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Application of catastrophe theory to transient stability analysis of multimachine power systems Parsi-Feraidoonian, Raiomand 1990

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Application of Catastrophe Theory to Transient Stability Analysis of Multimachine Power Systems Raiomand Parsi-Feraidoonian B .A .Sc . (Hons . ) , Southern Il l inois University U . S . A . , 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1990 © Raiomand Parsi-Feraidoonian, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department Date DE-6 (2/88) Abstract Transient stability analysis is an important part of power planning and operation. For large power systems, such analysis is very time consuming and expensive. Therefore, an on-line transient stability assessment will be required as these large power systems are operated close to their maximum limits. In this thesis swallowtail catastrophe is used to determine the transient stability regions. The bifurcation set represents the transient stability region in terms of power system transient parameters bounded by the transient stability limits. The system modelling is generalized in such, that the analysis could handle either one or any number of critical machines. This generalized model is then tested on a three-machine as well as a seven-machine system. The results of the stability analysis done with the generalized method is compared with the time solution and the results were satisfactory. The transient stability regions determined are valid for any changes in loading conditions and fault location. This method is a good candidate for on-line assessment of transient stability of power systems. ii Table of Contents Abstract ii List of Tables iv List of Figures v Acknowledgments vi 1 Introduction 1 2 Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory 6 2.1 Introduction 6 2.2 Catastrophe Theory Applied to Single Machine Infinite Bus System 7 2.3 General Dynamic Equivalent Method for Multimachine Power Systems 10 2.4 Application of Catastrophe Theory to Multimachine Power Systems 16 2.5 Identification of the Critical Machines 22 3 Numerical Examples 23 3.1 The Three-Machine System . . 23 3.2 The Seven-Machine System 31 4 Conclusion 37 References 39 Appendix Fortran Program Listing of Swallowtail Catastrophe Applied to Multimachine Power Systems 41 iii List of Tables 1.1 The seven elementary catastrophes 5 3.2 Network parameters of the three-machine system 24 3.3 Three-machine generator data and operating conditions 25 3.4 Cases of faulted generating buses 27 3.5 Cases of non-generating buses 28 3.6 CIGRE 7-machine bus data 31 3.7 CIGRE 7 - machine branch data 32 3.8 CIGRE 7 - machine voltage and power summary 34 3.9 Cases of CIGRE 7-machine grounded buses 35 List of Figures 2.1 Single-machine Infinite-bus power system 7 2.2 The energy function stabihty criteria a : Stable b : Critically Stable c : Unstable .... 8 2.3 The transient stabihty limits given by the swallowtail catastrophe for a multimachine power system 21 3.4 Nine-bus three-machine power system 25 3.5 Three-machine system load flow showing prefault conditions ; all flows in MW and MVAR 26 3.6 Results for the 3-machine power system with faulted generator buses . All cases are stable 29 3.7 Results for the 3-machine power system with faulted load buses. All cases are stable. . 30 3.8 CIGRE 7 - machine test system 33 3.9 Results for the CIGRE test system. All cases except for faults at buses 5, 9 and 10 are stable 36 v Acknowledgments I would l ike to thank Dr. M . D . Wvong for his continual encouragement, invaluable guidance throughout the course of this thesis and most of al l as being so understanding, listening and guiding me during difficult periods. I would also l ike to thank my parents Feraidoon and Homai for their incredible patience, my brother Hootash for his constant encouragement and my aunts Parvaneh, Go l l y and Kianoosh for having faith in me. vi Chapter 1: Introduction Chapter 1 Introduction Instability in electric power systems, leading to loss of system synchronization, is a very sensitive problem for power utility engineers. In assessing power system stability there are two separate criteria to be considered, v i z : • Steady-state stability, for small perturbations, i.e., leading effectively to linear system analysis. • Transient stability, for large system disturbances and involving non-linear system analysis. The stability problem of power systems became very important fol lowing the famous power blackout in north eastern U .S .A . in 1965 . Planning, operations and control procedures of power systems had to be revised to ensure secure and reliable operation of power systems. Considerable research effort has gone into the stability investigation both for off-line and on-line purposes [1] . A stable power system implies that all its interconnected generators are operating in synchronism with the network and with each other. These generators start to oscillate when a disturbance occurs due to a transmission fault or switching operatioa Loss of synchronism must be prevented or controlled because it has a disturbing effect on voltages, frequency and power, and it may cause serious damage to generators, which are the most expensive components in a power system [2]. The generators which are losing synchronism due to the disturbance should be tripped, i.e. disconnected from