived for the case where m machines are accelerated rather than one as obtained in chapter 2. A form of the stable manifold is again hypothesized. The hypothesized form of the stable manifold is then shown to be a portion of the actual stable manifold for that in machine u.e.p. The m machine u.e.p. is then shown to be a type in u.e.p. The stable manifold for m machine u.e.p. is shown to embody the constraints of m "1 machine u.e.p.". This 21 stable manifold for the m machine u.e.p. then lies within the closure of all "1 machine u.e.p." stable manifolds. Chapter 4 derives a peak angle predictor. This peak predictor attempts to predict the peak angle excursion of the faulted power system over all time after the fault is cleared. The predictor depends on the acceleration developed during the fault on period and the fault clearing time. The predictor also depends on the state of the sys- tem (angle and speeds) at the fault clearing time. The state excursion from the state at the fault clearing time x(tc) to the state Jrp when peak angle excursion occurs is predicted. The prediction is based on a mean square measure derived from a linear- ized power system model. The linearized power system model is linearized at some state along the states trajectory between x (re) and xp. The peak predictor is shown to be very accurate based on comparison with simulation results on a number of fault C3888. Chapter 5 develops a method for assessing retention or loss of transient stability. This method depends on the characterization of the boundary of the region of stability derived in chapter 2 and 3. The method for assessing retention or loss of stability predicts the group of generators that lose stability based on the peak predictor. The peak angle excursion obtained from the peak predictor for a fault, fault clearing time and fault clearing action is then used to decide whether the peak is on the stable or unstable side of the stable manifold associated with the group of machines predicted to lose stability. If the predicted peak angle excursions are on the stable side of the stable manifold, the trajectory is decided to be stable for that fault, fault clearing time, and fault clearing action and vice versa. This method of predicting retention and loss of stability is then applied to the New England 39 bus system. The results for different fault cases are then compared with the real time simulation to indicate the accuracy of this method. These results indicate the method is quite accurate. 22 Finally, Chapter 6 summaries the contribution of this investigation and makes some recommendations for future research. CHAPTER 2 MODEL AND APPROACH 2.1 POWER SYSTEM MODEL This chapter derives a closed form equation that describes the stable manifold for each of the n-l type 1 u.e.p. These stable manifolds can be obtained without having to compute the post fault stable equilibrium points for each of the stable manifolds as required in [25]. The equation describing the stable manifold for each unstable equili- brium point can be written knowing only the post fault stable equilibrium point. The n-l stable manifolds for each of n-l type 1 u.e.p. constrain the power across the cutset of branches that encircle each of the n-l non swing machine generators. A stable manifold is not developed for the swing machine. This is the first time that the n-l Stable manifolds for type 1 u.e.p. have been associated with the stability of the n-l. non-swing generators. For a point to lie on the stable manifolds of a specific u.e.p., the sum of power flows across all branches in the cutset that surround the specific gen- erator that corresponds to a specific u.e.p. must equal the sum of the power flow across this cutset at the post fault equilibrium point. This result is intuitive. If the kinetic energy accumulated during the fault-on period is to be exhausted during the post-fault period so that trajectory is stable, the power flows across the cutsets that surround all of the n-l non-swing generators at any point in the region of attraction must exceed the power flows across these same cutsets at the post fault stable equilibrium point. A point that is outside the region of attraction will have power flows across one or more of the cutsets that surround the n-l non-swing generators that are less than the power flows across these cutset at the post fault stable equilibrium point. This would result in continuous acceleration of the generators. The equal area criterion could be used to characterize the region of stability for a single machine infinite bus system in the exact same manner. The hypothesized characterization of the stable manifold for a specific type 1 u.e.p. and associated generator not only constrain the power flows and angle ’7'? 24 differences across the cutset that surrounds this generator but also constrains its speed. The speed of the inertial center of all of the non-accelerated generators in the system would equal the angular speed of the accelerated generator if the particular point in state space is to lie on the portion of stable manifold for a specific type 1 u.e.p. that is characterized by the hypothesized form of the stable manifold. The hypothesized form of the stable manifold for each type 1 u.e.p. and generator places two constraints that contain all the states of the system. The two state constraints assures that the genera- tor will not accelerate or decelerate with respect to the inertial center of the other machines in the system at every point on the stable manifold for a specific u.e.p. and generator. After the hypothesized form of the stable manifold of a type 1 u.e.p. is shown to belong to the stable manifold for that type 1 u.e.p., the actual stable manifold for that type 1 u.e.p. is given. The stable manifold for a type 1 u.e.p. is the set of all trajectories that converge to the hypothesized form of the stable manifold for a type 1 u.e.p; This actual stable manifold does not require that the generator associated with a specific type 1 u.e.p. not accelerate or decelerate with respect to the inertial center of the other machines. Lifting this constraint allows the stable manifold for type 1 u.e.p. to be described by a single equation, as it should based on the definition of a stable manifold for a type 1 u.e.p. The portion of the stable manifold that does not lie on the hypothesized form of the stable manifold can not be characterized by a closed form expression. Thus, the hypothesized form of the stable manifold for the type 1 u.e.p. will be shown to be subset of the actual stable manifold for that type 1 u.e.p. that can be characterized by a closed form expression. In this chapter, the CSG power system model and the individual machine energy function based on a CSG model are derived; the form of the stable manifold is hypothesized; and the hypothesized form of the stable manifold is shown to be a por- tion of the stable manifold for a type 1 u.e.p.. 25 2.1.1 Definition Experience indicates that when a fault occurs in the system, the transient stability is dictated by the group of generators which are accelerated and are generally electri- cally close to the fault location. The longer the fault remains on the system, the larger the group of machines which experience significant acceleration will become. This group of machines is called the acceleration group; the remaining generators in the system is called the stationary group. Hence, we define a stationary group as a group of generators in which the difference in any pair of generator internal bus angles is less than or equal to 90°. An acceleration group is a group of generators where the generator internal bus angle difference of any bus "i" in the acceleration group and any bus "j" in the stationary group exceeds 90°. It is also known that the generators initially forming the acceleration group do not necessarily remain in the acceleration group forever. Some of the generators that are initially in the acceleration group may decelerate and join the stationary group and then stay in that group. 2.1.2 Loads Each load is represented as a constant impedance model which adds an adnrit- tance between these nodes and the ground given by _ PLi ‘1' QLi u- Ith2 where IV,- | is the magnitude of the phasor voltage at the 1"” bus obtained form pre- fault load flow and PL,- + j Q“ is the load at the 1"” bus. Pu + j Qu is converted into an admittance that is reflected into the system Y bus admittance matrix. For sys- tems with transfer conductance, a nonpath dependent integral expression for total sys- tem energy is impossible without some approximations. Since a constant impedance load model is assumed and since the network will be reduced back to generator inter- 26 nal buses, transfer conductances in the off diagonal elements of the reduced line admit- tance matrix will occur and will not necessarily be small. These transfer conductances have been neglected for the sake of ease of analysis in deriving new theoretical results. Thus, we also assume that the transfer conductance of the reduced network after aggre- gation are zero. 2.1.3 The Network The power network consists of n generator buses connected by lossless transmis- sion lines. It is represented by its admittance matrix Y = [Yij] = j[B,-j]. For i a: j, By- is the susceptance of a fictitious branch connecting internal buses i and j in the post fault network. Let IE,- |<8,- denote the internal voltage phasor of the 1"” gen- erator. 2.1.4 Generator Model For the purpose of analysis in transient stability and easily computed transient assessment, a well-known classical model is considered here to capture the behavior of the generator in the power system. The classical model has the following simplifying assumption: 1. Mechanical input is assumed constant and equal to the prefault value during the time interval of interest 2. The voltage behind transient reactance of the synchronous machine is assumed to be of constant magnitude determined from the steady state conditions existing prior to the fault. 3. Loads are represented by constant impedances based on the prefault voltage con- ditions obtained from a load flow. 4. The network is assumed to be in the sinusoidal steady state. The motion of generator i is governed by the swing equation 27 8, = to,- - (Do (2.1.1a) Mid),- =P,,u- -D,-((o,- - coo) - i EU E,- E} sinfiij (2.1.1b) j=l¢i where a); rotor speed (21tf rad/sec) too synchronous speed (377 rad/sec) M ,- moment of inertia (211/010 sec) D,- uniform damping coefficient Ei the generator internal voltage (pa) 5,- the generator internal voltage angle (rad) Bi} susceptance connecting bus 1 and j (pu) Pm,- mechanical power (pu) Each portion of the boundary of the region of attraction is characterized by deter- mining a closed form expression for the stable manifold associated with the u.e.p. that lies on that portion of the boundary of the region of attraction considered. A set Of constraints is hypothesized as a possible description of a portion of the stable manifold for several classes of u.e.p. considered. These classes of u.e.p.’s are by no means exhaustive and thus no effort is made to describe the entire boundary of the region of attraction. The classes of u.e.p.’s for which stable manifolds will be established correspond to u.e.p. where m generators are accelerated and ultimately lose stability and where the remainder of the (n-m) generators have angle differences of less than 900 for the portion of the trajectory beyond some time :3 > rc , where tc is the clearing time. The u.e.p.’s in this class are called "m machine u.e.p." and may be inferred and will be later proved to be u.e.p. of type m, where the dimension of the unstable mani- fold is m. It will be shown that the class of type in correspond to the u.e.p. where m machines are accelerated and (n-m) machines remain stationary for m = l,2,...,n-l. There are possibly similar u.e.p. classes for the case where m machines are decelerated 28 and (n-m) machines remain stationary. There may also be classes of u.e.p. where m1 machines are accelerated, m2 machines are decelerated and n - m1 — m2 remain sta- tionary. The (n-l) classes of u.e.p. for which stable manifolds are derived in this thesis are possibly the most important since they correspond to u.e.p. for which the existing direct stability assessment methods are thought to be capable of handling. Lit- tle work has been reported for these other classes of u.e.p. using existing direct stabil- ity methods The following set of assumptions are made [2] regarding system (2.1.1) in order to make a theoretical description ( 1.3.5) of the boundary of the region of stability valid. (A1) A stable equilibrium point exists; (A2)A11 unstable equilibrium points on the stability boundary of (2.1.1) are hyperbolic, which implies they are isolated; (A3) The intersection of W’ (xi) & W‘ (xi) satisfies the transversality condition, for all x,- & xj on the stability boundary, in other words, the stable and unstable mani- folds constitute the entire space; (A4) There exist a C1 energy function v tan—>12 for (2.1.1) such that (i) V(¢(x,t )) S 0 at x e E; where E is the set of equilibrium points; (ii) if x e B, then the set{t e R : V((x,t)) is bounded implies O by lemma 3.1 and Vsa(.) is bounded below by Vm(x‘), we have V“ (x‘ ) < V“ ( 0 for all i,j at 1. we have j=l¢i n n "‘ 2 3;]: E; Ej C0560 + B;1 E; El C089;1 "' g Bi] E; E} C0590“ S l; J: 3“ 44 n n S- 2 8;}: E; Ej COSGU - B;1E;E1C059;1+ z 3;; E; E]: COSGU j=l¢i j=2¢i Thus, I: ‘2 2 8;]- E; E}: C089;j $115 28;1 E; El COSG;1 j=2$i i = 2,3,...,n (2.4.3b) Since cosfiil < 0 for i = 2,3,...,n, then 3.,- S 0 for i = 2,3,...,n. Thus, there is one posi- tive eigenvalue and n-1 negative eigenvalues of F 0- From [23], it has been shown that Jo has the same number of positive eigenvalues as F 0. Thus, F 0 has one positive eigenvalue when evaluated at a 1 machine u.e.p. Thus, the 1 machine u.e.p. is type 1 in an unreferenced model. To show that the referenced system has the same eigenvalues as the unreferenced system, we could prove that the two systems are equivalent by showing that they are related by a similar transformation. Linearizing the referenced swing equation, we have Aé _ , A9 [..] -, [a] where 1:, o I lM'lF —ol M = Diag (M1,M2,...,Mn) F = [2, g] (2.4.4b) and, the i j‘h elements of F is expressed in (2.4.5) 3ft " Mi " . 39—.- = - z Bij E; Ej C086;j - 7-28,“- Ek E; C059,“: 1 S 1 (2.453) I j=l¢i sa k=2 45 8f,- Mi n . . . 36—- = 3‘} E; Ej C089;j " 7231;} Ek Ej COSij j S 19] #1 (245b) j 3a k=2 3ft " Mi - -5—6— = - 2 BU E; Ej COSOU + rB;1E; El C059” 1 .>. 2 (2.4.50) i j=1¢£ 50 3f,- Mi . . . W =Bij E,- E; coseij + fi—le E; E1 costl J 2 2. J 3* l (245d) 1 30 where n ft' = .2. .(Bij Ei 5," sineg} -Bij Ei Ej Sineii) ‘ j=l¢ M. n so [:82 Consider a nonsingular transformation T, such that 6 = T8, since 9,- = 5,- - 5,0, we have T=l-[OO e ..... e]! (2.4.6) where I is a n square matrix, 0 is a zero vector and e is a column vector with all ele- ments being 1. What we are going to show is that those two Jacobian (J o) and refer- enced (J) are similar. THEOREM 4.1 The two Jacobian matries Jo and J of the unreferenced and referenced system respectively, are related by a similar transformation; thus, they have the same eigen- values. Proof: From equation (2.4.6) we could construct a state transformation matrix between equa- tion(2.4.1) and (2.4.4), thus, we have 46 ll '7]: 8 00 l.___l [2,]= [2; anal and therefore, (a = [r o]_ o ' z [a] d) O T M-IFO ‘0'] C0 = [a 2]- H thus, J = o T I T'1 T M‘lFo T-1 T -o 7-1 o 1- M4): -01 = fjof-l o I ,T-1 M-IFO —ol 0 (2.4.7) (2.4.8) 2.5 JUSTIFICATION OF A CSG REFERENCE BASED ON SIMULATION RESULTS The use of a CSG reference rather than a COA reference leads to very significant differences between the results obtained in this section and those obtained in the great body of literature on direct stability methods. The differences are summarized bellow (1) a unique energy function must be used to describe the portion of the boundary of the region of attraction associated with each combination of machines in the accelerated and stationary group. (2) use of a COA reference requires using only one energy function to describe the boundary of the region of attraction; Most of the terms in the energy function using a CSG reference are path dependent whereas most of the terms in the COA (3) 47 reference energy function are not path dependent. A path dependent CSG energy function would require selecting the correct generator that loses stability and would make accuracy of the entire transient trajectory important if a PEBS approximation to the boundary of the region of attraction were used to estimate the boundary of the region of attraction. Furthermore, use of the potential energy at a closest u.e.p., controlling u.e.p., or mode of instability u.e.p. would not accu- rately describe the CSG energy on the boundary of the region of stability since the CSG energy function is path dependent and the trajectory that brought the system to the particular u.e.p. must be known to evaluate the potential energy at the u.e.p. Thus, PEBS, closest u.e.p., controlling u.e.p., and mode of instability u.e.p. methods can’t be used to describe the true boundary of the region of stabil- ity if a CSG energy functions is used. a u.e.p. need not be computed to approximate the boundary of the region of attraction since each portion of actual boundary of the region of attraction associ- ated with a type 1 u.e.p. is determined as a closed form expression that does not require computation of a u.e.p. Since these results center on using a CSG rather than a COA reference, the fol- lowing simulation results will show that a very sharp change in acceleration can be observed if a CSG reference is used. A far less sharp change in acceleration is observed with a COA reference since both the COA reference and the critical machine that loses stability begin to accelerate if a COA reference is used. The CSG reference is relatively stationary and thus clearly shows the change in the sign of the acceleration at the point where the trajectory crosses the acceleration manifold. Thus, the CSG reference model critical generator acceleration will clearly indicate the exact time the generators trajctory crosses the acceleration manifold (2.2.8a) where the trajectory enters the unstable region. The angular speed of the critical generator in the CSG reference clearly shows the point where the angular speed of the critical generator 48 crosses the velocity manifold (toi=cu,a). The time at which the trajectory crosses the acceleration manifold (2.2.8a) and velocity manifold (2.2.8b) should be identical when the fault clearing time is close to the critical clearing time. TIME CURUE (88-29) 49 18 0600 14 9013- 10 7425 ‘ 7 08375 ‘ 3 h92500 ‘ 233750 " A a no anode 507. 3 003 a ma ) Figure 2.5.la - Acceleration vs time curve (CSG model) 2605 TIME (sec 1 506 we 1 A 5088 0100 -3 89250 " 55125‘ -11 2100 -7 QCCEL (p u ) 50 QCCEL vs TIME CURUE (88-89) 17 2100 "fi W W l ll ll? 14 7400 ‘ P A H l ”i r l ill ' 'r 12 2700 . l l l ‘ l l 9 00000 1 l .l l i 7 33000 ‘ l'l 9" 3: l 4 86000 - 3 l . l l ;! lié 2 39000 a = U ll w l i llll. l H :l' 1 W Ill ll 1|? - 080000 - l y 2 .l :3] 2'11 Vii“ . ll ‘8 55000 4 l TIME (sec > Figure 2.5.lb - Acceleration vs time curve (COA model) 0100 5088 i 008 l 506 2 005 8 504 3 003 3 501 4 ml (.) () SPEED (rad ) 51 SPEED vs TIME CURUE (28-29) 32 5800 ij 28 2025 ‘ 23 8250 4 l 19 4475 l 15 0700 l 10 6925 6 31500 ‘ l 1 93750 -2 44000 I 0100 5088 l 008 l 506 2 005 2 509 3 003 3 T I ME C SEC ) Figure 2.5.2a - Angular speed vs time curve (CSG model) Ul C) t r) (z () PEED (rad ) ) ( 52 SPEED vs TIME CURUE (28-29) 146 970 128 650 4 110 330 ' 92 0100 a 73 6900 q 55 3700 ‘ 37 0500 ‘ 18 7300 ‘ 410000 f 0100 5088 l 008 l 506 2 005 2 509 3 003 TIME (sec > Figure 2.5.2b - Angular speed vs time curve (COA model) _-__ --.. .—__V_—.—.-—_~—- _4_. ( ) .L_.-_.... C) () QNGLE (deg ) 53 QNGLE vs TIME CURUE (28-29) 1278 69 I u03104 /i l l l 927 505 l 751 912 l 576 320 l 400 727 l 225 135 E i 49 5425 d i l i I l -186 050 T T r I j ‘ - ' 0100 5088 1 008 l 506 2 005 2 50% 3 003 3 501 4 CC: TIME (sec > Figure 2.5.3a - Angle vs time curve (CSG model) : (deg ) QNGLI- 54 QNGLE vs TIME CURUE (28-29) 17293 1 ,] 15135 1 4 12977 1% 10819 1 ‘ 8661 03 4 E l 6503 02 l 9395 00 l 2186 99 28 9700 NRA (x) Ul . < J J 0100 5088 1 008 1 506 2 005 2 509 3 003 TIME (sec ) Figure 2.5.3b - Angle vs time curve (COA model) .T Vvv 55 Figure 2.5.1a and 2.5.1b are the plots of angular acceleration of the critical gen- erator for CSG and COA reference, respectively. Figure 2.5.2a and 2.5.2b are the angular speed curve for the critical generator for CSG and COA reference. Figure 2.5.3a and 2.5.3b are the plots of angle for the critical generator for CSG and COA reference respectively. The acceleration of the critical generator for the CSG reference (Figure 2.5.1a) show that the acceleration is zero at t = O and at t = 2.2. Note that the velocity of the critical generator in the CSG is zero at r = O and t = 2.2 seconds. The point (r = 2.2) is also the point where the velocity manifold (2.2.8b) of the stable manifold is crossed. This is the point where the stable manifold of this generator is crossed. The angular speed and angle curves (Figure 2.5.2a, 2.5.3a) clearly show a rapid increase in speed and a very pronounced increase in angle at approximately 2.2 seconds. Thus, the CSG model clearly indicates the point at which the stable manifold is crossed in the angle, speed, and acceleration curves when the fault clearing time is close to but just above the critical clearing time. The acceleration curve for the COA reference (Figure 2.5.1b) is very positive at r = O which indicates that the COA does not clearly indicate acceleration. The acceleration goes slightly negative and reaches zero at approximately 2.1 seconds. Zero acceleration in the COA reference only means that the critical generations is not being accelerated with respect to the center of angle reference that includes this gen- erator. Zero acceleration does not indicate when the trajectory crosses the boundary of the region of attraction since the trajectory is shown to cross the region of stability boundary at 2.2 seconds in Figure 2.5.1a The angular speed curve Figure 2.5.2b is never zero for all t > O and is a rather straight line. The velocity curve on the COA reference does not show the velocity manifold and does not clearly show the accelera- tion change after crossing the velocity and acceleration manifolds. The angle trajec- tories of the critical generator in the COA reference (Figure 2.5.3b) do not show any 56 indication as to when the stability boundary is crossed. The angular acceleration, speed, and angle of the critical generator in the CSG model very clearly indicate when the boundary of the region of stability is crossed. These results indicate that the COA model hides the crossing of the stability boundary in the acceleration, angular speed, and angle trajectories. These results confirm the theoretical results that the COA reference hides the stability boundary in the global or individual machine energy functions. 2.6 SUMMARY This chapter derived a closed form equation for the stable manifolds of all type 1 u.e.p.’s that are associated with acceleration of single generators. The closure of the union of these stable manifolds is a description of the boundary of the region of attrac- tion in the first quadrant of state space under the assumption Al-AS. Stable'manifold for type m u.e.p. for m 2 2 will be derived in chapter 3. This derivation requires developing a new model and energy function for each class of u.e.p. The form of the stable manifold for "m machines u.e.p." is hypothesized and then proved to be a por- tion of the stable manifold for that class of u.e.p. The "m machines u.e.p." will then be shown to be a type m u.e.p. The portion of stable manifold for a m machine u.e.p., that can be characterized by closed form expression, has 2m constraints. The first m constraints require that the power flow across a cutset that surrounds each of the m generators on the stable manifold is identical to that power flow on this same cutset at the post fault equilibrium point. The last m constraints are that (0,- = tum for the m machines in the accelerated group, where (03,, is the inertial center of the n-m machines in the stationary group. The 2m constraints that specify the portion of the stable manifold, characterized by closed form expression, for a "m machines u.e.p." is the intersection of the two constraints that describe the portion of the stable manifold characterized by closed form expression for each of the m generators in the accelerated group. The stable manifold for the m machines u.e.p. does not lie on intersection of 57 the stable manifold of the "1 machine u.e.p." but clearly lies in the closure of these "1 machine u.e.p." stable manifolds since stable manifolds of different u.e.p. can not intersect from the theory of dynamical systems. Although this description of the boundary of the region of stability has been derived for a classical transient stability model, the method could be used to describe the boundary of the region of stability models that have 1. generator flux decay dynamics; 2. generator exciter and power system stablizer dynamics; 3. combination of constant power, constant current, constant impedance real and reactive load models; 4. topological network models; 5. network models with transfer conductances Thus, the methodology used to establish the stable manifolds and thestructural form of the equations that describe the stable manifold for a particular model may be the most important contribution of this thesis. The importance of the result is that a portion of the stable manifolds for u.e.p. have been characterized as closed form algebraic equations. The equations describing these stable manifold are not approximations and thus there is no error in the charac- terization of the boundary of the region of stability. All previous Lyapunov or differential topology based methods for deciding retention or loss of stability for a par- ticular fault trajectory had error. The accuracy of any particular method for deciding retention or loss of stability has been in doubt partially because there was no method for deciding precisely which method for deciding retention or loss of stability should be best for any fault trajectory. Since this method has no error it obviously can be used as a standard for the classical transient stability model. CHAPTER 3 CHARACTERIZATION OF THE BOUNDARY OF THE REGION OF STABILITY FOR M MACHINE U.E.P. 3.1 INTRODUCTION The characterization of the boundary of the region of stability associated with any group of m machines losing stability is derived in this chapter for any m satisfying 2 SmS n-l. The steps involved in establishing the boundary of the region of stability for "1 machine u.e.p." must be repeated here. We must first derive a new model for the case where m machines belong to the accelerated group and m-n machines belong to the stationary group. We will assume that the first m machines belong to the accelerated or critical group and the remaining n-m machines belong to the stationary group. These is again no loss of generality in making this assumption since one can always relabel any combination of m machines out of n that lose stability as the first m machines in the model. Note that there is a unique model that exposes the stable and unstable manifold for every possible combination of stable and unstable machines. Previously referencing the power system model did not depend on which combination of machines were stable and unstable. The n’h machine reference and center of angle reference were used to describe the region of stability without regard to which machines lose stability. This is a fatal error that prevented these methods from accu- rately describing the boundary of the region of attraction. Furthermore, an energy function must be derived for this model, which implies there is a unique energy func- tion for every combination of stable and unstable machines. This energy function that is unique to the specific acCelerated group of machines and stationary group of machines, is the only energy function that can be partitioned into energy on the sta- tionary group of machines (V30) and the energy associated with the critical machines (V6,) and their movement toward or away from the stable manifolds. The 62 59 hypothesized form of the stable manifold for the m machines u.e.p. is then proposed and the proof that it is a portion of the stable manifold will proceed in a manner simi- lar that in Chapter 2. The "m machines u.e.p." is then shown to be a type m u.e.p. Finally, the set of 2m constraints that specify a portion of the stable manifold for the "m machine u.e.p." is shown to be equivalent to intersection of the constraints that Specify a portion of the stable manifold for "1 machine u.e.p." for each of m machines that lose stability. Since stable manifolds of different u.e.p. can not intersect, the stable manifold for m machine u.e.p. can be seen as the corners of edges of the stable manifolds where m type 1 u.e.p. stable manifolds come together. The CSG model and associated energy function are now derived. 3.2 MATHEMATICAL MODEL Under the same assumption as outlined in chapter 2, the swing equation for each - machine are rewritten again for completeness. &=%-% Qua 04,6»,- =P,,,, -D,-((o,- - 000) - i 3,, E, E}. sinfiij (3.2.1b) j=l¢i where 00,- rotor speed (21tf rad/sec) coo synchronous speed (377 rad/sec) M ,- moment of inertia (ZI-I/mO sec) D,- uniform damping coefficient E ,- the generator internal voltage (pu) 8,- the generator internal voltage angle (rad) Bij- susceptance connecting bus 1 and j (pu) Pm,- mechanical power (pu) Here, we assume that the entire power system breaks into two groups of machines 60 when the system loses its stability: the acceleration group has m machines numbered l,2,...,m, and the stationary group has (n-m) machines numbered m+1, m+2,...,n. Define the center of angle 85,, as n 2 Mk8]: 30 MS“ I: Msa = 2 Mk k=m+l The center of stationary angle satisfies the following differential equation Mussa = i Minsk k=m+l II o n = 2 (-Dk6k 4' PM - z Bu Ek El SIDS/d) k=m+l l=l$k fl . II II = 2 (43,5, + 2 3,, 3,, E, sin8,f, — 2 3,, 3,, E, sinfiu) k=m+l l=l$k I=l¢k ’3 since PM = E Bu 5,, E, meg. Assuming (=1 D D —E- = —m k=1,2,...,n Mk Msa I o n m n M3083“: .050834 + 2 (z Bu Ek El SIDS/f) + 2 Bid Ek El 81118:!) k=m+1 (=1 l=m+l¢k n m n " 2 (E Bu El: E1 SIDS/d "l' 2 Bid Ek El $1118“) k=m+l (=1 l=m+l¢k , n = -D:a5.sa + 2 (Bid El: El 51115:) - B“ E" El Sins“) k=m+l since 61 n n 2 2 Bu El: E1 sinfiu = 0 k=m+lI=m+1¢k The system equations in the variable 9, = 6,- - 5“ now become .. n m MiG,- = -Did),- + 2 2(3)] E‘- Ej sine; ”BU E; Ej sineij) j=1¢il=1 M‘- n m - Z 2(31‘1 E]: E) sine}; "Bu El: El Sine“) M30 k=m+ll=1 i = l,2,...,n (3.2.10) Using state space representation, the CSG model becomes 0,- = 0),- (3.2.2a) O n . . M;6),- =-Di(°i + “2(3‘1' E; E} 51116;; ‘Bij E; 5} smeij ) 1:109: Mi :0 m --A—ll- 2 2(Bu Ek El sine}; -3” £3 E! Sine”) SC k=m+ll=l i = l,2,...,n (3.2.2b) and, the manifold describing the boundary flow condition for stability is n M 2 2031: Er 5: sine)“; -Bu El; E: sine“) = 0 (3.2.3) k=m+ll=l 3.3 ENERGY FUNCTION The energy function for the CSG models (3.2.2) where damping is omitted is now derived by first multipling the i ‘h swing equation in (3.2.2b) by 8,- and form the sum from i = l,2,...,m for the critical group, ”0 . n o . Z[M,-Cbi " 2 (BU E) Ej Slnefj -Bij E; Ej srneij ) i=1 j=1¢i 62 +M 2 Emu Ek El sinef, " Bu Ek 5: Sine/4)]é; (3.3.1) 50 k=m+ll=1 Using the inequalities 3,-1- =8},- and sin8,-j =— sin8,-,-, the following relationship is obtained 0: m . . m-l m . . 2 2 BU E; Ej srneij 9,- = 2 2 Bi} E; Ej srneij 9i} i=1j=1=t i=lj=i+1 which is used repeatedly in deriving the energy function. Integrating (3.3.1) with respect to time, the energy components for the critical machines are VKE = [234, e, 0, d: = 2— -M, 0), (3.3.221) =1 i=12 VPE =I2 21 a“BU E; E} sine; 61' dt i=1j=l m m n . =1£< 223:,- E. E,- sinezfi + 2 8,,- E,- E, sine); )9, dr i=1 j=1¢i j=m+l -l m . =rz 2 31] El Ej Sine; éij dt+I2 z Bij E; Ej 81116:, 9,- dt i=1 j=i+l i=lj=m+l m-l m . m n . . e = 2‘. 2 31'} E; Ej 8m91'j(9zj-9:}) + )2 2 8,-1- E, 51- srnij 9,- dr i=1 j=i+1 i=1j=m+l (3.3.2b) m n . , VPE; =I2 2 Bij E; E] srneij 6i (It i=1j=l¢i m m =IZ(£B,-j E,- E- sin9,j+ z BijE, Ej sine,- )9,- d: i=1 j=1$i j=m+1 film 2 Bi] E, Ej Sine”- éij dt'l'Iz 2 Bij E‘- E} sine” 6; d! i=1 j=i+l i=1j=m+l -l m . =2 2 3,,- E, E (cosef- -cose,j)+jz z; 3, E,- E, sinOU 9, dr i=1j=i+l i=lj=m+1 63 (3.3.2c) m M- n v”): 1274— 211 (3,, 5,, E, sinek, —B,,, E, E, sine“) 6, dt k=m+ u :Ms (3.3.2d) The total energy for critical machines is the sum of kinetic energy and potential energy, Vcr = VKE " VPE, + VPE; + VPE, m 1M =,.E,M 2 03-1 at . .r s - 2 2 Bij E,‘ Ej smGU (9,7 - 9,7) i=1 j=i+l -1 m - m2 z 3,,- E, E,- (c056,, — cosefj) i=1 j=i+l In M‘- n m . + I(1+2 ) z 203,, 13,, E, sine}; -B,,, 5,, E, sine“) e,- d: (3.3.3) i=1 M34 k=m+ll=l The last term in the critical energy is accounted for the energy flow between two clus- II m ters of groups. If 2 2(Bk, E, E, sinef, - 8,, E, E, sine“) is positive, energy is k=m+ll=l transferred from the stationary group to the critical group, causing a destablizing effect. I: ”I If 2 2(Bu E,‘ E, sineg', - Bk, E, E, sine”) is negative, energy is transferred from k=m+ll=l critical group to stationary group, resulting a stablizing effect. Hence, from an energy point of view, the boundary between the critical and stationary group must be able to dispose of the kinetic energy developed in the critical group during the fault on period If the boundary can not dispose of this kinetic energy the critical group will be accelerated further. 64 Similarly, the energy components for stationary machines are .1. 2 M; 63: (3.3.4a) VKE =I 2 M,e,é,- dt = 2 i=m+l i=m+l n n . VPE, =1 2 2 Bi,- E,- Ej sin65~ 6,- dt i=m+lj=l¢i n m ,, . =1 2 (2.3;, Er 5; sin6,j- + 2 sz E,- E, smog) 9,- dz i=m+l j=l j=m+l¢i " m . n-l n _ =I 2 228:,- E: E,- sinei} 9,- dr +1 2; 2 3,, E,- E, sinefj 9,,- d: i=m+lj=l i=m+1j=i+1 n m . . n-l n . =l 2 2sz E: E,- smez‘j 9.- dt + 2: 2‘. B,- E, E, smegma—6:3) i=m+lj=l i=m+1j=i+l (3.3.4b) n n , Vps,= I 2‘, 28,-,- E, E,- sine,,- 9,. d: i=m+lj=1$i n m . n . . =I. z (23,}- Ei Ej $1n9,j + . 