the system before any serious damage occurs, and afterwards brought back to synchronism. Loss o f synchronism may also cause some protective relays to operate falsely and trip the circuit breakers of unfaulted lines. In such cases the problem is very complicated and may result in more generators losing synchronism. Therefore, an understanding of system stability requires a thorough knowledge of both the mathematical modell ing of the system and effective numerical techniques. In most cases, the model 1 Chapter 1: Introduction consists of a set of linear or non-linear algebraic and/or differential equations depending upon the type of study that is to be performed. It is important to select a numerical method which will provide accurate results, but the rapid growth of power systems makes it extremely difficult, expensive and time consuming to carry out these careful and detailed stabihty studies through solution of the system equations. Two possibilities for improving the speed of transient studies are : [3] • Reduction of the total system to a smaller one, that could be solved faster but has the disadvantage of inaccuracy because of approximations. • Improvement in the numerical solution techniques such as the trapezoidal rule of integration method which has already been used successfuhy for the solution of switching transients at the Bonneville Power Administration ( BPA ) [4] . Much work has been done to find ways to reduce the amount of computation required for stability studies and to find direct methods to solve the transient stability problem that do not require the solution of the system equations . One direct method used is Lyapunov's direct method of determining stability [5]. However, this method is still not suitable for on-line applications [6] and has practical difficulties such as : • The method is conservative and, although some stable points are readily identified, others are inconclusive. • This method has difficulty deducing the Lyapunov's function which defines all possible stability regions, and it cannot predict instability, therefore, it may produce false alarms. Other Direct methods of stabihty analysis are currently under consideration and investigation [7] . Catastrophe theory is a new way of thinking about changes such as in a course of events, a systems behavior, or even change in ideas themselves. Its name suggests disaster, and indeed the theory can be applied to literal catastrophes. The mathematical principles we are used to are ideally suited to analyse smooth, continuous, qualitative change, [8] but there is another kind of change, 2 Chapter 1: Introduction that is less suited to mathematical analysis : such as the discontinuous transition from ice at its melt ing point to water at its freezing point or the transition from stable to unstable state for a power system fol lowing a disturbance. The foundations of catastrophe theory were developed by the French mathematician Rene'Thorn and became widely known through his book Stabilite' Structurelle et Morphogenese in which he proposed them as a foundation for biology. A catastrophe, in the very broad sense Thom gives to the word, is any discontinuous transition that occurs when a system can have more than one stable state, or can fol low more than one stable pathway of change. The catastrophe is the jump from one state or pathway to another [9] . The elementary catastrophes are the seven simplest ways such a transaction from one state to another state can occur. This is true for any system governed by a potential, and in which the behavior of the system is determined by no more than four different factors, then only seven qualitatively different types of discontinuity are possible [10] . The qualitative type of any stable discontinuity does not depend on the specific nature of the potential involved, merely on its existence, i.e. on the existence o f cause-and-effect relationship between conditions. N o w we can see how the elementary catastrophes are comparable to the regular forms of classical geometry. Just as we can say that any three dimensional object, i f it is regular ( i .e . al l its faces are identical polygons ), must be one of the five solids, so the catastrophe theory asserts that any discontinuous process whose behavior can be described by a graph in as many as six dimensions, i f structurally stable, must correspond to one of the seven elementary catastrophes [11] . The seven elementary catastrophes is shown in Table 1.1. Catastrophe theory is used for stability analysis of multimachine power systems. This theory has been applied previously to the steady-state stability o f power systems, one-machine infinite bus system, as wel l as multimachine systems [12] with the worst case approach i.e., only one generator becoming crit ical for a three-phase fault. Catastrophe theory has also been used as a tool for determining synchronous power system dynamic stability [13]. The application of catastrophe theory to steady-state and transient stability of power systems is attractive because it provides comprehensive 3 Chapter 1: Introduction stability regions with minimal computation. The transient stability regions have been shown to be applicable for changes in loading conditions and fault locations [12] . This thesis generalizes the use of catastrophe theory to the case of multimachine power systems, with more than one machine being critical (likely to go unstable ). Chapter 2 of this thesis derives the mathematical ideas involved in reducing the multimachine power system and the application of catastrophe theory by first briefly going through the single machine infinite bus system. In deriving the equations for the multimachine system, the general dynamic equivalent approach is used, grouping all the critical generators as one equivalent machine and grouping the rest of the system as another single equivalent machine. Then the swallowtail catastrophe is applied to this general system. Chapter 3 contains test results from two power systems, and the conclusion of the application of catastrophe theory to multimachine power systems is in Chapter 4. The program listing for the catastrophe application is given in Appendix. 