2 Bl] Ei E]. smeij) 9‘. dt ‘=”'+1 [=1 j=m+lztt " m . . n-l n . , =1 2 23:7 Ei Ej smeij 9i 4‘ +1 2 2 3,,- E,- E,- srn9,, 9,,- d: i=m+lj=l j=m+1j=i+1 (3.3.4c) n m . . n-l n 3 =1 2 23,7 E,- Ej Sine”- 6,- dt + 2 z Bij E; Ej (COSBij‘COSGij) t=m+lj=1 j=m+1j=l+l l n n m . VPE, = j 2 — z 203,, 5,, E, sine; -B,,, E, E, sine“) e,- d: i=m+1 :0 k=m+1l=l (3.3.4d) The total energy for stationary machines is the sum of kinetic energy and potential energy, 65 Vsa = VKE - VPE, + VPE, + VPE, ilMi _m.+12 n-l n - z Z By E; Ej 81119;} (9,!- "' 9,3") i=m+lj=i+1 n-l n " z 2 Bi} E, Ej (COSBij - C0565“) i=m+1j=i+l ’3 Mi '1 m . +1('1+ 2 _) Z 2(3u 5;; E, sinef, -B,,, E,‘ E, sine”) 6,- d: i=m+l MSG k==m+ll=1 (3.3.5) The positive definiteness of the energy function based on zero transfer conduc- tances can be verified in a number of ways. From physical point of view, since V represents the energy integral, it has to be greater than zero. Mathematically speaking, it can be shown that V is equal to the sum of a quadratic term and the sum of integrals of nonlinearities, each of which lies in the first and third quadrants in a region around the origin. Hence, V > 0. This energy function must be positive definite in order to show that all trajectories on the hypothesized stable manifold con- verge to the unstable equilibrium point. 3.4 THEORETICAL RESULTS The following procedure will now be followed to establish the stable manifold W‘ (xm) for the "m machine u.e.p.". (1) hypothesizing the form of the constraints characterizing the stable manifold. The hypothesized constraints are p n z (Bu Ek El Sine}; "Bld Eh El Sine“) = O l = l,2,...,"l k=m+1 S o W (xm) ' L0), = (Dsa I = l,2,...,fll (3) (4) (5) 66 Note that these constraints are 2m dimensional and require that the flow across the cutset that surrounds the m generators satisfy n m X 2(Bkz 5k E: sine/f1 -B,d Ek E, sine”) .-. o k=m+ll=l This constraint implies the boundary flow on the cutset encircling the m machines in the accelerated group is the same at the post fault stable equilibrium point as it is at every point on the stable manifold of the m machine u.e.p.. showing that the "m machine u.e.p." lies on W‘ (xm) showing that excursions on W‘ (xm) remain on W‘ (xm) and asymptotically con- verge to xm. This result indicates W’ (xm) is a portion of the stable manifold for x", . showing that x", is a type m u.e.p. and thus has a m dimensional unstable mani- fold. Lemma 4.1 and The energy function for (3.3.3) and (3.3.5) m 1 var 213M m-l m . " z 2 81'] E; Ej smej (9U — 65) i=1 j=i+l m-l m s "' .2 2 Bi} Ei Ej (C0860 - COSOU l=1 j=l+1 m Mi n m . + I<1+z ) z 2(Bkl El; E1 Sine/é -B/d Ek El sine“) 9i dt i=1 M30 k=m+ll=l n 1 Vsa = 2 EMi (bi i=m+1 67 n-l n - 2 Z Bij E,“ E}- Sine"; (9U "' 95) i=m+lj=i+1 n-l n - z 2 Bi,- E,- E,- (c059,,- -cosB,-j-) i=m+1j=i+l 5 Mi n m . + I(-l+ z ) z 2(Bkj E]; El Sinai] " Bu Ek El Sine”) 9“ dt i=m+l MSG km+ll=l have time derivative that are negative definite for all 6,- and (0,- ¢ 0 along the trajec- tory. Proof: Observe that d VPE "“1 "' . . - -— =-2 z Bij Ei E} (31110;;- " smGiJ-WU d ‘ i=lj=i+l m Mi n m . 4' (1+2_) 2 2(31" Eh El sinef, “Bu E]: El sinGu) 6i i=1MM k=m+ll=l m m . . . = .2: Bi] 51' Ej (sme'j- " 51119009,- i=1j=l 03 Mi n In ' . +(1+Z—) 2 2(Bu £1: £1 sine; “Bu Ek El sine“) 91' i=1 MSG k=m+ll=l = (-D.- co,- -M (b.- )é.- (3.4.1a) d V . (11:5 =M,-a),-Co,- (3.4.1b) Differentiating Vc, along the trajectory of (3.3.2) using (3.5.1) gives . dVKE dVPE V = + . . cr d: d: (342) "' 2 = 201-6),- < 0 i=1 68 Observe that 8V . 36s: =‘Di63i-Mimi 3V J. = MiG): 6(1),- Differentiating V,“ along the trajectory of (3.3.2) using (3.5.3) gives n 3V” . 3Vm . 2 (—9 +—53i) V . m i=m+l 86,- ‘ 363i 2 [(‘Di 6),- -M" 63") é; +Mi 6),- 6);] i=m+l " 2 =- 2 0,6),- <0 i=m+l The derivation of (3.4.2) and (3.4.4) complete the proof. The following two lemmas prove r ll 2 (Bu Ek El sine}? “Bu Bk E, sinGu) = 0 k=m+l 3 o W (xm) ° (0, = a)” l = lgzru’m is the stable manifold for a "m machine u.e.p." xm Lemma 4.2 "m machine u.e.p." xm = (9“ ,0) defined by O = 6% n O = —Di6)i + 2 (BU E; Ej Sinai; —Bij E“ Ej Sinai?) j=l¢i i n m M 2 2(31‘1 El; El Sifleé -Bkl Ek El 5111913) 30 k=m+ll=l (3.4.3a) (3.4.3b) (3.4.4) l,2,...,m 69 lies on W3 (xm) Proof: We use the fact that the u.e.p.’s are obtained by setting M,9, = 0 and 6,- = O for all i’s For critical machine, the unstable equilibrium xm satisfies n 2 (Bi). Ei E} Sines “BU Ei Ej Sines- ) j=l¢i Mi n m = 2 2(Bk1 Ek E! Sine/fl ‘31,: E, E, sinefi) i = l,2,...,m MSG k=m+ll=l For stationary machines, the unstable equilibrium x", satisfies n z (B‘j E, Ej Sine}; —Bij E; Ej sineg ) j=1$l Mi n m = 2 2(3):: £1: E! Sineé '31:: E, E, may) i =m+1,m+2,...,n M30 k=m+ll=l Summing both sides of the above equations for i=1,2,...,n the following equation is n n 2 2 (BU E,‘ E} Sinai;- '30 BI Ej sine}; ) i=1j=l¢i M‘- n m = z 2(Bu E}, El sine}; '3” El: E1 smefi) M30 k=m+ll=l i = l,2,...,n (3.4.5) Since the L.H.S. of equation (3.4.5) is zero due to cancellation of sine terms, the equilibrium xm lies on the manifold described by the boundary flow condition n m 2 2091:; Ek El sine}; '3“ E, E, sine“) = O k=m+ll=l 70 Lemma 4.3 Assume the set {x : V((D(x,t)) = O, t e R, x sex," } has measure zero, then W3 (xm) is an invariant set and any trajectory starting on W3 (xm) converges to xm Proof: Every point x e W ‘ (xm) satisfies the acceleration manifold condition n 2 (Bu Ek El Sinef, ‘ Bu Ek El Sine”) = O k=m+l and the velocity manifold condition to, = (use for l = l,2,...,m. The invariant property is proved by establishing that (3.2.2) is identically zero for i = 1,2,...m on the mani- fold. Note that the acceleration manifold and velocity manifold conditions require that d),=0 and £19,]. E, E}. sineg‘j - 3,]. E, E,- sine”. = 0 Thus, alljniachines in the critical group are stationary with respect to each other at 9,]: f} and are stationary with respect to 63,, and to“ since é,=0 and 6), =0. for i= l,2,...,m. The critical energy (3.3.3) is identically zero for trajectory on the manifold W‘ (xm). The total energy can be split into two parts, the critical energy is zero, and the stationary group energy V” is positive definite due to the angle difference being less than 90° in the region considered. Since Vm( AB 0: C0 D0 and, the i j "' elements of F 0 is expressed in (3.5.2) 3 i n i- = - Z BU Ei Ej COSBU l Sm (3.523.) 39, j=l¢i afi . . . —.— =Bij E‘- Ej C0893]: ] S m, j #1 (1.521)) 39, , afi " . 8—9- = j: _Bij E, Ej cosG,j z 2 m+1 (3.5.2c) I j=l$t 8f- Sé'i- =Bij E; Ej COSeiJ‘ j 2 "7+1: j $1. (3"52d) J 72 where n f, = z (BU E, Ej smBS -Bij E,‘ E] sin9,,-) j=l¢i Using the Gershgorin’s theorem, 3 a - n a - f; _2 I3 :MISK L+ Z I ft fori Sm n ”I n " 2 Bi] E,- E] COSGU - z I Bij E, Ej COSOU I - z I 8,, E, E} C086,,- I 5 7L, j=l$i j=1¢i j=m+l S- 28,,E, E cos6,,+ 2'31“} E E c059,, |+ Z lngE; 5, COSBz,| j=l¢i j= —1$i j=m+l Noting that for ij belonging to the stationary or critical group has cos6,,- > O, and for ij between groups has cos9,,- < 0, thus, I: — 28,-,- E, E,- cos9,,- + 21 8,, E, E,- cose,,- + 2 8,,- E, E,- cosO,,- SA, j=l¢i j=l ati j=m+l n ”I n S " z 3,} E, Ej C089,} " 2 Bi} E,‘ Ej C056,? - 2 Bi} E,‘ E} c086,, j=l¢i j=1¢i j=m+l Thus, I! 0 S l, S - 2 2 Bi} 5, Ej C089,]- 1 = l,2,...,m j=l¢l The above equation shows that if the angle difference in the critical group is greater than 90° then its eigenvalues are bounded below by zero, i.e. it has m positive eigen- values. Fori 2 m+1 n m n 2 8,,- E, E,- cos9,,- - 2| 3,,- E, E,- c059,, |- 2 IB,,- E,- E, c056,,- | s l,- j=1¢i j=l j=m+l¢i 28,,EEcos9,,+£|B E,E,cosG,-,|+ z |B,,- E,-,---EcosO tj I j=l$i j= j-m-i-laei 73 Since c056,, > 0 for ij belonging to the critical and stationary group, and c059,,- < O for i j between groups. n m n - 2 8,} E, Ej C089,, + z BU E, Ej C086,, "' 2 BU E, Ej C089,, S. A, i=1w' j=1 j=m+l¢i n m n - z BU E, Ej C086,, - z 8,} E, E, C059,,- 4' Z 3,}- E, Ej C089,, j=l¢i j=l j=m+1¢i Cancelling terms, the equation becomes ’1 m — 2 Z BU E, Ej C089,j S 2», S - 2 28,, E, Ej C089,, 1 = m+l,m+2,...,n j=m+l¢i 1:1 If 3,,- = 0 along the boundary then the eigenvalues are bounded above by zero, i.e. it has (n-m) negative eigenvalues. To show that the referenced system has the same eigenvalues as the unreferenced system, we first look into the linearized referenced system. Linearizing the referenced swing equation (3.2.2), we have A6 A6 . = 1' 3.5.4 [Am] 96)] ‘ ) where _ 0 I J - [M'IF «5] M =Diag (M 1,M 2,...,Mn) _ A B F ‘ [C D] and, the U” elements of F is expressed in (3.5.5) aft“ " M,- " 36—. =-.E.B,j E, E, C089,, -' M— 2 Bid Ek E, COSGk, l j=l¢t Sd k=m+l i S m (3.5.5a) 74 af‘ 8 E E e M‘ f; B E E 9 —= .. . 'C05 l“ _— k. k 'COS k. 39], U I 1 I Msa km“ 1 J 1 j Sm, j ¢i (3.5.13) 8 . n M m 'Sgi' = — z 8,, E, Ej C089,]- 4' #28,, E, E, C089,, i j=l¢i 5'0 (=1 i 2 m+1 (3.5.5c) af‘BEE 9+M‘MBEE e — = .. . . .. — . . cos . aej l] l 1 C08 1] Msa [E1 ,1 j l ,1 j 2 m+1, j :i (3.3.5d) where n f, = z (3,} E, E] 81119;} "BU E, Ej sine”) j=lzti Mi n m ‘ .— 2 2(Bld £1: E, Sinejz, ‘Bk, E]: El Sine/d) Msa k=m+ll=1 Consider a nonsingular transformation T, such 9 = T8, since 9,- = 5,- - 8,0, we have T=I-[OO e ..... e 4 (3.5.6) where I is a n square matrix, 0 is a zero vector and e is a column vector with all ele- ments being 1. THEOREM 5.1 The two Jacobian matries J 0 and J of the unreferenced and referenced system respectively, are related by a similar transformation, in other words, they have the same eigenvalues. 75 Proof: From equation (3.5.6) we could construct a state transformation matrix between equa- tion(3.5.l) and (3.5.4), thus, we have [slaw-[2.1412,] and therefore, [2] [2; HIM-9p. 3013)] = [g fl'lM-(BFO ’10 ] [T61 791].[g] thus, I = o T I T-1 T M-IEOT-l T (—o) T-1 = o I M’IF -o =T‘JOT" (3.5.7) Equation (3.5.7) shows that those two Jacobian matrix are related by a similarity transformation, therefore, have the same eigenvalues. To prove that stable manifold "m machine u.e.p." is the closure of the stable man- ifolds of "1 machine u.e.p.", we first notice that the equations describing the stability boundary for the I "' machine to lose stability is F n 2 (Bid Ek E, sinef, -Bkl E/c E, sine/d) = O ‘k=l¢l WS(x1) : f0: = O)... (3.5.8a) Adding equation (3.5.8) from [=1 to I=m, we have 76 n m 2 2(3u Eh El Sineiz ‘Bu Ek El sine“) = 0 (3.5.8b) k=l¢ll=l Equations (3.5.8b) could be reduced to equation (3.5.8c) due to sine cancellation. I! ”I 2 2(Bkl £1: E, Sinai, " Bu 51; 51 sine“) = 0 (3.5.8c) k=m+ll=1 The stable manifold for the "m machine u.e.p." is defined as the simultaneous satisfaction of the constraints associated with the stable manifolds of the m "1 machine u.e.p." Noting that stable manifolds can not intersect, it is clear that the stable mani- fold for the "m machine u.e.p." does not belong to the stable manifolds of the "1 machine u.e.p." stable manifolds but is included in their closure. It should also be noted that the hypothesized form of the stable manifold for "m machine u.e.p." is only a portion of the stable manifold of type m u.e.p. The trajec- tories that converge to the hypothesized form of the stable manifold also belong to the stable manifold. Thus, the constraints a),- = M, i = l,2,...,m would not satisfy on the part of stable manifold of the type m u.e.p. that converges to the portion of the stable manifold that includes them. CHAPTER 4 PEAK PREDICTOR DERIVATION AND COMPUTATIONAL ALGORITHM 4.1 INTRODUCTION A fast accurate method of determining retention or loss of stability that has been sought for over forty years. The differential topology characterization of the region of stability raised serious questions on the accuracy of energy function methods. It has been shown that critical energy associated with any type 1 u.e.p. on the boundary of the region of attraction is the only point on the boundary of the region of attraction with that energy value. In the "controlling u.e.p." or TEF "mode of instability methods" the energy at clearing is compared with this critical energy when it is known that the faulted trajectory will not cross the boundary of the region of attraction exactly at the u.e.p. and that the potential energy at the point where the trajectory crosses could be quite different than at the u.e.p. The assumption that the kinetic energy between inertial centers of the stationary and critical group should be used in the TEF and controlling u.e.p. methods is justified since the results in Chapter 3 indicate this kinetic energy will be zero on a portion of stable manifold associated with the u.e.p. and not just at the u.e.p. The justification for using kinetic energy between inertial centers in obtaining the total energy at clear- ing and assuming it was zero on the stable manifold portion of the boundary of the region of attraction could not be justified previously. The results in Chapter 3 also indicate that the total system energy is not a correct energy function for assessing retention or loss of stability on the unstable manifold for a specific u.e.p. The critical energy function V6, for that u.e.p. is the energy of critical machines as they approach or cross the stable manifold. This critical energy function Va, is path dependent. Thus, the error in using the total system potential energy evaluated at the u.e.p. to characterize the total boundary of the region of attraction has two types of errors 77 78 (a) the total system energy function is not the energy function associated with trajec- tories that approach or cross the stable manifold for a specific u.e.p.. The energy function Vc, associated with trajectories that approach specific u.e.p. is path dependent but the total system energy function is not path dependent. The total system energy function is written with respect to a different reference frame and thus the terms in the path dependent Vc, do not appear in the total system energy function. (b) the critical energy Vc, of n different points on the stable manifold for a specific u.e.p. depend on both the path and the point which the trajectory on the unstable manifold crosses the stable manifold. for that specific u.e.p. The most important error is that the wrong energy function is used. The correct critical energy function changes depending on the u.e.p. which is associated with the stable manifold the trajectory crosses when it enters the region of instability. The correct critical energy function could change with the fault, fault clearing time, fault clearing action, and post fault s.e.p., stable equilibrium point, and operation conditions. The fact that each critical energy function is path dependent prevents one from using the critical energy value at the u.e.p. as the critical energy that describes the energy on the portion of the boundary of the region of stability associated with the stable mani- fold of that u.e.p. The consistent accuracy of controlling mode and TEF mode of ins- tability methods using a global system energy function for every possible fault case seems unlikely based on these errors. The accuracy of PEBS methods is also questioned regardless of whether they util- ize the system energy function or individual or group energy functions. It in known that the PEBS does not describe the boundary of the region of attraction. The PEBS can be inside or outside the true boundary of the region of attraction. The individual or group energy functions [17,18] approximate the critical energy function of the unstable manifold for each u.e.p. but do not use the CSG reference and utilize the 79 mechanical power out of the individual or group of machines rather than the energy across the cutset surrounding the individual or group of machines at the post fault stable equilibrium point. This use of cutset flows at the post fault equilibrium to represent the generator mechanical power and the use of the center of stationary group reference (CSG) rather than the COA reference is required if one is to determine a boundary flow description of the stable manifold. The accuracy of PEBS methods is also questioned since the critical energy associ- ated with the trajectory that approaches or crosses a particular stable manifold is path dependent. The global system energy is not path dependent. Thus, one did not need to be concerned about the path and a fault on trajectory could be used to determine the critical energy. The critical energy for the fault on trajectory could then be used to decide retention or loss of stability based on whether the global system energy at the clearing time is less than or greater than this critical energy value. There is a different critical energy function associated with the trajectory that approaches the stable mani- fold of each u.e.p. and these critical energy functions are path dependent. Thus, one can not utilize a global energy function or the value of it on a fault on trajectory to characterize the boundary of the region of attraction and be sure that it would provide accurate results for every fault on every system. The results in the previous chapter provide an exact method of characterizing the boundary of the region of stability without computing a single u.e.p. or calculating a fault on trajectory as required in all energy function methods of assessing retention or loss of stability. Calculating the u.e.p. or calculating the fault on trajectory required using a Taylor series or cosine series for the trajectory. The computation required to sinmlate the fault given a desired clearing time tc or for a fault on trajectory (tc < 0.6) was approximately that of a dc load flow solution. The Taylor or cosine series approximation could be used to compute the global system energy at the clearing time and to select the controlling u.e.p. or "mode of instability" for u.e.p. methods. The 80 assumption made is that the post fault network has no effect on the trajectory direction and thus that the proper controlling u.e.p. whose stable manifold is crossed could be selected at re before the effects of the post fault network has any effect on the trajec- tory. The TEF mode of instability u.e.p. is selected without regard to which u.e.p. is associated with the stable manifold that the actual fault trajectory crosses. It is obvi- ous that an accurate determination of the proper stable manifold crossed by the actual trajectory is important and that this decision could not be made accurately without simulating the trajectory in the post fault network. Simulating the trajectory in the faulted network by one Taylor series and then in the post fault network by Taylor series approximations over each .4 to .6 second time segment in the post fault period would undoubtedly greatly improve the accuracy of any PEBS method and would aid in proper selection of the proper controlling u.e.p. or mode of instability in u.e.p. methods. The computation associated with this approach would be prohibitive not only because it requires simulation of the trajectory which direct methods were intended to avoid but the Taylor series is not a computationally attract simulation method if it needs to be updated. A peak angle predictor will be developed that will predict the peak angle excur- sion based on the fault, fault location, fault clearing time. The peak angle predictor is based on a linearized power system model evaluated at some time after the fault is cleared. This peak angle predictor is accurate and can be computed at the computa- tional costs of load flow simulation. A fast accurate peak angle predictor could be used to improve the controlling u.e.p. or mode of instability methods by allowing one to more accurately determine the controlling u.e.p. by using the peak angle predictor that estimates the peak angle excursion based on the post fault network, prefault s.e.p., post fault s.e.p., fault location, and fault clearing time. The use of the peak angle predictor should also more accurately determine the critical value of potential energy using either the individual or group energy functions 81 or the global system energy function. The use of the peak angle predictor could be used to evaluate any specific potential energy function for each value of clearing time. The critical energy value for any energy function is the value where the potential energy function is maximum as a function of clearing time as was actually investigated in [18]. The results indicated that the retention and loss of stability could be deter- mined with very small error using this method and the individual machine energy function. The peak angle predictor should produce the most accurate results with the least computation cost if used directly with the descriptions of the stable manifolds derived in the previous Chapter. The peak angle predictor will be derived in section 4.4. The algorithm and computation requirements for computing it will be presented in section 4.5. A direct stability assessment method that can act as a fast screening tool for determining retention or loss of stability is becoming of increasing interest in system planning, operation planning and on-line control center applications. In this chapter, a fast, accurate method for determining retention or loss of stability without requiring time simulation for a particular fault, clearing time, clearing action, and operation con- dition is developed. This method is shown to require approximately the computation time of three d.c. load flow solutions and can potentially be computed using distribu- tion factors. The method required development of a linear peak angle predictor based on a linearized power system model and a pulse acceleration model for the fault. The linear predictor is shown to have the accuracy in predicting the peak of the trajectory as an infinite term Taylor or n-l term cosine series. The linear peak angle predictors are shown to be far more accurate than the controlling u.e.p. in estimating the angle and angle differences at the peak of a transient stability simulated trajectory. The justification for these methods is that a particular fault at a particular location and cleared with a particular clearing action excites generally one or at most possibly a 82 few modes of oscillation. These modes of oscillation are or are approximately sinusoidal even though the system is nonlinear. 4.2 LINEARIZED POWER SYSTEM MODEL The linearized power system model can be derived from the nonlinear swing equation of the classical model and a set of algebraic equations for the power flow in the network. It can be shown [12] to have the form x(t)=A x(t)+B u(t) (2.1) where x = [22)] u = [APM] (2.2a) - 51-5,, .1 e = STE," Aé = A6) (2.21:) Ban-1.6:: . A = [in 6] B = [[8,] (22¢) Mfl 0 0 --M;1 0 M51 0 —M;1 M = (2.2d) 0 0 M31 -M;1 .J b The A matrix depends on the synchronizing torque coefficient matrix T that depends on the operating conditions at which the nonlinear transient stability model is linearized. The choice of the operating condition at which transient stability model is linearized will heavily influence the accuracy of the peak angle predictor that will be derived. 4.3 DISTURBAN CE MODEL 83 The input u (t) composed of the deviations in the mechanical power APM on the generators, can be used to model electrical faults, loss of generation contingencies, line switching and loss of load contingencies. The fault contingencies can be modeled by an input u (t) that has the following form. 4.3.1 Pulse Disturbance A pulse of duration T1 and amplitude u, can model the effect of electrical faults where T 1 represents the fault clearing time. F 0 t >Tl up“) = *u,=APM O S t 5 T1 (3.1) 0 I <0 4.3.2 Impulse Disturbance An impulse disturbance can be used to approximate the effects of electrical faults if the clearing time is very short. The impulse disturbance, that will approximate the effects produced by a pulse model of a fault having a very short clearing time, is; u,(t)=APM T18(t) (3.2) where 5(t) is the "Dirac’s delta function". The change of mechanical power, APM , which is equal to the accelerating powers on generators due to a particular fault is computed by the ACCEL program [24], where it calculates the accelerating power on each generator for a fault on a specified bus. The ACCEL program converts the generators into equivalent current source and parallel admittances, calculates voltage at each generator terminal bus for a specified bus fault, and computes generator power output during the fault and the gen- erator accelerating power APM. A linear equation solution is required to compute APM. 84 4.4 A PEAK ANGLE PREDICTOR FOR FAULTS WITH SHORT CLEARING TIME A peak angle predictor has the form 6,}..‘(1’ ’T1) = 9,- (T1-) ‘3‘ (6,32 ‘- 9,31) 4' Aeide' ’TI) 1 = l,2,...,n-1 (4.1a) where 9101—) is the internal bus angle of generator i just before the clearing time T1, 6(T1“) can be computed using a Taylor series approximation MAPMTIZ 2 6(T1") = 931+ (4.1b) APM is the acceleration of the generators due to a fault computed from ACCEL and M is the generator inertial matrix and 9’1 is the prefault stable equilibrium point. 9.32 - 9:” l l is the change in the internal generator bus angles due to the action of the fault clearing action that change the internal generator bus stable equilibrium angle from prefault value 9" l to post fault stable equilibrium internal generator bus angle 6‘ 2. 9(T1+) = 6(T1')+(9‘2 — e“) is the angles just after fault clearing. ABiM(t',T1) is the predicted peak change in the internal bus angle of generator i over the time interval [T 1,:‘] due to the fault acceleration APM that occurred over time interval [0, T1]. The time t’‘ is chosen so that the amplitude of every mode excited by the fault is maximum and the predicted change is maximized based on the fault acceleration APM , fault clearing time and the response of the linearized post fault 85 system. The first two terms are clearly defined and easily computed. The last term A9,?“ (t' ,T 1), the maximum change in angle over the post fault period t>T1 that can occur due to the fault acceleration over [0,T1], is based on an rms peak predictor l A9,M(t*,Tl) = sgn (APM,-APM,, )VZ -I:'I[A0, - A0m12dl‘ 1‘ 0 = sgn(APMi-APMn)\12$x(t*),-i (4.1c) where t' 1' e _ 1 T _ SI 52 5,0 )—7-£x(t)x (:)d: .. [52 S3] (4.1d) is a 2m-2 square symmetric matrix. Note that the sqrt {2} factor is used since it relates the rrns value of any mode to its peak value, The value t' used to evaluate the predictor when the amplitude of every mode is maximum. The factor 1 if APM,-APM,,>0 sgn(APMi-APMn) = {—1 If APMi-APMnTl where upz = (1)0“). The matrix [MT] in A is evaluated for the post fault network at the state at some time t > T1. The matrix 5’90.) used to obtain A6(t"T1) is 7.1+! . Sx,(t')=-:£- iI x(t)x(t)T dt 1 1.1+! . T =_._ 11 cut-T.) 0 O aux-Tod: t ‘ “[12 “[12 where _ O 0 VP - [0 upz “32] A matrix of the form of (4.1d) can be derived if t = m-Z— j = l,2,...,n-1 The matrix S2 = 0, S3 can be approximated by S3 = upz upTz, and 94 x 1 _ _ 51: 2. [[MT] lap, a}, + up2 .4}, [MT] T] (4.17) The derivation of these results is similar to that for an impulse approximation to the fault. The derivation is omitted here. The pulse peak angle predictor is the impulse peak angle predictor if the fault clearing time T1 is small so that eAT‘ = l + ATI. The linear pulse peak angle predictor, where aewofirl) = sgn(APM,-—APM,,)\/2 [S1(T1)],~,-, is actually implemented in the com- puter program developed. 4.4.3 Accuracy of the Peak Angle Predictor The accuracy and computation requirements of the peak angle predictor are now compared with that of cosine and Taylor series approximation that have often been used to approximate the transient trajectory. The accuracy of a Taylor or cosine series approximation to the transient trajectory over a particular interval increases with the number of terms evaluated. The number of terms evaluated over the fault-on period need not be large since the interval is short. A Taylor series is used to approximate the fault on trajectory in [6]. However, use of the Taylor or cosine series over the post fault interval may result in large errors in approximating the trajectory if the time interval required to approach the boundary of the stability region is long. The coefficient of the Taylor or cosine series for the post fault period are usually evaluated based on samples of the fault trajectory at or near the fault clearing time. Each additional term in the Taylor or cosine series requires one (or two) additional independent samples of the trajectory and an additional linear equation solution to evaluate the coefficients of the series. If the Taylor and cosine series are evaluated using the same samples of trajectory, they should both converge to the same trajectory approximation. If all the samples of the post fault trajectory are taken close to the fault clearing time, the infinite Taylor series and n-1 term cosine series approximations should produce a trajectory with the same maximum angle 95 excursions as predicted by the linear peak angle predictor as long as the approximation made to derive the peak angle predictor is accurate ( the inner product of any two different eigenvectors is small compared to the inner product of the eigenvector with itself). Experience indicates this assumption is valid since generally only a few local modes are excited and since they affect different machine or machine pairs. The peak estimated by the linear peak predictor may be larger than observed on the first swing since the linear peak angle predictor estimates the absolute peak obtained when all modes are in phase. Thus, the peak predictor will be conservative relative to the Tay- lor or cosine series trajectory approximations that only approximate the trajectory out to the peak of the first swing. The conservativeness of the peak angle predictor is exactly what one desires. This peak angle predictor would not just determine retention or loss of stability for the first swing peak but rather whether the system would retain or lose stability during the first or any ultimate swing of the trajectory. It should be noted that the infinite Taylor series, n-l terms cosine series, and linear peak angle predictor do not capture the nonlinearity of the power system model if all the samples are taken close to the clearing time. The Taylor and cosine series can be make more accurate by spreading the samples out uniformly through the post fault interval. However, spreading the samples uniformly through out post fault inter- val actually requires simulating the fault trajectory. A numerical analysis evaluation of perforrrring a Taylor series approximation updated at uniform intervals during the post fault interval requires more computation than a trapezoidal integration of the differential equation over this interval. The linear pulse predictor have significant theoretical advantage over Taylor or cosine series due to accuracy as well as the small computation (three d.c. load flow solutions) required. This linear peak angle predictor has the accuracy and computational speed to be implemented on-line in an EMS con- trol center rather than off-line as is contemplated with the transient energy function. The linear pulse peak angle predictor estimates the peak based on the fault 96 acceleration, fault clearing time, fault clearing action, prefault load flow and post fault clearing time dynamics. The pulse peak angle predictor is also a function of the initial acceleration and a linear approximation of the post fault unstable equilibrium [MT TIM APM . The nonlinear peak predictor [25] replaces the linearized approxima- tion to the u.e.p. by the actual controlling u.e.p. that is the best approximation to peak angle excursion observed in the trajectory. 4.5 COMPUTATIONAL ALGORITHM From equation (4.12). The computational algorithm for evaluating the peak angle predictor can be summarized in the following order. (1) Compute APM using ACCEL program that requires one linear equation solution (2) Compute 9‘2 using a d.c. load flow and compute 9(T1‘) and u)(Tf ) using Taylor series approximations (4.16). (3) Compute [MT J'Iupz from the following d.c. load flow model M6 = J11112, APL 121122 where the post fault network load flow Jacobian is linearized around [65(t),6[(t)] A65 ' A9,, (4.17) for t 2 T1, where 60 = (61.92,...,6,,,_1) is the generator internal bus angles and OL = (9m+l,6m+2,...,9,,,+,,) are the load bus angles. [MT ]'lup2 is computed by multiplying both sides of the equation by M O O [(n-m) letting M APG = “p2: (0(Tf' ), and APL =0 and then solving equation for ABC = [MTJ‘lupz where MT = Mun-11252112,) (4) Compute [MTrl up2 upTz, $10") from (4.17), and A6(t"T1) from (4.1c). (5) Compute the peak angle predictor from 4.1a 97 The peak angle predictor requires a linear equation solution to compute APM , 9‘2, and [MT ]'1up 2. These calculations are very small compared with the computation required to reducing the network to internal generator buses, compute the post fault stable equilibrium point, determining the mode of instability, and computing the u.e.p. required by the TEF method. The computation associated with the three linear equa- tion solutions could be further reduced using the matrix inversion lemma. The compu- tation required to compute the peak angle predictor would then be comparable to that of determining the effects of three line outage contingencies using distribution factors. The peak angle predictor is a fast method that could determine the peak angle excur- sion for a fault at relatively the same level of computation required for load flow solu- tions. Thus, dynamic security assessment for fault contingencies would be feasible. No other computational method for determining the stability requires so little computa- tion to a fault trajectory to the boundary of the region of stability. It should be noted that a careful error analysis was performed to establish the source of errors and their magnitude in the peak angle predictor. (1) APM computed by ACCEL is an accurate representation Pm(t) - P, (t) computed by the Philadelphia Electric Transient Stability Program. (2) 6‘2, 9(T1‘) and 03(Tf ) computed by Taylor series approximations (4.13) and (4.14) are very accurate representations of the angles and speed at clearing time T1. This errors in the Taylor series approximations increased with T1 but were never large. (3) the computation of [MT ]“up 2, 6‘ 1 and 6‘2 are based on an unaggregated network model with constant power load models, There is significant error between that model and (a) that used by the Philadelphia Electrical Stability program and (b) that used in deriving the boundary of the region of stability in Chapter 2 and 3. This error could be addressed in future research. CHAPTER 5 SIMULATION RESULTS OF THE DIRECT METHOD 5.1 INTRODUCTION The primary purpose of this section is to develop and test an algorithm for deter- rrrining retention or loss of stability based on the description of the boundary of the region of stability given by BA (x3) = U W‘(x,-) x,- e E naA the description of the type 1 stable manifold for the case where generator i belongs to the accelerated group n 2 Bh' Ek Ei sine}; -Bki Ek E; sine” _ k=l¢i W3 (xi) - mi = 0):“ and the peak angle predictor. The algorithm is (a) compute the peak angle predictor for some given clearing time T 1. (b) determine the accelerated group of machines j e J such that GjM(t',T1)>9O° (c) for each j 6 J test whether the peak angle excursion is on the stable side or unstable side of its acceleration manifold n kzlat'Bkj El: E] sinesz -Bkj Ek Ej Sinekj = 0 = I The generator j is on the stable side of the acceleration manifold if n 2 Bkj Ek Ej sinesz -Bkj Ek Ej sinGkI-(t*,T1) < O k=l¢j and is on the unstable side of the acceleration manifold if n kzl Bkj Ek Ej Sine/:12 -Bkj Ek Ej sinGkJ-(t‘Jl) > O = at} ' 09 99 (d) the system is stable for clearing time T, if all generators belonging to the acceleration group are on the stable side of their acceleration manifold for the peak angle predictor evaluated at clearing time T1. If one or more of the genera- tors jeJ are on the unstable side of their acceleration manifolds at the peak angle prediction for clearing time T 1 , the system is unstable at clearing time T1. (e) if the critical clearing time is to be determined, steps 1-3 are repeated for different clearing time values until the largest T1 is determined for which the sys- tem is stable for that fault and fault clearing action. The accuracy of the algorithm does not depend on the accuracy of the description of the boundary of the region of stability since the the description has no error for the given model and the assumptions A1 - A5 . The accuracy of the peak angle predictor thus is tested. A second purpose of this section is to investigate the accuracy of the peak angle predictor. Since the peak angle predictor depends on linearized power system equa- tions to obtain Sx(t' ), A60" ,T1) and thus predictor 9(t' ,Tl), the accuracy of the peak angle predictor will depend on the point on the fault trajectory where the linearization is carried out. If the linearization is carried out close to 9‘ 1, the network will be stiffer than at some point further along the faulted trajectory. Thus, it would be expected that the peak excursion will be larger when the linearization is carried out when the trajectory is closer to the boundary of the region of attraction. It is hoped that the accuracy of the linearization performed at the fault clearing time will be accu- rate for most fault cases. If linearization at the fault clearing time is not always accu- rate, the ultimate algorithm would need to automatically select the point along the tra- jectory at which linearization should be performed. No effort is made in this thesis to develop an automatic procedure for selecting the point along the trajectory where the linearization is to be performed. The objective of the results presented is to show that the linearization at the fault clearing time generally provides an accurate assessment of 100 critical clearing time for most faults. If linearization does not allow the algorithm to obtain a sufficiently accurate assessment of critical clearing time compared to that obtained from a Philadelphia Electric Transient Stability Program, effort is made to obtain a more accurate assessment by moving the point at which linearization occurs closer to the boundary of the region of stability. The results indicate that the algo- rithm for predicting retention or loss of stability can assess the critical clearing time as accurately as desired if the point at which linearization is performed is properly chosen. The adaptive selection of the point at which linearization occurs is a subject for future research. To investigate the accuracy of the peak predictor as a function of the point where linearization is carried out, the post fault Jacobian (4.18) is evaluated at four different state conditions - prefault, post fault, fault clearing state and a point between fault clearing and initial peak angle prediction. This Jacobian is used to compute [MT ]"lup2 in S 1 that is used to compute A90. ,T1) and ultimately 6(t' ,Tl). The pre- fault condition is a load flow solution from base case which is shown in Table 5.1(a), and 5.1(b), the post fault condition is obtained by a contingency case load flow solu- tion where a line is removed to clear the fault and for the specific fault contingency studied. The load flow datum are presented in Appendix B. The fault clearing state 9(Ti') and 03(Tf ) is the state at which the clearing action is applied and is computed by the Taylor series approximation given in (4.16). The angle 6(Tf‘) is computed from (4.14). The state between fault clearing state 6(Tf ) and initial peak angle pred- iction 9(t‘,rl) is obtained by multipling fault clearing angle em“) and initial peak angle prediction 6(t’,T1) by weighting factors A and B respectively. The formula used in this approach is to evaluate the linearized Jacobian matrix at angle 92-- 2“ 25 24 1 --—- 27 O 15 a. 4-- -UJ—J -—- 21 3 --l—‘- 15--—- .....?.Z ‘ ‘T 5E5 101 29 [- 14 -r——— 35.. ,__£'—— 11- 10 26 19-— 12 33 M m MA 20 - I3 - 34 Figure 5.1.1 - The 39 bus New England System 102 FNM'IO'DNOO 000000000000000000000000000000000000000 0000 0000 0000. 0000 0000. 0000 0000. 0000 0000 0000. 0000 0000 0000. 0000. 0000 0000. 0000 0000 0000. 0000 0000. 0000. 0000. 0000 0000. 0000 0000, 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 000000000000000000000000000000000000000 000000000000000000000000000000000000000 00000000000000000000000000666660000000o 00.00~- 00 0-- 00 00w. 00 00w. 00 000- 00.000. 00 000. 00 000- 0 0 000000000000000000000000000000 2 00.000 00.000 00.000 00 00v 00.00~ 00 000 00.000 00_000 00.000 0 c 00000000000000000000000000000 A53. 2:: .438 932. 533? ._n:w:3 >52 00 - 0000. 00~0 00~0. 0000. 0000. 0..0. 0000. 0000. 0000. 0010. 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000, 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 00000000000000000000000000000P000P"PP"0 0.000) 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00.00. 00000 00000000000000000000000000006 .v.~ .00 .0. .00. .0- .00. .v0 .~0. .00. .00. 00000000000000000000000000000 .000. .000 .0v0 000 .000 .000 .000 .000 .0~0 .00~ 00.0 0.. 0 0 000000000. 0 0 '09) 000 NNN 00.00. 00.v- 00.000 00 5'0 00.vs~ 00.000 00.00. 00.0«0 00.0~0 0 0 000000 a" -. — .mw nu_«_.w.—. 0 0 PMQNDONOO Pv-v-Pv-v-PP pp P 0 0 wtw0w~0000vv'~~0~—o~00000§0060 0000 0000 00~0 0000 0000 0..0 0000 0000. 0~00. 00.0. 0.~0, v0~0. 0000. 00.0. 00~0. 0500. 0.~0. 0-0. 1000. 0000. 0000. 0.00. 0000. 0000. 00~0. 0.00. 0.00. 0000. 0000. .000. 0000. 0000. 05.0. 0000. 0000. 0000. .~00. 00.0. N000. PP000000F0000000000000v-Ov-v-v-v-v-v-oco—u-v-c-v-v- O. 00000000000000000000000 )- .- OOOOOONNNNNNNNNM 000000000000000000000000000000000000000 pp'-"'FP-"“PF9'Fpr'FF-PFpFP-'PPPPPPF 03¢: 0:0 0. FNQ'OONQ. 2 103 0&H60u00hu00 0000 0000 0000 0000. 0000. 0000 0000 0000. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000 0000. 0CH30CH30CH3OCHDOCHDOCHDOCHDOCH30CH30CH30CH30CH30CMDOCHDOCHDO 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000. 0000 0000 0000 0000 0000 0000 0000 0000 0000. 0000. 0000. 0000. 0000 0000 0000 0000. 0000 0000 0000. 0000. 0000. 0000, 0000. 0000. 0000 0000. 0000. 0000 0000. 0000. 0000 0000. 0000 0000 0000. 0C“:0CHO0CNDOCH30€HDOCH30CH30CHDOCH30CH30CHDOCH30CH30CHDOCHDO 00000 00000 00000 00000 00000 00000. 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000. 00000 00000 00000. 00000 00000 00000. 00000. .00000. .00000. .00000. .00000. .00000. .00000 .00000. 00000. .00000. .00000. .00000. .00000. .00000. .00000. .00000. 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 .00000. 00000. 00000. .00000 00000 00000 00000 00000 00000 00000 00000. 00000. .00000 00000. .00000 .00000. 00000 .00000. 00000. 00000. .00000. 00000. .00000. 00000 00000. 00000 .00000. 00000. 00000 00000 00000 00000000°0000000000000000000000000000000000000 A33. 2:: on? was: 53.9? __n__w:.w._ 252 00 - 0..—.0 93:. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000. 0000 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000 0000. 0000. 0000. 0000. 0000. 0000 0000. 0CHDO¢H30¢H30CHDOCH30CH30CH50CH30CH30€H30CH30€H30€MO0CH50CH30 0CH00€H30C”:0C“:0CHO0CH=0€H30CH30¢H30€HDOCH30CH30CHDOCH30CH30 '00€HDOCH30CHWO0CHO0CH30€HWO0CH30CHDO€H30€H=0CH30CH=0€H30CH30 ‘00CHSOCHSOC“:0CH30CHDOCH30CHDOCH30€H30CH30CH300CH30€H30CH300 00CHDO€H300CH30CHD00€H30CH30CH3000H300CH300CH000CH300CH300 0CH30CH30CH30CH30CH30CH30CH30CHDOCH30€H30CH30CH30CHDOCHDOCHDO 000000. 000000. 000000. 000005. 000000 000000. 0000.0. 000000. 000.00. 000000. 00000.. 000000. 000000. 000000. 000000. 000.00. 000.0. 000000. 000000. 000000. 00000.. 000.5.. 000000. 00005.. 000000. 000000. 000000. 000050. 000050. 000000. 000000. 000050. 000000. 00000.. 0000... 00050.. 000000. 00000.. 00000.. 0000.0. 000.00. 000000. 00000.. 000500. 000005. 005000. 0‘0 0CH30CH30CHD0¢H30€H30FW30CH30CHD0€HD0CH§0CHD0CN30CH30CHD0'ND0 000.0. 0.0.0. 00000. 00500. 050.0 00000. 00000 00500. 00000. 000.0. 00000. 000.0. 000.0 000.0. 000.0. 005.0 00000 00000. 000.0. 000.0. 00000. 00000. 05.00. 0.0.0. 00000. 00000. 00000. 00000. 00000. 00000. 00000. 00000. 00000. 00000. 00000. 00..0 00000. 000.0. 000.0. 000.0. 00.00. 0.0.0. 00000. 0.0.0. 00000. 0..00. 0000000000000000000000000000000000000000000000 00000, 00.00. 05000. 00000. 00.00. 00000 00000. 00000 00000. 00000 00000. 00000. 00000. 05000 05000. 00.00 05000. 00000. 00000. 00.00. 05000. 00000. 00.00. 00000. 00.00 00.00 00000. 00000. 00000. 00.00. 00000. 00000. 00000. 05000. 00000. 00000. 00000. 00000. 00000. 0..00. 00.00. 00000. 00500. 00.00. 00.00. 00000. 0CH30CH30€HDOCH30CHDO 0 indicates the peak angle predictor is on the unstable side of the acceleration manifold for generator i if 9,- (Tf ) > 90°. Nine cases are summaried in this section, 5.2.1 F0‘7-06"I case A three phase fault is applied at bus 6, and is removed by clearing line connect- ing buses 6 and 7. The computational result, given in Table 5.2(a), show that the acceleration group are generator 2 and 3 (load flow bus no. 40 through 49 corresponds to terminal bus for generator 1 through 10) since only these two generators have angles greater than 90° at the fault clearing time. Note that at a Tl==0.26 fault clearing time the accelerating power is negative for both machine. At a T 1:0.27 fault clearing time, the accelerating power on generator 2 changes from negative (stable) to positive ( unstable) indicating the acceleration manifold for generator 2 is crossed. 106 000. 000 000. 000. 000. 000. 000. 000 000. 000. 000. 000. 3.25.2... 093 50-00.... .5. 05.30 3.5.03.3:50 - :0...0 93:... .0.. .... 00 00 00 00 00 00 00 00 00 00 00 00 00..0.. 000.0.. 000.0 000.0 050.50 005.00 000.0.. 000.0.. 0.0. 0.0. 000. 000. 550. 000. 000. 000. 000 0.0. 00 00 00 00 00 00 00 00 .0 0 000.0. 050.0. 00: 00 000 .00.0 000.0 002 00 000 005.0 000.0 002 50 000 050.0 .00.0 00: 00 000 000.0 000.0 001 00 0:0 00..0 000.0 00: 00 020 050.0 .00.0 04: 00 000 000.0 .00.0 001 00 000 000.0 005.0 00: .0 000 50..0 000.0 00: 00 000 005.00 50 005.5.. 00 00..0.. .00.00. 00 000.05. .0 000.50 0000000 2. 0.0.0 00.0.0000 00..0 50 000.0 00 000.0 000.0 00 000.0 .0 05..0 0000040 .40 00.00 20.000000 000.00 50 000.00 00 055..0 .00.00 00 000.00. .0 500.00 050.0 I 500.0. .0 .000004020 000.0- 050.0. 00: 00 000 055.0 000.0 00: 00 000 000.0 000.0 00: 50 000 000.0 .00.0 00: 00 000 000.0 000.0 00: 00 000 000.0 000.0 00: 00 000 05..0 .00.0 002 00 000 005.0 .00.0 00: 00 000 505.0 005.0 00: .0 000 000.0 000.0 00: 00 000 00..00 50 00..0.. 00 ..0.... 000.00. 00 000.05. .0 000.00 0000000 2. 0.0.0 00.0.0000 05..0 50 000 0 00 000.0 000.0 00 000.0 .0 00..0 0000040 .40 00.00 20.000000 000.00 50 .00.00 00 000.00 050.00 00 000.00. .0 000.50 000.0 I 000.0.. .0 .000004020 00 00 00 00 00 00 00 00 00 00 00 00 107 000. 050 000. 000. 000. 000 000. 0.0 000. 000. 000. 000. 3.25.80. 343 50-00... 5.. .2050 .0.. .00 00 00 00 00 00 00 00 00 00 00 00 00 5.0.... .00.0 .00.0 005.00 050.00 00..00. 050.0.. 000.0 000.0 050.00 000.00 .00. 000. 000. 000. .00. 000. 00. 00.. 0... 500. 00 00 00 00 00 00 000. 050. 000. 5.0. 00.. 000. 000. 000. 50.. 000. 00 00 00 00 00 00 cu0<30v--c:0v-0 CVOCDOv-PCDOv-Q p p 000. 005. 000. 0.0. 000. 000. 00.. .00. 0.0. 00.. .00.00 000.00. 00..0 000.0 000.00 050.00 000. 000. 0.0. 000. 0.0. 000. 000. 000. 500. 0.0. 000..0 00x .0. 00..0 000.0 000.00 050.00 3.5.3.3350 - 40....“ v.23... OG’OGDGMDO‘IQ‘I OIDCNDGHOISQIDV 050. 000. 000. .00. 000. 000. .00. .00. 005. 000. p hiOflDOflflOIDOCDO 50 000.0.. 00 .00.05. 00: 002 002 00: 001 001 000 002 000 000 00 .0 000 000 000 000 000 000 000 000 000 000 000.0.. 500.00 0000000 2. 0.0.0 00.0.0000 50 000.0 00 000.0 00 .0 000.0 05..0 0000040 .40 00.00 20.0000x0 50 .00.00 00 000.00. 00 .0 000.00 000.50 000.0 I 500.9. .0 .000004020 050.0. 000. 000. .00. 000. 000. .00. .00. 005. 000. 50 000.... 00 000.50. (MOHBOMDOMSOC’ 000 002 000 002 000 002 002 00: 000 00: 00 .0 00 00 50 00 00 00 00 00 .0 00 000 000 000 000 000 000 000 000 000 000 00..00. .00.00 0000000 2. 0.0.0 00.0.0000 50 500.0 00 000.0 00 .0 000.0 05..0 0000040 .40 00.00 00.0000x0 50 500.00 00 000.50 00 .0 .50.50 05..00 000.0 I 500.0. .0 .000004020 00 00 00 00 00 00 00 00 00 00 00 00 108 n.6, .oc« can 000. man '00, CON n—a. 'pN ova, .No— Ops cco. can. can. can. own. o—o. Caz—uh. i; .433 2.59.. .5.— ...:_..:c _:._:::...:_:.33 - .54.... :33.— av '1 0' 1' av Q' 01 '1 0' Q' 0' 1' 0v '9 0' CV v—N.0Op NhN.——— nvn.o non.o moo cm was av p< Ow—<:a(>w coo.~o QON..0 N00 090 N51 Qua pNQ v—o no— 516. NC. can. 0' MC 0' NV 0' a! CV 0' NOOGNPOO’M Qh' N00. MON 00' '00 can. a. DON Nvo you a. o~o.o. m¢z a. man «on o can a m¢z a. man o~a . oo._n m<= s. was “No v .oo o m¢z a. man o~o n can o m¢z av mam one.» one a mm €00.0a unn.ov 00¢. NNv. own. 0pm. NNN. ova. n—o. can. QON. saw. 0' n' 0' n1 0' av 0' a. cuOGDO'-'¢m .. 842:3. n.a. .. on..=z.. a.. 4.2:. .p.=.-u4_u.c-h_2:. zaeo o a. ...n.a.a..a...\.:..~.a.+.~.a...~...p.~.pz.onr.o..a.¢~r.... «steps «.0. ~.. ..aao.1m:h..oan.xuz>..on...xup .43. «n 3 Zea—4.3.... 5... 3 . . . .3 . .. . x. o.i...5.5..e.5.5.4.3....youuuuucoauuuuuuuuuuuouuuuuuunvuuuuuu no 0360:: 32.31.. . 30 .92 #42. 3mm .30.: . Sam .00 . 3mm .0.. 45.... .n on can no cameos—a .OAn..aa..on..uao:<..enn.>.:oam..on..p::n..on..e.uma 4 we. 44aho¢ ue=.ua...n.p. .. o.nao3_a o a. oaa.oz.. .\..<.. - azure; u.:boaau s:.n~..n_r. a. canoes—a racks—x. . O .. o.o.oz.. .\..¢.. - u.x»¢: z<.aoo huaahmzoo o v- .maao.a.=oam won v 09 on can ...222p-...2299m2 no. a 32.222 op. 2.2-...2229 .2222 .22. .. 2. can .-pna2w up. m:nn¢t..-. an on. map p. oh :3 .. .oa. hma22. .. p~. ..... «2: 2o 922.2 222p.u us:pn-----u «e. um.» .«p .z.:=...::.<2 n:=2.:2.2n.. 242.29. 222. >.:a.2. um..=. ...:oa2-...>.:ouaa...>.=ou2 opp omv~o2.. u o.n .=...2n .z._ .::_.2_ .22.. 22.22... 9.. p2 u.02<.....2:....2¢2-...2¢2 mp. coopoz.auouuuuuuuououuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu no. ...2.< :. .2; 2:.n2:.x. 2.222 xae<2.2;u. ~x2.:ou2u....2: vpn 322.9230 o.~ can m:22<¢...m=22u. on app so 22. pop -----.. . ._ . . --.- -.---------------------------u----u ....... --------nu «pm .....an.....2.uwa.... ...p. we.22 ncm go. o2.u2usa..a2 224» u .pn ea. or so mop ............... - - ---------------------------------------uu--------u opp .aza.hxazwozu can .n:=x.:Oa2. .. .nno~.o.. up.23 «on no. a? on .o .32. .azu.exa2. 2. oo~ nap 2<2409 .. .... 29.22 22p .. :22 «on o...=2.. u::.hxou c~. «on cow :9 so .0n 2<..49....2.2.9=.~.\.pn...:.uu<2u.m=aa.>.:oam . «on ma4u can .2.m:22.m:¢2 no. .azu.p=2...azu.hata.s.. ...p. uh.22 oop ax. o. 3: .o .ou. .2.n:ax. 2. .2. 222a .maa2au .bu. a. .. oa~ o.a.:2.. 22oz..-t ca. on non .ozm.2<.aa oo. po~ m::.hzou no. ~on ...hm..naza cow ....uu~...2<2 .2. nap Oh so .o .o.. ...pm... .. nap a.o-...u¢oan2 co. m:ou¢2.m:nzuu4. no cap o...=2.. o.o-...>.:ou. on. ..... 22o. 22:.ou 2. 2.292: 2<.aau2ou .429 u p.~ c-n.oz.. ~n~o~ moan .2au2 .pu. 2.2. «v. a..no2.. .ha2..2..u2.aanx¢2.2zou2.cao2o2 . op~ o.n~=2_. .n~o~ moan .c .24. 2... .vn conno2.. .ap2pn2.242.9222.~m...halu...uaa.u.u2<.>.2022.uu=amau 2.20 app o.n~o2_. .. or so .224222 .22. u<..2... eon out—.22 vpn annp22.. .n.o..2~.o...2.h¢2ao. n~o~ op— :a2.¢aoo a. . ouuaaus 2o-h.:<. 2 ~22» an. 2<2azupm<. n.~ =.r~:2.. 2anu2. .u...:o...ua¢an2 .pp Enact... . 3.... 2.... .9: . ... . . . .,.............J........._..........UUUUU...uUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU van uuuuuuuuuuuuuuuuuu ctuna- uuuuuuuu an- nnnnnnn un-nnc.. ccccc U 2.“ o~n~az.. 0 man ..ao.. .a... use. aou o.npo2.. ._ p.22 ”22292 a2.p¢2u.aoo¢ 2422 u .p. .maa aahaa22 2. .22. ..... 0...: a.” o¢n~=2.. u use a... . . .02. .n..c..ac. .. sen 30.2.22... . 5.5.... . .. .h. v. . ... . . ...M5.,5......5..you”...UUUUUUUUUUUUUUUUUUUUUUUUUUUOUUUUUUUU «an 338.325 0. . own .. 22¢ .np 2o.h¢2.2..a raoap.2 mamas 4422292. u>2uma2. .1..2.n=az.u+¢sm2 nop o 8...... .2223... 8.8. S 8.... a. mwm m 88 238.022 8.8. : 8... a. mam 123 3-n32.. 3-34.32 .3 .33. 2.332... 333 3.~n32.. z:_.33n n. 3333 3 «3. 3»..32.. .33333 33343.2..323 333 33.3.3333 33 .33: 3333 3 ... 3...:2.. .33343.2..3 33333 33 333232 32. m. 2.332 3 33. 3...32.. .3..42_2... 33.3 33333 33. 33.32 3. 33333 32. .3 93.. 4 3. 333332 3 3.. 3a..32.. .a.34..3> 333 323 333 uh3h.hwnam 3343 Oh 32.h3383=a 3 on. 33..32.. ..34..2.333.n333.22333.333332 . ... 333.32.. ...<932.34.....2..n...33:».3333..3333.m33..333333 32.r333333 on. 3.3.32.3 3 as. 323 ... 3333 3.. as. a. 35 33 «n. 3.. 3» 33 ...3 .33. 343339. 3. ... ..333.-33h. 3.. 33.33.343.3ha24333h ... .333.34:.3.m3334z.333333. ......4rn 3443 3.. 33. -- 1- - - .......................... u ................... 3 so. 3. 323 3.. 3.33:2.332..3333..24332.343.pu32.um.3 ...33. 3433 an. 33 33.233 .3. 33.3 no. 3 333.. 3.3.3.33... 8.8. : 38. 3.3.3 393332.332..3.33..24233.24..3333.pm.. ...3.. 3433 3.. 232. .3 .33. .3333. .. ... .334332.333. 3.43 3.. ............................................................................... 3 an. .32 33 3.34.3.3. 23.3.3 3. 333 33 32.333333. .3....3433. 3343 3 33. --- ............................................... u ............................ 3 so. .....3.3.n..n..xx..n332333 2. 3.3.3 33p3.333...3.s.\. .4233. n.3~ .n. .333342...3332\......x33.....342... .naom... 39.33 on. ----z3332333 3. .3.3.3 3333.333. 333 92.33 ..... 3 .n. 33 323 no. ...342-...3243. an. 3. 323 .n. .333.+2an....34t-...342 .3332 .33. .. 3. on. 33.3 3.. .333-.hzow-...2484...x4: .3.. 23:. .3 .3.. m3. .324. «322 .33. .. 3. 3.. ...33333.....3.324-333 3.. 333442.... 33 3.. 3.. ----n .33324 333. 333 33.333233 33. 23.3333 ..... 3 ... ------ ......................................................................... 3 ... 333m-.33.4332.m.33324-...33324 3 3.. unuuc .23_34.32.m 33.3.34hn 2333 3.324 2. 3433 3 ... ----- .33.3.3333 3433.3232 33. 23..33m 3 3.. ............................................................................... 3 an. ... 3. 33 3.. .....3.3.n..n......3334333 ... 333.4 23.n33333..3.a.\. .4333. .n.. 3.. .333342...m332-.....33324... ..n~.... 33.33 3.. 3.3.33.3 .3343.2..32.333342.22332.333332 . mm. 33nn33.. .334332.343.9332.33.3.3323.3333:.33324.>.3333.333343 3.43 ... 3.3.33.3 3 an. 33.n33.3 ..34..2..33.n33342.22332.333332 . «n. 3...33.. .334.32.34...332..n...3323.33333.33334.>.3333.333333 ..43 ... 3.3.33.3 3 3.. 3.232... 2.34.. .2333 £33.42 .22332 .333332 . o3. 33nn33.3 .334332.343.px33.33....333.3333:.3.324.>.3333.2.2..3 3.43 33. 3.2.332 p~.. 33 323 .3. 33342.2..3 33 39 33333 3423332. 333. 3....334332 33. «33342...3332-_.x. ... .3. ------ .......................... - ................ ------ ........................ 3 n3. 33.333333 3.324 3.33233 3. 32 3332333333 33: 3 3.. --n-- .......................................................................... 3 .3. .2332.bm4..333342.3332.32.23.. 33332 ..43 on. 24332-3343 ... ............................................................................... 3 3.. .3334.-3333. 9: 3332333333 3933233 33 32.9333333 333+: 33 33 3 3.. ..... ----------------u----------------------------------------------------n----3 3.. 3. 323 3.. 3. 323 ... 32.34333 .333 34 .3343. .3.4.3.32433..33...3n.»33342.333333. 3.343 .343 ... 33.3 3.. 32.34333 .933 p4 .3343. .3.4.3.34t..33...33.333342.n33233. 33343 .343 ... 2333 .3 .33. 33... 3. 3.. 334332.332..3333».34332.343.3333.p3.3 ...33. 3433 33. «a 32.333 33. 33.3 .3. 33..33.. 3.4.33.3323.33333.34333.343.3333.~3.3 ...3.. 3433 .3. 3...33.3 3. 32.333 33. 333. .3 .33. .3333. 3. .3. .334332. 333. .343 3.. 33. ..---- ......................................... - ............. 3 «3. .3333432 333332. 3.33333 35 34.33343 23-33343 333 3 .3. .3333.3333 3433.. 33. 23.3333 3 33. ........................................... ------------------------------------3 3.. a- ................. - ......................... ----- ............................. 3 33. 33 323 33. 3 83.. 3.3.3.3333 8.8. 2 38. 3. wow 124 39°23 3.22.25.22.22. .... a: 3.3.... .3522 0.232.. .§.hu.....§ on. 2.232... “32:23... on 22...... 42.32.21... 2!.- u on. 3.32... no a... on 32...... ...): :2... ._ ...... :32. a... .2 £223.. 3...... ..o 2282 2...... . 322.53 u ..3 9332... .32»..S:.2.2..2u 33.2.... n: o... 3 2 .o... 92.2.... '3 3232... an a... on 3.0....32275227: a. 31:2... ......22 2...... .52 2.2.. u as 3.32... .3zu.t¢.»-.....ou» 3:32... 2 :2. ... 23 2.2.2. _. 23:. ...... a... ...... 95.2.2... 52...... 2 .....— ....2 u on. 2.2.2... .....za...a!-...:2o.: 2.3.... 7.22323252 .3 ...-32.. 12:22-32. :52. .3. 222...... .352... 32:28 2: ,2. 3.32... 7.233.722... 9.30:... 332:3”. as :o 60:63... ~\:.8:zu. -4535! 33...... 7.382.228: 332.2282 8. 3.32.. 2.82.2.3: 3.332.. 7.532.223 .2. 2302.22.52 a: 2.32.... 232.92.... 5 .532... 2. 2. 3 : .3. 92.2.... a: 3.32... 2.5.5.. 23...... .39..-.32222. ..3 3.32... 3.52232... 3...... .... 95.3223 ...:33. 32 33...... 322.332....2. .3 3.2.2... $92.32.}: 33:23 ......cz....m......o2u:~uz n3 92.3.... .v 2 3 53.3.2... 3.82... 3...! .82.}... .3 02.32... .... 2.32.. 32......53232. no. 232.... v ...— ou ..-.ou..2ua=2.u.—<..n2.... 3.2.2... .....az.._.2...:.2u.=...2 «3 02.32... 32.22.23. 2...... 9... 2.32 ... .52.... u 23...... ....-.23322.» :2 0332... .32238722232 3 32...... 2». 232.75.: 03 3.32.. .22..az....m.....2.2.. 2.32: 7.9.2-3.... .3 2.32.. ..-. 2.32... 7.223.233-32... on. 3.32.. .a....<2.:.... u 23...... 52.7.2922. ..3 2332... tau: .52 23... 20...... 23:2 3... 3.52228 3.32 ..o ......2. 2 .95.. U 3205:... gal—FIDO «Na 0n. edema—.— 4233: .5 23:22—11; 808: u:.—. 5.5.1.: U 33...... 222.5. «3 e332... dun... 3.22.23.22 u 3.32... 3... n. 5.52.2.5 3.52 . .3 9332.. o... «23:822.... .... 2.952 222.2... ...... 2...... 2.62 ...... n. 2.25.. o 2332...... 22.222 .... a... 22......3... n. 23.2.23 20.22. as... 7:53.... .2 n: 3.22.... 3.422.225.3230: .. .0“. 9322.... 3:32... 2...”... 22.33.22: «3 8.2:...— xuacz..t...uz.2u:2:z S .0.... 93.3.... 3232... a: 9. 8 ...2 ...... .33.... .3 9332... ......52- 23.2 8.3.... 753.52.. a. 23 3.32.. .32....on 3 23:2... ......222... a... :22... 423...! 2...: 2 0:4 “iguana... u an: 232.... o: o... 8 3.3.2223... 23...... .3.... ._ : 32 .23: 5...... 322......23 .22.! ....» ... 9.2.. 228 u .3 33923 .3223...» 2.... 2...... 3.52 32.3.... .... t 29.5 o 2.32... ... o... 8 .... 3.392... .52....28 ... 3302... .2. .3575... .3223» :2 332.... 332.2332 2.3.... 27.323.217.3255» «3 9332... 22332232232 3.32... 732.732.. .32... .3. 322.... .3 232.... 3 0.. 8 .§.hu...§2.2282..: 2.3.... T .233; 32.. :2 9332.. 3 a... 8 3324.33.22.52... 3.32.. 22.322. .....E! 2. «3 3.32.... .5»:.:...... 2...... m5. .... ... .5 :22... as o.— n. woo: ... 20...... u 3232.. .52... 2:...2... ...... 522.5 .2953... .2.2.... 35.2.... tau... 42.5! ”...... u .3 3.32... 3 2. cu .o.ou......o..2:.w.i... 33...... no a... 8 2. .3232... «32.7.33 3 o: 2.32... 2.2-2.52.7.3 a: 3.32.... 0.22.52 23...... n. o... 8 .o.ou...iu.=a2.... a: 332.... 82.5.2: 23...... .... o. 3 .....aoz.ou.:n.a.2<....: a. .... 3.32... .c a... 3 2 .3. 92.2.... 8.32... 2282.53.32... .3 3232.. ”.23 .32.... 32.2.5.5 ... .....m ... 23...... .3222 ...... ...:32 2.4.2.... 2.2.2 .32.... 22”.... 42:52 2 .. 29...... o m; 3.32... #2252228 .3 2......32 .55.. ......— .....2 mac: ...: ...... o 2532... !»\.2..22-.:.» I. 2232... .32....23 on 23...... 3:37:23... 2. ...-32... 2202922 u...— 2. 3.22.2... .... 3 .32 .55. .5... as... .53 u 33...... .....52 2.2 ...aoz 2.22.52 322......32 .23.! 292:“. u «S 8.32... ”32:28 ...n 3932.. sat—.58 3 :9 0.32.... an 0.. 8 339:. on o.— 8 a: 3.32... .322......22..32u 23...... .0....2..2.T..u32.....um..:32...dn~ .3 2.3.2... 2 2. 3 .o.ou.....2u....xuz.... 23.2.... .232 .3 8.2.5.3. ...un 3.2.6 u .3 3:92... 1.333223213832282 ... 23...... I 2 3 .32... .22. 33...... S. 32.2... 2232.53.32”. 98.32... :..:z<=. .... .... m. 322......22 .....E! .3 ....un ... .625 u .3 3.32... 31.302.22.52 92.2.2... 3387....» 3. 23.2.... 2 2 8 3.22.33.95.52... 22.2.2... .....372; .3 9232.. .3552 8.2.2.23... 22.... n5. 3.... .... 3 9.3.... u. 2. n. u82 ... 23.6 u 22...... .2327?» S. 22.2.... on 2. 3 8. cu. ...aoz....n.:.: Goons-.... .33022212 «a. canal—.— nan: _ :82 an S 3.32... ...-32.382 . .3 9332... en 2. 3 2 6.... 92.2.... 3.82.. ....2r2-.302 8. 322.... .229. 35.2.... 02.23.... .... .....n u .2932... ...-2...... 2: 3 non 2.22... .3235: H... 2. 382 us... .3 30.3323 .2352 ”...... ...233 u 23:2... pint... .2: 8 .3 e322... 32:83....“ 5222.. 2.... 2.5.... 382 32.532... ...... u...<2.2_..u u 2.22.... .53....22JS.» .3 32.2.... c-2232 3,32... o2 e... 3 3 .9... 522.... .... 333.... 2.32.2232 2.22.2... 222...... .232 ... . 32.5223”. 423...! 02 2.... 28...“. 0 non 9:32... 702...... 3 ....u. 232...... . . . . . 9 can... 8.8.: mom. 9 m Sam — .3230. 8.8.: m8. @— 125 239:4 3.4.2:... 3... 54.4.. .32.! 34.. 4 :3. a.5..a..u..... u a: c-oaa.4 .4444. 4.» —:z n.o: xx.# u;2¢hh.t:< 4¢$hax ask 3. ans: £890 0 .0. 6.3.1... 2. 0h 8 0... oc~.e:.4 .au..5aza.h:!x-.a:xu.paxu 0.. 00.2.1... ...». 2.85.5.3». .25....3» ..n ...—2.8... 7.5.5-1283 3:2... .0”... 50.5.... Fla 0..-a3... .ud<:§.~-§ o... 0:03... ~\.—..iu. «45...... an a... on..oa.4 .t14— :z_.n_.. .z. .;z<=; 4:3..z.:y .mpn.nu ra4334¢ Isak 4cahax ask o .0. ...-'8... we a... 8 «.9 2.33.4 3383...:qu ...-.0 «0.. 022.3... n. O... 8 .o.ou...g.hxul..= —.n 2.03... a. a... 8 ...uao...¢u.:g...<.=h— no 0... 02.3... ..aooa.hn.....alu 0:. 62.2.2... 4.4.52 22¢ 4.5:: 8343...... 95.5. 2.436.: 8.3.... .3.—.3. d ... '08:". U 0.... 0.9.9.... inx.!».§.uua.fiu a: 0:301... ‘.~\3.ac...»14..-r 0:. 02.09.... 74:32 024 .393: 2.43:”... ugh-:54 459...! go 0 0:. cacti... 832.50 .0 z... .3623... on Oh 8 n: 362...... inx.£u.l.>n..u§.3unun33008.38“: «.... 62.2.8... .339. .... Hit—g .....um 30 u «.... 0.02.8... v. Oh 8 .3508 .6... .8308...— 2... cocoaa.4 :.;z<=4 4: 3. w. auz<: =;.:1 29:9: 09 carouzxou «use: no man¢m a. on m. mac: n. auuxu u .op o..noz.4 n. as co .o.oa.._4aoz.um44... no» cannon—4 n:¢:..u.maoz n. on so. oa~moz.4 .n 99 cu .. .04. u<4.z.u_ no. common—4 .uzoo ua4uc4< uz.¢ua:a .4 n.4a u no. on~noz.4 .aup¢s mane: ous—aaan 44¢ 24:: man: «sou u .9. ooo0o3.4 u32.hzou o~. one anoooa.4 on as 00 no. o~ooaz.4 .uooa has: «an ua<14244u op auca .304 u ... o...oz.4 o~4 as so 4u>¢mz .0.. 2.42: .az<. . .0.. u<4.z.u. ... oooooa.4 a4k¢s nuns: 44.4394: 44¢ .— ;Ua:u u ... canooa_4 mat—hzou n.. no. oooooz.4 o._ op so ... canoes—4 .xazu.rxuz-zazu no. omnqoz.4 ... o& co .o.oa..t¢zu.pxuz... «a. annooz.4 .-.zuaoz.zzouz-.zuooz.zzouz .o.oa..zuaoz.uk.242>..p4.2. 4424 :0 2232.4 .529232-532 .m- ........... .---------------------------u-n---u----unu-a-u-u ... o~0002.4 t.4a2..-2 a“. on on. o. .7332 on .... ...5... 3.52 o 2. ...-32.4 9.5825232 no. a: . . _. ..a..._ a. 4. 424:: ....22.2-> ..922.2.p2. uhaaxau u ... ooo.c2.4 «322....4402 4.. ca 2.. ......... - ,-----. .-.-...!...-......--..............-......u 2. 0232.4 ...-.2422...» 443.42.... ...:- 4-4342 u ..2. .23....2222 4.443.322.4253. 32...: 444B «.2 ...-302.4 2.2.2.8524 2.43.4.2. :2 .......... - - _ --------------------I------n------2-!.-l-u ... 3392.4 ...-5.2.43.3: 24.4.4.2. m: .2 .....42424.42 4:32.34 3... 42.59.23 9...... 445. u 0.. 3:62... .omn.2zouz..ana.¢4aoz . :- ------- - - ................................................... 0 ac. oonoo2.4 ..onn.4»¢pmz..am..2<...onn.hxaz..onn.hm.4 244492. ... 2323.23 4.4,»... 25...... £34.54... 3.4 .. .3. 9.44. 3. 23.5.4 .24...342.52».8422443.342m§4..3324... 45... a: 44.2.52. onus 4.20 3. 9232... 6.24223 3.. 23 22.52.42 9:32.... 2.5-...”... .... 42.59.23... .4 .3 .: 62.24.. 30 02.2.2.4 ...m4..2.442.n322.22ou2.24:232 . a: ...-......2 i . , .. ----------------------- - u 3. 3.32.4 .....2..m2.2<...=42...m.4...-3:...442...-34.32.3223... 422.323... a: p 4.4-4.2.42 2.23.44... 9.42 u 3. u a; ------------_--; - ------I----------------------I--usu---!i-!.!.!i.innu a... 0:32... :2. a: 3 :2... 3. 2.32.4 4325......22324 34.4.. u on. a .o- 4 . ...-.0.. .3 3302. 4 222...... m: .33.»...2. 2:72.223 3. 3232.4 .94....52444 244.. 42:. n43: 9.2.2.4.. 442 2422 224.. 4:8 0 :- A........4=..... 3 :- 3.32.4 4.42.25“. 2. an. 7224.; 3. 0332.4 4. o... a... «no .................. - .---------:---------------------..---a ----3-3-1:-.. no. 0332.4 .482 .542 4.... ”...-«2.2.4.. 2. 2.4.2. .3.. u ... 3424.243 44.42.22. u 4: 3.32.4 2. a... on .4332 .34. 2.442 .224. . .04. 92.2.... 22 ----iii- ----.- ............................ - ................. u no. 3.32.4 24.22.54. 2.4.42 42... 94.52 442.342 44.. .4. 54.5 o a... 43.222342. 84224952 32.22.34! 4.3492. :- 0332.4 “.22....28 4.. a... 32.....5-w2.3..4:2.. 32.542.33.524 4444.2. 2- 3432.4 3. a... 3 2... 39.2442.8422.22.33.34.- 4<42 a: ...-302.4 .2924.§42.-2:24 .2. 32”.)..cn...22........cnn.a4a>.853222.. 45... .00 30002.4 n... a... 8 36a. 3:23.542... 0: .....xa...4.4wa R234... 2<....§42...m.4 22,434.: 2325.4 3. 0332.4 742.32.223.42 424.52.223.42 3.3. 3432.42.52... .2. .24....532.22.322.4252 213...... 42...... 4.2.59.3... :- ovnoo2.4 .32u.¢<.-2aao2 o: :2 u .2. 0.32.2... .582.kn.4.tozu «.4... 02... .... 03292.4 54.22.54... 2m: 0 .2 222.42 .2. 0332.4 ...:2 .42... 3.22 24.52 on 34.5.4223 4.4.52 .3 2232 22... 4 55.2.3 u 2. o: 024 o... no. 82.2.2... a: o... 3 .. .94. u<....2..... o.- c. a... a... :- 3.2.2... .43.: 2.44.. 202 4.2.. u a... .34....222.a24 no. 332:4 2 4.82 ,3 2222.23... 22... 4.; o.- ..S: 2235.99. 2.2.3. » 4.... 442 u .... ...: o... 3 3 .3. 32.4....242. .. :2 ...-302.4 .5 224.492.4252 .3 ...-23.443.33.824532 $3.92.... u .2. 3.32... .4223on 8. m... m:......m:o..242> .....4222 u 3. 33°23 .4224...on ... v.2 3332.3? 32475:... ...-3.....42..- .3425»; a.- 33023 .. 4332.228213322282 ... 32”.. 24......33 ... 2.. 0:32... .. 24.52.2209:- ..u.52.22..02 .4... 33253524 .... 0332.4 ea c... 8 .. .04. 024.42.... ... .5. o.— 3 8 ...... .m:n....—m.4. ... :- 2.32.4 .§2.432u.2<.. o.- .naa... :43. ...-32.....43..- 3.2.2.242) 2- 8.32.4 324.. 23.52.: .4 3. ...o. 3:2...2422 :- 3302... 3482.934. 4324.542 3- 32242.73... 8 :- 3.32.4 332132224... 3- 303362. .own.2u2> 44.... «... 2.302... .3....cozzm... on. Swazi-183.342. 44...... .3 3.32.4 3432.53.52.32: 32 a..44<2.n:23...24.m:42.342.2...32....242..33...n.4 3°32. 2.. 3.2.2.4 ......324.....!» 3- ..__.E......_._.3.2¢.4...422..5....n:ax<2.30>.242>. 34.! 22.52....“ 3. 3302.4 27.32352. 3- u 3. 3302.4 .. 324-324 «a. 334.2... 92“. ..3 2. 32.4 745.922.133.324 82 3.32... 43922....220232 33.. o 3- 2.32.4 324.4393. 82 23.42... 33.5.. a: 8302... ”52:20“. -~ «2 3.2.2... 32......on on. :0 0.392... 223...... a... 33:2... 522.32.42.52 3. 3. 3302.4 3... u. 9.52.2444 332 . .2 3.32.. 3. 9p 8 «on 23.2.3.3 24.252 ”.2... 4... ...-43.4.4... 2 20.22.23 2925... 42.. 7:45.... .2 on. 4.2.32... 4296.94.42.21... .4. 3392.4 2.442 22.34.52: ...: 23:2... 3. o... 8 .o.aa...82a...xu2.... a: 3. 0332.4 «2 o... 3 .42 ....4. 51:... v: 2.152... .22...4324.57.24.82321248233. a... 3333 754.9524 2. no. I 3.... .4223... 8.8. 2 8... a. mom a. 8.... . 3.2.2.... 8.8. 2 8m. ... mom 127 0-...0242 0.0....0022 0.0....24402 0.222042 0....240202 0-...402042 0....pm.4 0-...22. 0....-. 00 .04..p02...0nn..440. 244.200 .04..»022..04...44«2 4242 .04..22002..0an.240202 240402. .0:..4r<+n ..0»..22...0n..h242..0n..pw.4 2404b2. .52....402 :20042 208.00 .220.._.ns \x2004ax 22..h~42.hm.4 \.20042\ 202200 .492042. 024. 42.900220» 024 220942 00 024 ....0..~.n.\. .2220. .n:02<2.n00 1..0...h22. .00n.p. 4h.22 «00.22.00... 00 00 024 ....4400-.....»22 0. 00 00 .024.~242-024 00~ 00 00 .0 .04. .024.0242. 2. .:24.+:20 .0...+¢t .200 .40. 0. a. .024.22.-n ...pm.4.024 004 0+ 00 .0 .04. ...hn.4. 2. «02222.00... 00 00 024 00 024 0.0-.0...h22 002222.002-0 00 002222.00.-. 00 n:02<2.n00.024..0n..224..0nn.~202..0an.hm.4 240402. .0»...024..04...4440..04.04.u22 .no. ....tn..44..4 .3304... 224.02.42.04... 2200... 20.0.00 .n:ax<2.w0a.~<2. 0240200 42.0002000 024 220042 002.0200 ...0..... m2: ...... m0:..0.~.\. 022200 .44-22. .4. .2. .. .0.... 4a.22 4.422 222 .02_n .2423. :400.042.. ..42h0.00200.04.4<42-4. 2.3:; .0204 .:<..n 04:2. +402. ..Iahu.003004...4.4.0040.2.2.40 4420 0.2.442 ---‘l-'--l-"-|'-I.--‘-"Io--"‘-'---' -' ‘-'--|l'-l-'|l annuafl I‘ldahlu OF '04. I‘OIFQI 060308.. b ahzmtou 2.442..0nn.22002..0nn.200202 240402. .0nn.4p.004...04n.2402..0nn.0022..0an..44ma 4242 U l---|-.l"Illl'|-"-I""|"-'-‘I‘l-|lul'--III-"II-I||'--'-'|--'---‘-'I-I'-I-|'-lv 03. U an O: 3. av On .-l--|"|¢'III--Inl'l-"-l|"-|-l"ll"l'lI.lllII.-al|"--"-'nll"l|"I'-III'I'-IIllu|'V 0 ---0 0 o~ I." UUUUUUUUUU U nn3~ n~o .... .0.-... 2.0.0.00... 8.8” 3 08. ... mom 128 azu zxzyua ano~oz_a o._-.maax.z;uaa .o.c .ou. .mzaa.uauma. @— c~v~on.4 .n Oh so a.v~cz_4 .czu.puuz-ozu o0v~oz_4 an ch 00 .o .02. .ozuvhxuz.u~ oon~ox_a ems-.maaa.54amau.maaa.uquna oun~ox.4 cma‘.oza.p:xn osn~ox_a ...naao.xu n. .ou h OOOOOO0000000009000OOOOOOOOOOOOOOOOOOOO p"p-"--—-p”"p-—pp,’p-'-'-""F'-’F’p 0102 0:0 "Nfl'U’O-‘OOO’NH v-wv-v- 8 131 000006 .OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO "NM'OQNOO 35th a. 2-0. 2.: 3:3 803% 25: 33 - m0 039—. 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.000 55.000. 00.000 00.00.. 00.0 0000.. 0 0 . 0000 0 0 0 00 000- 00 000 0000.. 00 00. 00 00 00.000 00.0 00 0 00.5. 0000 . 0 0 . 0000 0 0 0 00 ONN- 00 000 0000.. 00 00. 50.00 00.000 00.0 00 0 00 .. 0000 . ~ 0 . 0000 0 0 0 00 000- 00,000 0000.. 00.00. 00..00 00.000 00.0 00 0 00 5. 0000.. N 0 . 0000 0 0 0 00 000. 00 000 0000.. 00.00. 00.000 00.000 00.0 00 0 00 0. 0000 . ~ 0 . 0000 0 0 0 00 00~. 00 000 0..0.. 00 00. 0..00. 00.000 00.0 00 0 0. .. 0..0 . ~ 0 . 0000 0 0 0 00 000. 00 000 0000.0 00 00. 00 00 00.000 00.0 00.0 00 ~— 0000 0 N 0 . 0000 0 0 0 00 000. 00 000 0000.0 00.00. ...00. 00.000 00.0 00.0 0..0. 0000.0 0 0 . 0000 0 0 0 00 000- 00.000 0000.0 00 00. 00.00. 00.050 00.0 00.0 00.0. 0000 0 ~ 0 . 0000 0 0 0 00.0 00 000 0000.. 00.00. 00..0. 00.000 00 0 00 0 00 0 0000 . ~ 0 . 0000 0 0 0 00,0 00 0 0000 0 00.00. 00.0 00.0 00.00 00.000 00.0. 0.00.. 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.50 00 000 0. 5 00.0.. 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.05 00..00 0... 0000.0 0 0 p 0000 0 0 0 00 0 00.0 0000 0 00 00. 00.0 00.0 00.5. 00.00. .0.0 05.0 . 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.0 00.0 00 50 00.000 00.0 5500 . 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00 00. 00 0 00.0 00.00- 00 000 00 0 0000 0 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00 00. 00 0 00.0 00.00 00.50~ 0m 0 0000.. 0 0 . 0000 0 0 0 00 0 00.0 0000 0 00 00. 00 0 00.0 00.0 00 0 00.0 0-0 . 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00 0 00 0.. 00.050 05 0 0000.0 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00 00. 00 0 00.0 00 00. 00.000 50 0 0000 0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 00 5 0000.0 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00 00. 00.0 00.0 00 00 00.00. 50.0 0000 0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 00.. 0000.0 0 0 . 0000 0 0 0 00.0 00 0 0000.0 00 00. 00.0 00.0 00 00 00.000 5..~ 0000 0 0 0 . 0000 0 0 0 00 0 00.0 0000 0 00 00. 00 0 00.0 00 00. 00.000 00.0 0500 0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00,00. 00.0 00.0 00.0 00.0 00 0 0000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 00.. .000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00 00 00.0 05.0 5000.0 0 0 . 0000 0 0 0 00,0 00 0 0000.0 00.00. 00.0 00.0 00 0 00.0 00.0 5000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 00.0 0.00.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00.0 00.0 00.0 0000.. 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00.05. 00.000 00.0 0000.0 0 0 . 0000 0 0 0 00.0 00 0 0000.0 00.00. 00.0 00.0 00 00 00 000 00.. 0000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 00.0 0.00.0 0 0 . 0000 0 0 0 00.0 00 0 0000 0 00.00. 00.0 00.0 00 0 00 0 50 0 0000.0 0 0 . 0000 0 0 0 00,0 00.0 0000.0 00.00. 00.0 00.0 00 00. 00 000 00 0. 0.00.0 0 0 p 0000 0 0 0 00 0 00 0 0000 0 00.00. 00.0 00.0 00 N 00 ~00 .0.0 .000.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00 0 00.0 00.0 00.0.— 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 00.. 0000.. 0 0 . 2.10 x030 0.000) 00000 00 00 00 00 020 > 00>~ 0102 0:0 FNM'OQNQO S 132 'NM'OONOG OOOOOOOOOOOOOOOOOOOOOOOOOOOOGOOOGOOOOOO 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQD OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOCOC OOOOOOOOGOOOOOOOOOOOOOOOOOOOOOOOOGOOO00 00 000. 00 000. 00 000. 00 000- 00.000. 00 000. 00 000 00.000- O O OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 00. .000 .000 00 00000000000000000000000000000 000 000 000 5 .53th 0000. 0000. 0000, 0000 0000. 0..0. 0000, 0000 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000, 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. OOGOOOOOOOOOOOOOOOOOOOOOOOOOO‘GOO""F""° 0.000) a. 3-2 2... =2? 8:28 33. 33 - ...m 03:. oc.co. oo.oo. oo.oc. co.oo. ao.ao. oo.oo. oo.co. co co. oo.co. oa.cc. oc.oo. co co. oo.oo. oo.oo. oo.oo. oc.oo. ca co. oo.oo. oc.oa. oo.oo. co.oc. oo.oo. co.oo. oo.co. oo.ao. oc.oo. aa.oc. oo.co. oo.oo. oo.aa. oo.oo. oc.eo. oo.oo. oo.oo. ao.a°. oo.oe. co.oc. co.co. c¢.oo. 00000 00000000000006000000000000066 00000000000000000000000000000 H000. GQONQOOOO QNOGOOOVG 0100000000. '0 N o 0 0 0 30000000000 0 0 0 0 0 0 005505 GOV-~64 0.. 0 0 0000000000 .0- 0 0 -0-0-000000000000—050000000000 020 0000. 0000 0000. 0000. 0000. 0..0. 0000. 0000. 0000. 0000 0000. 00.0. 5000. 00.0. 0000. 0.00. 0.00 00.0. 0000. 0000. 0000. 0000. 5000 0050 0000. 0000. 0500. 0000. 0000. 0000. 0500. 5000. 0000. 0000. 0.00. 0.00. 0000. .5.0. 0000. «wooacconaooooodéoooo—-0-—0———0oo-————— OOOOOOOOOOOOOOONNNNNNNNNfl l OGOOOOOOQOOOOO ’ .- 060°00060009000000OOOOOOOOOOOOOOOGOOOOO PP---“F-'-"--Fpv-Fw'-"’--p"’p"-ppp“ 0302 0:0 wnuo0un0~un0 O 2 133 "Nfl'00N0O OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000. 0000 0000 0000 0000 0000 0000 000°0°000000000000OOOOOOOOOOOOOOOCOOOCO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQOOOO OOOOOOOOOOOOOOOOOOOOOO000000000000GO0°C 00.000. 00 000. 00 00w. 00 00w- 00.00N- 00 000- 00 000- 00 000. 0 0 00000000000000000000000 00 O O 5 0000000 00 000 00.000 00.000 00 00v 00.00N 00.000 00.000 00.000 00.000 0 0 00000000000000000000000000000 00582 0 m~-- 0:: :23 .5033 :50 33 - 0.0 03:. 0000. 00~0. 00~0. 0000. 0000. 0pp0. 0000. 0000. 0000. 00'0. 0000. 0000. 0000 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000, 0000. 0000 0000 0000. 0000. 0000. 0000. 0000 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 0000. 00000000000000000000000000000—000—w—vv0 0.000) 00.00— 00.00p 00.00— 00.00— 00.00— 00.00. 00.00. 00.00— 00.00« 00.00w 00.00. 00.00. 00.00. 00.00— 00.00. 00.00— 00 00p 00.00— 00.00w 00 00' 00.00. 00.00p 00.00. 00.00— 00.00. 00.00p 00.00p 00.00p 00,00p 00 00— 00.00p 00.00— 00.00— 00.00. 00.00— 00.00— 00.00. 00.009 00.00p 00000 cooodédéédéééddéodéodddéddddd 906966édéddédédddédééddédéééé .000. 00°N00000 0N00000'fl “00000000 0 N 0 0 0'00000000 0000000000 0'0"0fl 0N0000 MNFNNN '666666 0 p p .500 .v- .000 .000 .0N0 0 0 0'NONN00 I-v-v-v-c-u-q- pp ’ n 0 'QPOPN0°00QV'NNGNPQN0POM0'floP0' 620 0000 00~0 00~0. 0000. 0000. 0p—0. 0000 0000. 0~00. 00'0. 0—00. «000. 0000. '0—0. 0000. .s00. 00.0 0p~0 0000 0000 0000 ——00. 0000, N000. 0.00. ——00. N000. p000. 0000. 0000. 0000. 05'0. ~000. 0000. 0000. 0p00. v0.0. .000. > "F000000*000000000000F“OFF"PP'OOO'F'P" Q 000000000000000000000000 )- .- OOOOONNNNNNNNNM 000000000000000000000000000000000000000 "'pppp'-P'-'-p"p-P’pp'-’-"’p-’pp”p“p 0202 030 “N0'00N00 C 2 134 'NH'OCNQO OOOOOOOOOOO0000000000000OOOOOOOOOOOOOOO 02688 a 3&0 0:: 02.3 .5233 300 03.0 - 0.0 03:. coco o o o cc o 00.0 0000.: 00.00. 00.10N n~.noo. oo.oo~ oc.vo.. oo a coma.. n o . coco o o o oc oa~. 00.000 oo~0.. 00.00. v~.o. 00.0.0 00.0 00.0 ~c m~ oo~o . ~ 0 _ 0000 0 0 0 00 0-- 00.000 00~0.. 00.00. .0.~0 00.0v0 00.0 00.0 cw v~ 00~0 . ~ 0 . 0000 0 0 0 00 00w. 00.0mm 0000.. 00.00. n..n0~ 00.000 00 0 00.0 0m..~ 0000.. ~ 0 . coco o o 0 cc on~. oo ecu coma . 00.00. on.0n~ 00.000 00 o 00.0 ow o. 0000.. ~ 0 _ coco o o o co om~- cc.om~ 0..0 . co co. m~.on. 00.000 00.0 00.0 .~.m. 0..0 . ~ 0 . 0000 0 0 0 00 000- 00 000 0000.0 00.00. 00.00 00.000 00.0 00.0 .0 0. 0000 0 ~ 0 . coco o o o oc con- 00,000 coco.o 00.00. 00.00. 00.000 00.0 00.0 mg... 0000.: ~ 0 . coco c o o ca 000- 00.000 0~o0.o 00.00. ov.~o. 00.”.0 00.. o~.0 .n N. o~oa.o ~ 0 . 0000 0 0 0 00 0 00.000 00v0.. 00 00. '0.0~. 00.00~ 00.0 00.0 00.0 00v0 . ~ 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0~ 00.000 v0 0. v..0.. 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00..~ 00 00~ .0.0. .0.0.. 0 0 . coco c o o co 0 00.0 coco a 00.00. 00.0 00.0 00.0. no .o~ 00.. .000.0 0 o . coco o o o oo o co 0 0000.0 00.00. 00.0 00.0 00... 00 an. 00... 0000.. o o . ocoo o o o 00 o 00.0 coco 0 00.00. 00.0 00.0 on... oo.v- so... n~.o.. o o . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00.00- 00.000 cm 0 0000 0 0 0 . 0000 0 0 0 00 0 00.0 0000 0 00.00. 00.0 00.0 00..0 00.~v~ ow n. v0~0 . 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 .0.0. 0.~0.. 0 0 . 0000 0 0 0 00.0 00 0 0000.0 00 00. 00 0 00.0 00.0.. 00.1.0 00.0 0.00.0 0 0 . ocoo o c o 00.0 00 0 0000.0 00 on. 00.0 00.0 00.no. 00.0-o ~o.c. aooo.o o o . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 av... 0000.0 0 0 . coca o o o co 0 00.0 0000.0 00.00. 00.0 00.0 00.0n 00.00. .0.. .oco.o o o . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 we 0 0000.0 0 0 . coco o o a co 0 00.0 0000.0 00.00. 00.0 00.0 on.~n 0v.0~n - o 0..0.0 o o . coca o o o oo o oo 0 0000.0 00.00. 00.0 00.0 00 no. oo.o~n on.w 0.00.0 o o . 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.0 00.0 00.0 00.0 00.v 0000.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00.0 00 0 00.0 0000.0 0 0 . 0000 0 0 0 00 0 00.0 0000 0 00 00. 00.0 00.0 00.00 00 0 00.0 0000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 0 00 0 00.0 0000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0 00 0 00.0 v~00.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0 00 0 ...0 0000.. 0 0 . occc c 0 o co 0 00.0 0000.0 00 00. 00.0 00.0 co 0.. 00 -o 00.0 ..v0.0 0 o . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 v0 00 nn~ 00.. ..v0.0 0 0 . ccoo o o a 00.0 no.0 0000.0 00.00. 00.0 00.0 00 o 00.0 n..n «000.0 a o . 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.0 00.0 00 0 00 0 n0.~ 0v00.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00 «0. 00.000 v0.. 0000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 0v 0 00.-n 00.0 .n00.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 0~.n 0000.. 0 0 . coca c o o 00 o 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 a... cone.. 0 o . 2.30 x um». w.a .- OOOOOGOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOQ00 Pv-Pv-Pv-v-c-v-w—v--0-Pc-v-u-v-c-v-v-Q-Pv-c-Pwv-v-c-v-PF-v-v-v-q— w3(2 000 P00000000 O 2 136 FNfi'O‘ONQGC - OOOOOO000°COCOOOOOOOOOOOOOOOOOOOOOOOOOO case 0:3: coco coco 0000 case oooo oooo ccoc .ccoo 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 OOOOOOOOOOOQOOOOOOOOOOOOOOOOOOOOOOCOOO’3 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOG0000O OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOO 00 O O OOOOOO OOOOOO GO GO OOOOOOOOQ OOOOOOOOO 3 OOOOOOOOOOOOOOOOOOOOOOOOOO GOO GOO OOOOOOOOOOOOOO OOOOOOOOOOOOOO 00 0 0n~- 0-. 00w. 00w. 0ow. 000. OOOO 00 000 00 000 00.000 00 00c 00 0mm 00 000 00 000 00 000 00 000 O O OOOOOOCOOOOOOOOOOOOOOOOOOOdOd 09582 0000. 00N0. 00~0 0000. 0000 0..0 0000‘ 0000 0N00, 00'0. 0000, 0000 0000. 0000, 0000 0000 0000 0000. 0000 0000 0000. 0000 0000. 0000 0000. 0000. 0000, 0000, 0000. 0000 0000. 0000. 0000. 0000 0000 0000 0000. 0000. 0000‘ OOOOOOOOOOOOOOOOOOOOQOOOOOOOOPOOQP—"“’6 3.000) 2 «.2: 2.: =2? 8:29. 32. 33 - 3. as: 00.00. 00 00' 00000— 00.00— 00 00— 00000— 00,00— 00 00s 00.00p 00 00p 00.00— 00.00p 00.00— 00.00. 00,009 00 00— 00 00— 00 00. 00 00— 00000— 00 00— 00 00— 00 00— 00.00— 00‘009 00 00— 00,00— 00.00— 00‘00a 00.00— 00.00— 00.009 00.00. 00.00— 00 00. 00.00— 00.00« 00.00« 00.00— 00(00 00000000000000000000000000000 .N- .0!— C V NOOOOFO CDfiGMQO ooooooooooaoodéédéé0600066666 .s-0— OMONOOOOO ONOMOVDO'M «IDOQOQOOO I, N G O O'GOOGGOOO 00 00 NO NN 0 0 ~n~o 'v-h O 0 'N 00 '0000000006 .QO’CM ONMOO. GNPNNN ééédéd 0p. .svm .vsm 000 00. .0~n .0~0 60060006 QNQQVCQO ~n~m0~00 .——-.PM 0 N unav- I~ ’ pp 0 0 wv—o—~0000vvvmwn~~o~baodm§~ 624 0000‘ 00~0 00~0 0000 0000‘ 0w-0 0000, 0000. 0~00 00.0. -00. ~000. 0'00. ~000. 0-0. 0000 50.0 00~0. ~000, 0000. 0000 0100 0000 ~000 0000 «000. v000. .000. -00. 0.00. 0000. 0010. «0'0. «v00. 0N00. 0~00. «~00. '0—0. 0100. "°°Q°°°’°°°°°°°°°°°°P"O'OOOPPQQQ—Pc-«Dc-v- OOOOOONNNNNNNNNQ A OOOOOOOOOOOOOOGOOOOGOOO ) .— OOOOOO00600060606000OOOOOOéOOOOGOOO6°00 pppppfippppfipp'fipp'ppppppppppppp’p—pppp— w8<2 0:0 PNQ'OONOO O 2 137 OOOOOOOOO00°GOOOOOOOOOOOOOOOOOOOOOOOOOO FNM'OQNQQ 02682 a. 8.5 2.: 5:3 .5239. 300 33 - 0.0 03:. 0000 0 0 0 00 0 00 0 0000.0 00.00. .0.00N N..100. 00.00N 00.10.. 00 0 0000 . 0 0 . 0000 0 0 0 00 00N- 00 000 00N0 . 00.00. 00.00 00.000 00 0 00.0 00 1N 00N0.. N 0 . 0000 0 0 0 00 0NN- 00 000 00N0 . 00.00. sN.01 00.010 00.0 00 0 0~.0. 00N0 . N 0 . 0000 0 0 0 00 00N- 00 000 0000.. 00.00. .1.NON 00.000 00.0 00 0 us 1N 0000 . N 0 . 0000 0 0 0 00 00m- 00 001 0000 . 00.00. 0N.00N 00 000 00 0 00 0 00 .N 0000.. N 0 . 0000 0 0 0 00 00N- 00 00N 0..0 . 00 00. 1~.00. 00.000 00 0 00.0 0~.0. 0..0 . N 0 . 0000 0 0 0 00 000- 00 000 0000.0 00.00. 0..00 00.N00 00.0 00.0 0..0N 0000 0 N 0 . 0000 0 0 0 00 000. 00 000 0000.0 00 00. 00 00. 00 000 00 0 00.0 0N 0. 0000 0 N 0 . 0000 0 0 0 00 000- 00 000 0N00.0 00 00. 0s..0. 00.0N0 00 1 0N 0 00 0. 0N00.0 N 0 . 0000 0 0 0 00 0 00 000 0010 . 00 00. 0. 00. 00.00N 00.0 00.0 0..1. 0010.. N 0 . 0000 0 0 0 00 0 00.0 0000.0 00 00. 00.0 00.0 00.0N 00.00N 10.~. 00N0.. 0 0 . 0000 0 0 0 00,0 00 0 0000.0 00.00. 00.0 00.0 00.~N 00.00N N0.1. 10.0 . 0 0 . 0000 0 0 0 00 0 00.0 0000 0 00 00. 00.0 00.0 00.0N 00..0N 00.0 N~00.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00.~. 00.00. 0N... 01.0 . 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 0N.~1 00.1NN 00.N. 0NNO.. 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00.00. 00.0 00.0 0N.N0- 00.000 10.0 0000 0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.0 00.0 00.10 00.N1N NN.0. 00N0 . 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 00 0. 0.N0.. 0 0 . 0000 0 0 0 00 0 00 0 0000 0 00 00. 00.0 00.0 00.0.. 00.1.N 0N.N. 0100 0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00 00. 00.000 .0 0. 1000.0 0 0 . 0000 0 0 0 00.0 00 0 0000.0 00.00. 00.0 00.0 00.0 00.0 N0.1. .000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00 00. 00.0 00.0 00 00 00.00. 00.. 1~00 0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00 0 00.0 .0.0 0000.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 N0 01 0N0 .N.0 1500.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00.00. 00.0 00.0 00 00. 00 0N0 00.. 0~00 0 0 0 . 0000.0 0 0 00 0 00.0 0000 0 00.00. 00.0 00.0 00 0 00.0 00.N N000.0 0 0 . 0000 0 0 0 00.0 00.0 0000 0 00.00. 00.0 00.0 00.0 00.0 N0.0 0000.0 0 0 . 0000 0 0 0 00 0 00 0 0000.0 00 00. 00.0 00.0 00.00 00.0 01.0 1N00.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 00.0 0000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 00.0. 0000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00.0 00.0 11.. N000.. 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00 00. 00.0 00.0 00 0N. 00.NNO 00.0 0110.0 0 0 . 0000 0 0 0 00.: 00 0 0000.0 00 00. 00.0 00.0 00 10 00.00N 00.1 0110 0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 00.N 0N00.0 0 0 . 0000 0 0 0 00.0 00.0 0000.0 00.00. 00.0 00.0 00 0 00 0 00.0 1.00.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00 10. 00 000 00 0 .N00.0 0 0 . 0000 0 0 0 00.0 00 0 0000.0 00.00. 00.0 00.0 01 N 00 NNO N0.~ 1000.0 0 0 . 0000 0 0 0 00 0 00.0 0000.0 00.00. 00.0 00.0 00 0 00.0 .N... 0N.0.. 0 0 . 0000.0 0 0 00.0 00.0 0000.0 00 00. 00.0 00.0 00.0 00.0 «0.0- 5000.. 0 0 . 2.10 3030 0.000) 00000 60 00 00 00 020 > 00>. 0102 0:0 F~0100N00 3 APPENDIX C 138 010. 515 000. 05N. 010. N00. 001. 00N 000. 05N. 001. 05N. 2.5.23 88 2-0.”. 8.. 33.5 ...—5:33:50 - 2.0 03a... .0N. 01 11 01 11 01 11 01 11 01 11 01 11 005 000 00N. 00N. 00. N10 105 005 0NN. 01N. 0NN. ..5 10. .0N. .0N. .05 .05 .5.. .NN. .5 0N0. .50. .N.. 505. N00. 000. 000. 000. N00. 500. 01 01 01 01 01 01 N.5. 005. 000. 0N5. 10N. 000. 151. 0N.. 100. 500. 01 01 01 01 01 01 0I0<30v-P0 0. 000.00 01 10N N01 10 01 000 0.0 1.1 0.0 055 550 000 500. 010 005 05.. .00 .00. 01 00 O'OON'Q'OM - 000 000 001 000 001. 000 .N0 000 .05. .00. NOOOOGOOOG 51 000..N. N1 0.5.00. 012 012 012 012 012 012 012 012 012 012 01 .1 01 01 51 01 01 11 01 N1 .1 01 000 000 0:0 000 000 000 000 000 000 000 5N0.0.. 055.00 0000000 2. 0.0 0 00.0.0000 51 N01 0 N1 510 0 01 .1 001 0 .00 0 00010.0 ..0 00.01 20.000010 51 N55 00 N1 00...5 01 .1 .N1..0 00...N I .002. :0?! ...-3°. 3 33.30.: .51 .00 sq can.au ~c one... 01 .1 100.N0 000.NN 20..01 20..30 02.0 00.0 00N.0 I 50..0. .1 200000.021 100. 010. 500. 110. 001. 0.0. 00N. 00N. 0.0 000. ' NOOO"*°°P' 000 00. 01 000 000 N.. 01 .00 110.0 01 .0N. 000 0 01 N00. N.0.50 01 000 0.0.00 01 010 .1 00.10.1>0 0. .0..00 01 105 001.N0 01 NNO 0N0. 000 000. 000. 000. 00. 5... 005. 1.1. 00. .00 ..0. .11 .05 OOQOHO'QOC 000.0. 000 001 000 001 000. .N0. 000 .05. .00. NOOOU’COOO 51 050.0.. N1 ....01. 012 012 012 012 012 012 012 01: 012 012 01 .1 01 000 0:0 0:0 0:0 000 0:0 000 000 030 000 .50.0.. 000.00 0000000 2. 0.0.: 00.0.0000 51 000.0 N1 001.0 01 .1 0001000 ..0 00.01 51 000 .0 N1 0.0.50 01 .1 20. 000.0 00N.0 000010 000 01 00N.0N .0010 20.23 ..0..u. .1 .00000d021 .01 .00 51 1.1.00 N1 000.00 01 .1 N.N..0 10. .N 20..01 20..30 02.0 0000 0NN.0 I 5.0.0. .1 200000.021 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 141 0.0. 500 000. .0N. 0.0. .00 N10 010. .0N. 000. 0NN. 010. 000. 0:502... 88 2-0.". 5.. 5055 35:25:50 - «N0 03.... 1 .00. 01 11 01 11 01 11 01 11 01 11 01 11 000. .00 005 .50.. 5.N. N00. .00 00. 501 00.. 0.N. .N0. .51. 50 .0 000. 01 01 01 01 0N0. N00. 01.. 010. 501. 000. 000. 00N. 050. 01 01 01 01 01 01 566.666.. No-0—oooo1 p O O p 550. 00N. N0.. 000. 100. NN5. 005. 00N. 505 001. 00N..0 00N.00. 00..0 00N.0 010.51 0.5.00 005. 010. 00N. 510. N00 055. 100. 000. 000. 0N1. 105.05 00..00. 00..0 00N.0 N51.01 00..00 OONOQOOON' OONOQCDON1 500.0. 012 01 000 00N.0 012 01 000 .01.0 012 51 000 .00 0 012 01 000 001.0 012 01 000 000 0 012 11 000 0N0.0 012 01 000 000.0 012 N1 000 005.0 012 .1 000 001.N 012 01 000 51 00. .0. 01 001.00 N1 1.0 N0. .1 N00.00 0000000 2. 0.0.0 00.0.0000 51 0.N 0 01 0.N.0 N1 0.N 0 .1 00. 0 0001000 .00 00.01 20.000020 005..0 00N..N 51 .N0.10 01 N1 000.05 .1 01N 0 I 010.0. .1 .000000021 500.0. 012 01 000 00N 0 010 01 000 .01.0 012 51 000 .00.0 012 01 000 001.0 012 01 000 000.0 012 11 000 0N0.0 012 01 000 000.0 012 N1 000 005.0 012 .1 000 001.N 012 01 000 51 0.5 50 01 000.00 N1 050.0N. .1 0.0.00 0000000 2. 0.0.0 00.0.0000 51 N.N.0 01 N.N 0 N1 1.N.0 .1 11..0 0001000 .00 00.01 20.0000x0 51 0.0.N0 01 000.01 N1 ..0 N5 .1 100.0N eo~.a u ....o. .¢ .oua.u.oz< 01 01 01 01 01 01 01 01 01 01 01 01 142 0.0. 01N 000. 0NN. 0.0. N10. 010. 10.. 000. .NN. 010. 000. 030.83 38 2-0.... 5.. 535 35:35:50 - 05.0 28.... .00. 01 11 01 11 01 11 01 11 01 11 01 11 N00. .00 000 00.. 0.N. N00. 000. .00 50N 00.. 00N. .N0 .51. 50 00 00 .50 .0 000. 010. 0NN. ..0. 000. 000. N01. 005. N.0. 100. 01 01 01 01 01 01 N05. .00. 00.. 000. ..0. 000. 0.0. .10. 00.. .00. 01 01 01 01 01 01 v D NOOOPOOOP' ' NOPOFGOOOQ 50. 000 00. .1N 010 0.5 005 00.. 50N. N51 00. 000. 00N. 05.. 000 .10. 5.5. 005. 00N. 0.5. 101. ..0 .00. .0 ,0 .51 .00 0N5. 000. 01N. 000. 000. 055. 010. .10. 0.0. 5.1. 05 .00. .01 .00 0051001003008. 005-00000081 500.0. 012 01 000 00N.0 012 01 000 .01.0 012 51 000 .00.0 012 01 000 001 0 012 01 000 000.0 012 11 000 0N0.0 012 01 000 000.0 012 N1 000 005.0 012 .1 000 001.N 012 01 000 51 0.0 .0. 01 NNO 00 N1 00..00. .1 000 N0 0000000 2. 0.0.0 00.0.0000 0.N.0 01..0 51 5.N.0 01 N1 .NN.0 .1 0001000 .00 00.01 20.0000x0 005..0 00N..N 51 .N0.10 01 N1 000.05 .1 oq~.o . .no.u. p< .ouo.mdoz< 500.0. 012 01 000 00N.0 012 01 000 .01.0 012 51 000 .00 0 012 01 000 001.0 012 01 000 000.0 010 11 000 0N0.0 012 01 000 000.0 012 N1 000 005.0 012 .1 000 001 N 012 01 000 51 000.50 01 000.10 N1 000.0N. .1 000.00 0000000 2. 0.0.0 00.0.0000 ..N.0 11..0 51 0.N.0 01 N1 0.N 0 .1 0001000 .00 00.01 20.0000x0 000.01 100.0N 51 0.0.N0 01 N1 ..0.N5 .1 on~.c - ....o. .4 .amo.mdaz< 01 01 01 01 01 01 01 01 01 01 01 01 143 .55 0N.. 000. 10N 005. 550 .55. 550. 5.5. .N0. 000. 10N. 005. 10N. 03.9... .5 88 2-0.0 8.. 30:5 3.5.3.3950 - u~.0 03¢... 01 11 01 11 01 11 01 11 01 11 01 11 01 11 01 11 0.1. 000 01.. 000 .00. 101. 101 050. N00. 00N. 005.10 01 000 00N 00 01 00. p NOOO'OOO" 00N 0 01 0.N NON.0 01 0.1 000 10 01 000 0n..51 01 000 .1 00.10.1>0 0. 50N.00 01 N1. 000.01 01 .01 80..01 :0..30 02.0 ' 55N. 000. 000. 0N0. 500. 0.0. 510. 10N. 110. O’C'OOOO' O O p N .N1.00 01 N00 000.00 .00 O Q 00N.0 01 50. ~0N.0 01 000 00..00 01 505 050.01 01 005 .1 00.10.1>0 0. 0N0.N0 01 000 100.01 01 0N0 000.1. 500 0. 01: 01 000 00N 5 00N 0 01: 01 000 00N 0 .01 0 01: 51 000 010 0 .00 0 012 01 000 0.0 0 001 0 012 01 000 010 0 000 0 01: 11 000 105 0 0N0.0 01: 01 000 05. 5 000.0 01: N1 000 000 0 005.0 01: .1 000 00N 0 001.N 01: 01 000 .05 51 00N 50 01 00N.10 50. N1 000 N0. .1 NNO 01 0000000 2. 0.0 0 00.0.0000 0 51 NON.0 01 .0N 0 ,0 N1 500.0 .1 52. 0 00010.0 ..0 00.01 20.0000x0 N1 51 05N.01 01 000 01 05 N1 N00 .0 .1 NNO 0. .002. :32! ...-.3 2. 33.05.: ..1 51 0N0.01 01 .00.11 .05 N1 001 N0 .1 505.5. 0000 00N.0 I 000.0. .1 .000000021 0NN 1. 500 0. 01: 01 000 .00 5 00N 0 01: 01 000 010.0 .01 0 01: 51 000 000 0 .00 0 012 01 000 000 0 001 0 01: 01 000 505 0 000 0 01: 11 000 500 0 0N0 0 01: 01 000 105.5 000 0 01: N1 000 150.0 005 0 012 .1 000 000 0 001.N 012 01 000 .15 51 0N0 .0 01 .00.00 .... ~. n~o..~. .. ~o~.o. uwucawo z. a.u.: awpo.au¢a .o s. ~q~.o o. .¢~.o .0 N1 0.0.0 .1 55. 0 0001000 ..0 00.01 20.000020 .01 51 0.0.01 01 000.11 .00 N1 000.00 .1 010.5. .0010 80.23 .500.0. .1 .000000021 .00 51 00..01 01 0.1.01 .00 N1 000.00 .1 N.0.5. 80..01 :0..30 02.. 00.0 00..0 I 500.0. .1 .000.0.021 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 144 «an ace. acm. 00¢. ovv. -O. ape coo. avm. smo. 000. 3.3..33 88 v~-m~& 8.. 393 38:23:80 - and 03:. .00v .vop av 11 av QC av vv av vv av 1' av vv mvp QNN cap. 00N. Ovv 0pm 00'. aNN. po— aOv .na .00p .00 can. ~u .oa ova. pow .vu .co vmv. Gap. 0.0. Gas. Noe. ~v~. poo. Gun. owN. ~00. 0v 0v 0' av av av «90. 0.6. 00v. 000. ”up. asp. Nan. ¢n~. 0N.. avo. 0v 0v 0v 0v 0v av Nc-v-OC-Oc-Qpn tan-Ov-Ov-Ov-n ' p van. pap. pao. Goa. Gav. vna. pwn. whp. O~o. ano‘ -O.¢s onu.m~ vop.o Do—.c noo.ov aao.~v Nva. new. woa. one. v00. Ova. Nah. «ON. 00‘. 00°. 000.00 oa~.- ov—.o an«.o 0.0.an 900,—v p aac. Oon. pov. aao. won. one. awn. can. ens. mav‘ xv ona.oop Nv vn—.~h CD'U’OGDOF~QF~G OUOGDOflifllimdflc “(2 at: «(I «(I m<2 m<2 mCZ «(I m<2 ”(2 av pv 0v 0v hv 0v av vv 0v ~v '1 av man man man man mna man man man man man oaa.anw ONv-~v mwNCGwG 2. a.w.: Omp0.0w¢a sq o-.o. «v no..o 0v pv ~0N.° pvw.o ow¢w «. .oou «. .ooa aaa .aa aaa. aa .av aa .va aaa. va av av av av av av 33.5 38:33:25 - and 03:. -o.o ~oo.c. m¢z a. was ~o~.o can a a¢z a. one v.o.~ can o «<: s. man n,o a can a «<2 my man nn~.o .on.o a¢z a. was vaa.. coo m aa a. aaa. vaa. .aa aa .av pa aa aa vpa.a aaa.a. aaa.a. ava.a. paa.a apv.a aaa.a- aa- a. vv. a. aaa.a av av av av av av av av aaa.a aaa a. av: av aaa aaa a aaa.a av: av aaa aaa a aaa.a av: av aaa ava a aaa.a av: av aaa aaa a vaa.a av: av aaa aaa.v aaa a av: vv aaa aaa.a aaa.a av: av aaa aaa a aav a av: av aaa vaa.a aaa a av: pv aaa paa a. paa a av: av aaa vaa.aa av uvv.va av vaa aa ava av av aaa.av pv a—a a aaaaaaa Z. a.w.a Oahu-aaaa aap.a av aaa.a av aa..a aaa.a av paa a pv ava a awcvaao pan awauv :O.a:awa aaa.aa av avv.aa av aaa.aa aaa.aa av pav aa pv aaa a aav. ava. .Ocvfi :O.:3 .....0— av Aaaavaaazv aaa.aa av aav.aa av av aav.va pv aaa a va av zo._0¢ :o_.:a uz.a mags aaa.a u ....op _< .oua.udaz¢ av av av av av av av av av av av av av av av av 153 22:33 88 3.3"— .8 39.5 35:23:80 - 8.0 03.5. aav.a aaa.aa aaa.a a-a.a aav.a aaa.aa aaa.a aaa.aa aaa.a aaa.a aaa.a aaa.aa av vv av vv av vv av vv av vv av vv aaa.av— aaa.aa aaa.a aa—.a aaa.aa aaa.av aaa.aa— aaa.aa aaa.a aav.a vaa.va aaa.av avp. aaa. aaa. ava. aaa. aap. aaa. ava. aaa. aaa. av av av av av av ava. vap. vva. paa. aaa. aaa. vaa. aaa. aaa. aap. av av av av av av OOOOPOOVON a v- p p OOCOPOOQO I I aaa.a- aaa.av av: av aaa aaa.a aaa a av: av aaa pav.a «av.a av: av aaa vaa.a aaa.a av: av aaa aaa a aav.a av: av aaa aaa.a aaa a av: vv aaa aaa.a apa.a av: av aaa aaa.a paa.a av: av aaa avp.a aaa a av: pv aaa vav.a- aav a av: av aaa aav.aa— av aaa.aa av aav.aa aaa.aa av aaa.aa pv aaa.aa amwaaaa :. a.a.a aabo_aa¢a aa—.a av aaa.a av aav.a va—.a av avp.a pv aap.a Guavaaa aaa ampuv :O.a¢30xa aaa.—a av aaa av av aav av vaa.vv av aaa.vv pv aa—.a on~.o u ....o_ p< .auovuacz< aaa.a. aaa.a. av: av aaa aav.a aaa.a av: av aaa vva.a aav.a av: av aaa aaa.a paa.a av: av aaa aaa.a aav.a av: av aaa aaa.v aaa.a av: vv aaa aaa.a aaa.a av: av aaa aaa.a vaa.a av: av aaa vva.a aaa.a av: «v aaa aaa.a. aav.a av: av aaa aaa.vap av aaa.aa av apa.va aaa.va av aaa.aa pv aaa.va aaaaaaa :. a.a.a aaa0.aaaa aav.a av aaa.a av aaa.a av—.a av pvp.a pv pap.a amavaaa aaa auauv :O.a¢anu aaa.aa av aaa.av av aaa av aaa.av av vaa.av pv aaa.p ¢-.o a ....op p< .cma.u4az< av av av av av av av av av av av av 154 aaa. aaa. aaa. aaa. aaa. aaa. aaa. ava. aaa. apa. aaa. pva. 33:83 88 8-3m 8a 35.5 38:33:25 - 5.0 03:. av vv av vv av vv av vv av vv av vv ava.avp aaa.aa aaa.a aaa.a vaa.va aaa.av aaa.av, aaa.va aaa.a aaa.a awa.pa vaa.vv vaa. aaa. aaa. aaa. aaa. aav. paa. aaa. aaa. aaa. av av av av av av «av. aaa. aaa. aaa. aaa. aaa. aaa. aaa. aaa. aaa. av av av av av av OOGOPOQQOV COOOPOO'OP p I P aaa. aaa. aaa. aaa. ava. vaa. aaa. ava. pvp. aaa. apa.aap aaa.aa aav.a aav.a aaa.aa aaa.av aav. aaa. aaa. vaa. aaa. aaa. aaa. aaa. apa. aaa. aav.aap aaa.aa aav.a aav.a aaa.—a aaa.av MDOQGDDCOF VDDOQODOQ" aaa.a. ma vaa.aa aaa.aa aaa aa— aav aaa. aaa. aaa. ava. aaa. pav. vaa. av av av av av av av av .ap .a. .v. NGOO'OO I aaa. aav. aaa. ava. vaa. “pa. aaa. aaa. aaa. aap. aaa.aa. aaa.aa vva.a pau.a aaa.aa aaa.aa 0000.00.00 aaa.a. av: av aaa aaa a av: av aaa aav.a av: av aaa paa.a av: av aaa aav a av: av aaa aaa a av: vv aaa aaa.a av: av aaa paa.a av: av aaa aaa a av: pv aaa aav.a av: av aaa av aaa.pa av pva.aa av apa.aa av paa‘a— aaacaaa :- a.a.a awp0.aw¢a av aaa.a av aaa.a av pap.a pv aa—.a awavuao aaa Caauv 3°.a830-w av va-.pv av vaa.aa av aaa.aa pv aaa.v a. .OOva :0.:3 .5...ua av .Gwaauaozv aaa.aa ava.aa av aaa.av av aaa aa av ava.aa pv aaa.a 8°.p0v :Oa.la a:.a aaaa aav.a I bIO.o~ av .aaaawaazv av av av av av av av av av av av av av av av av 156 apv. aaa. aaa. aap. apv. aaa. aaa. ava. aaa. aap. aaa. vaa. 3:823 88 aaa-aa ..8 33.5 35:23::5 - and 03a... av vv av vv av vv av vv av vv av vv aaa. aaa. aaa. aap. aaa. .av aaa aaa aav aap. aav. aaa. aaa. app aa aa .aap .aa aa av aap. aaa.. aav.a aaa.a aaa.a .a a av av av av apa. paa. vva. apa. paa. paa. aaa. aaa.a. vaa.a- vaa.a av av OOOON". I av av av av aaa. aaa. a—a. apa. aap. app. aaa. aav. aaa. aaa. aaa.aa aaa.aa aav.a avw.a aaa.aa vaa.av aaa. paa. ava. a—a. aap. aap. apa. aaa. vaa. aaa. aav.va vva.aa avp.a aa—.a aaa.aa aav.av “OON'UONON FUONQO0NCQ pva.ap av: av aaa aaa.a av: av aaa aav.a av: av aaa aaa.a av: av aaa aav a av: av aaa aaa.a av: vv aaa aaa.a av: av aaa aaa.a av: av aaa aaa.a av: pv aaa . aaa a av: av aaa av aaa.aa av aav.aa av a—a.aa pv aaa.aa aaaaawa 2. a.w.a aura-Oman av aaa.a av aav.a av av-.a wv aav.a awavwau aaa aaamv :O.a¢30xa av aaa.av av aaa.av av aav.av pv aaa.aa aav.a I sll.ua av .awaawaazv pva a— av: av aaa aaa.a av: av aaa aav.a av: av aaa aaa.a av: av aaa aav.a av: av aaa aaa.a av: vv aaa .aa.a av: av aaa aaa.a av: av aaa vaa.a av: wv aaa . aaa.a av: av aaa av avv.va av aaa.—a av aaa.aa pv apa.pa aawaaaa :- a.w.a awa0_aa¢a av aa—.a av aav.a av aa—.a pv aav.a awavudo aaa aaauv :O.a8302m av aaa av av ppa.av av aav.av pv pva.aa aa—.a I 500.0» av .Oaavwaazv av av av av av av av av av av av av 157 apa. aaa. aaa. .aa. awa. aaa. aaa. apa. aaa. aaa. aaa. aav. 23.383 88 No.5”: ..8 33.5 Ecotfizasoo - 5.0 03:. av vv av vv av vv av vv av vv av vv aaa. aaa. aaa. aaa. ava apv. aaa vva aaa. aaa. aap aaa aap pa pv .aa .aa .aa .av a a a aaa. aav. aap. aaa. aaa aaa. ava. a—a. vaa. av av av av av av aav. aaa. apa. aav. aaa. aaa. aaa. a—a. paa. aav. av av av av av av NOOOOOONO N°°°°°.°N°O aaa. aap. aaa. aaa. av- vap. aaa. aav. apa. vaa. aaa.va aaaVaa aaa.a a—a.a aaa.av vaa.—v aaa. aaa. apa. aaa. aap. aaa. aaa. a—a. aaa. .va. vaa.aa paa.pa ava.a aav.a aaa.av a—a.av oaaavaaaan pva.a— av: av aaa aaa.a av: av aaa aav.a av: av aaa aaa.a av: av aaa aav.a av: av aaa aaa a av: vv aaa aaa.a av: av aaa aaa.a av: av aaa aaa.a av: pv aaa - aaa a av: av aaa av aav.aa av aaa.aa av aaa va av pap.aa aaacaaa 2. a.a.: aa-O.aw¢a av aaa.a av aaa a av aaa.a pv ava.a aaavada aaa :wauv :O.a¢=02a av vaa.av av vaa.av av aaa.av pv aaa.aa o...o . ....up p< .oma.wdoz< a pva.ap av: av aaa a aaa.a av: av aaa a aav.a av: av aaa a aaa.a av: av aaa a aav.a av: av aaa v aaa.a av: vv aaa a paa.a av: av aaa a aaa.a av: av aaa a .aa.a av: «v aaa a aaa.a av: av aaa av paa.aa av awa.aa av aaa.aa pv aaa.av aaaaaaa 2. a.a.: aaa0.aa¢a av aaa.a av aaa.a av aap.a wv aaa.a aaavaao paw :a—mv :O.a¢30xa av vaa.av av ava.pv av apa.av pv aaa.—a oo..o - .oo.op p< .omo.mdoz< av av av av av av av av av av av av 158 aaa. aaa. aaa. .aa. aaa. aaa. aaa. .aa. aaa. .av, aaa. apa. aaa. ava. Cue-uh. .3 88 8.5". 3.. 53.5 35:22:50 - cad 03:. av— av vv av vv av vv av vv av vv av vv av vv aav.aa- aaa.aa. ava a ava.a ava.aa aaa aa av Oaavaav>a aaa.aa vvv.aa aaa. aaa .aa. aav. aaa. vap. aaa. vaa. aaa. ava. av av av av av av av av “NNGN’ONOC vaa vva aaa. aaa apa ava. aa. .aa vv.. .aa aaa. aaa. aaa aaa‘ ava, .va— .aa— —a va MGQORIOON.’ .va. aaa, aav aaa aav: aaa. .aa aaa paa. aaa: 01000000006 av p-a va. av aaa aa— av: av: av: av: av: av: av: av: av: av: av pv av aaa av aaa av aaa av aaa av aaa vv aaa av aaa av aaa pv aaa av aaa aaaaaap aaa‘v—p awaaawa :. a.w.: cape-Oman av ava a av .aa a av pv ava a aaa a aaavaao aaa capav :O.a¢:oxa av vaa aa av aaa va av pv aaa.va aaa.aa .004a 20.x! ...-.0. .4 .owo.waoz¢ aa .aa av aaa.aa av vva.aa av av avv.aa a—a.aa :O.p0v :Oa.3a a:.a aaaa aaa.a I 5.0.0— av Aaaavaaazv vaa.aap aaa.vv- aaa.a aaa.a aaa.va aaa.va —aa. ava. aaa. aaa. aaa. aaa, aap. vaa. aaa. aaa. av av av av av av 0"ON'ON0. vaa vav aaa. .aa. vav aaa bv aaavaavau a. aaa.aa aaa.aa av av vav ava Ovv. avp. aaa. aaa. aaa. aaa. aaa. aaa. aaa. aaa .aa- .aa— .aa .aa GOV-MOO...- pva a. aaa aav aaa. aav aaa. .aa. aaa paa aaa NOCCIDODO. av aaa.av- av ava.aa— av: av: av: av: av: av: av: av: av: av: av wv av aaa av aaa av aaa av aaa av aaa vv aaa av aaa av aaa «v aaa av aaa aaaaavp aaa.aa— aaaaaaa :- a.a.: aa—0.aa¢a av ao¢.o «q o~o.o av pv Guavaao aaa :apav av apa.va av aaa.aa av wv aaa.a aaa‘a 2°.a830xa vaa.aa aaa.aa .Oova :O.:3 .sla.op av .Gaaaaaazv .va .aa av aaa aa av paa.aa av pv avv.aa .aa.aa 8°.a0v :Oa.3a a:.a aaaa aaa.a I .0..0a av .aaaaaaazv av av av av av av av av av av av av av av av av REFERENCES References 1. J. Zaborszky, G. Huang, B. Zheng, and T.C. Leung, “New Results on Stability Monitoring on the Large Electric Power System,” Proceedings of the 28th Conference on Decision and Control, pp. 30-40, Dec. 1987. I-I.D. Chiang, F.F. Wu, and PP. Varaiya, “Foundations of Direct Methods for Power System Transient Stability Analysis,” IEEE Transactions on Circuits and Systems, vol. GAS-34, no. 2, pp. 160-173, Feb. 1987. F.M.A. Salam, A. Arapostathis, and P. Varaiya, “Analytic Expressions for the Unstable Manifold at Equilibrium Points in ynamical Systems of Differential Equation Time ,” Proc. 22rd IEEE Conf. Decision Control (Las Vegas, NV) , pp. 1389-1392, 1983. F.M.A. Salam, “Power System Transient Stability: The Critical Clearing Time,” Proc. 23rd IEEE Conf. Decision Control (Las Vegas, NV), pp. 179-184, Dec. 1984. J. Zaborszky, G. Huang, B. Zheng, and T.C. Lueng, “On the Phase Portait of a Class of Large Nonlinear Dynamic Systems such as the Power System,” IEEE Transaction on Automatic Control, vol. 33, no. 1, pp. 4-15, Jan. 1988. N. Tsolas, A. Arapostathis, and RP. Varaiya, “A Structure Preserving Energy Function for Power System Transient Stability Analysis,” IEEE Transactions on Circuits and Systems, vol. GAS-32, no. 10, pp. 1041-1049, Oct. 1985. H.D. Chiang, M.W. Hirsch, and RF. Wu, “Stability Regions of Nonlinear Auto- nomous Dynamical Systems,” IEEE Transaction on Automatic Control, vol. 33, no. 1, pp. 16-26, Jan. 1988. MA. Pai, in Power System Stability, vol. 3, North-Holland. A.A. Fouad and SE. Stanton, “Transient Stability of a Multimachine Power Sys- tem; Part 1: Investigation of System Trajectories,” IEEE Transactions on Power 159 10. 11. 12. 13. 14. 15. 16. 17. 18. 160 Apparatus and Systems, vol. PAS-100, no. 7, pp. 3408-3416, Jul. 1981. Q]. Tavora and O.J.M. Smith, Stability Analysis of Power Systems, pp. 1138- 1144, 1971. RS. Prabhakara and AH. El-Abiad, “A Simplified Determination of Transient Stability Regions for Lyapunov Methods,” IEEE TransaCtions on Power Apparatus and Systems, vol. PAS-94, no. 2, pp. 672-689, Man/Apr. 1975. T. Athay, R. Podmore, and S. Virmani, “A Practical Method for the Direct Analysis of Transient Stability,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, no. 2, pp. 573-584, Mar/Apr. 1979. N. Kakimoto, Y. Ohnogi, H. Matsuda. and H. Shibuya, “Transient Stability Analysis of Large Power System by Lyapunov’s Direct Method ,” IEEE Transac- tions on Power Apparatus and Systems, vol. PAS-103, no. 1, pp. 106-167, Jul. 1984. K.D. Demaree, M.A. Pai, and P.W. Sauer , “Trajectory Approximation and Itera- tive PEBS Method in Direct Stability Analysis,” Proceedings of the 1982 Midwest Power Symposium, Nov. 1982. AN. Michel, A.A. Fouad, and V. Vittal, “Power System Transient Stability Using Individual Machine Energy Function,” IEEE Transactions on Circuits and Systems, vol. CAS-BO, no. 5, pp. 266-276, May. 1983. P. Rastgoufard, A. Yazdankhah, and RA. Schlueter, “Multi-machine Equal Area Based Power System Transient Stability. Measure,” IEEE Transactions on Power Systems, vol. 3, no. 1, pp. 188-196. P. Rastgoufard and RA. Schlueter, “Application of Critical Machine Energy Function in Power System Transient Stability Analysis,” Electric Machines and Power Systems. P. Rastgoufard, “Local Energy Function Methods for Power System Transient 19. 20. 21. 22. 23. 24. 25. 26. 161 Stability,” Ph.D. Thesis submitted to Michigan State University, Dec. 1982. A.A. Fouad and SE. Stanton, “Transient Stability of a Multimachine Power Sys- tem; Part 2: Critical Transient Energy,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 7, pp. 3417-3424, Jul. 1981. Stewart E. Stanton, “Transient Stability Monitoring for Electric Power Systems Using A Partial Energy Function,” 'IEEE Winter Power Meeting, 1989. DJ. Hill and AR. Bergen, “Stability Analysis of Multimachine Power Network with Linear Frequency Dependent Loads,” IEEE Transactions on Circuits and Systems, vol. GAS-29, no. 12, pp. 840-848, Dec. 1982. G. Sigari, “Lyapunov Stability Based Network Conditions for Characterizing Retention and Loss of Power System Transient Stability,” Ph.D. Thesis, submit- ted to Michigan State University, 1985. H. Yee and ED. Spalding, “Transient Stability Analysis of Multimachine Power Systems by the Method of Hyperplanes,” IEEE Transactions on Power ' Apparatus and Systems, vol. PAS-96, no. 1, pp. 276-284, Jan/Feb. 1977. R. Podmore and A. Germond, “Development of Dynamic Equivalents for Tran- sient Stability Studies,” in EPRI Report, vol. 1, Nov. 1977. A. Yazdankhah, “Power System Security Assessment for Faults Using Direct Methods,” Ph. D. Thesis, Michigan State University, June, 1984. J.H. Chow, “Natural Modes and Their Stability in Power Systems,” Preceeding of 24th Conference on Decision and Control, pp. 799-803, Dec. 1985. "rill/71711111111111.1111?