4 Chapter 1: Introduction Catastrophe Control Dimensions Catastrophe Mani fo ld Fo ld 1 x 2 + u Cusp 2 X 3 + UX + V Swallowtai l 3 X4 + ux2 + vx + w Butterfly 4 6x° + ux* + vx2 + wx + r El l ip t ic 3 3x2 — 3j/2 + 2ux + v — 6x3/ + 2uy + w Hyperbol ic 3 3x2 + «j/ + v 3t/2 + ux + w Parabolic 4 2xy + 2ux + w 4y* + x 2 + 2vy + r Table 1.1: The seven elementary catastrophes 5 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Chapter 2 Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory 2.1 Introduction The transient stability analysis of multimachine power systems is more complicated than that of a single machine infinite bus system because the behavior of each machine is effected by and has an effect on the behavior of al l the other machines coupled to it. During a large system disturbance, usually there are two switching done, one during the occurrence of the fault and the other at the time o f clearance of the fault. For transient stability analysis the fol lowing assumptions are made : • Each generator, i is modelled by a constant voltage, | E,; |, behind its direct axis transient reactance, x'a. • Turbine dynamics are ignored so that the mechanical power input to each generator, P m t , is assumed constant. • Mechanical damping is ignored. • The loads are modelled as constant impedances. In this chapter we briefly review the application of catastrophe theory to a single machine infinite bus system [14] . Then we apply the General Dynamic Equivalent Method [15] [16]to multimachine power systems such that a suitable energy function can be defined for the application of catastrophe theory. This general method requires the identification of the crit ical machines involved for each fault being considered. The group of critical machines is then replaced by an equivalent machine and the rest o f the system which is not significantly affected by the disturbance is also replaced by an equivalent machine. 6 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory The transient stabihty regions are found by use of catastrophe theory in terms of system parameters. 2.2 Catastrophe Theory Applied to Single Machine Infinite Bus System Consider the one machine-infinite bus system [17]in Figure 2.1 which has two transmission lines. \ \ \ Figure 2.1: Single-machine Infinite-bus power system The swing equation is given by where = Pi - Pmax sin = P* M = inertia constant of machine Pe = electrical power output Pi = mechanical power input Pa = accelerating power ip = rotor angle of the machine Pmax = maximum power for post fault condition (2.1) (2.2) 7 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory If a three-phase fault occurs on one of the transmission lines near the generator bus, the rotor w i l l start to accelerate and hence the machine would gain kinetic energy. If the fault is cleared at a clearing time such that the kinetic energy produced by the fault is absorbed by the potential energy produced after the clearance of the fault and the gained energy is less than zero then the system is stable and, i f exactly zero the system is crit ically stable. This is shown in figure 2.2 > > > (c) t i m e ( s ) a : Stable b : Critically Stable c : Unstable 8 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Consider the crit ical clearing case for a three-phase fault initiated near the generator bus. Then kinetic energy = potential energy (2.3) Equation 2.3 can be derived by mult iplying equation 2.1 by and integrating with respect to time once between fa, fa and next between fa, ipm to obtain ^Mtf = Pm COS fa + Pi fa - Pi fa - Pm cos Vm (2-4) where fa = critical clearing angle Pm — maximum power of post-fault network (2.5) ipm = maximum angle Using Taylor series expansion to approximate fa and fa as a function of time we get = ltc and 1 where 7 = acceleration at instant of occurrence of fault Pi - Pe(*0+)] ~ M Replacing cos fa in equation (2.4) by cosine series expansion and denning we obtain Pm 4 Pm i 3 . /2 — Ipf + - t i ^ O ~Pm- Pith + k) = 0 (2.6) i>c = fa + ^Itl (2.7) (2.8) x £ ± 7 * c 2 (2.9) k = PiTPm + Pm cos ipm (2.10) 24 6 r v 4 + {M1 + pmfa - - Pi) * ( 2 - n ) 9 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory For the above equation to be in the form of swallowtail catastrophe manifold divide the equation by and eliminate the cubic term, by setting x = y-rpo (2.12) Therefore, y - 12s/2 + 24 Pi - M 7 y + 24 — ( M 7 V 0 - k) + 24 •* m = 0 (2.13) This is in the form of the standard swallowtail catastrophe manifold namely: j / 4 + uy2 + vy + w = 0 (2.14) where u = -12 24 v = w — Pm 24 Pi - M-/ Mjtpr, - k (2.15) + 24 The bifurcation set can then be defined by 4y3 + 2uy + t; = 0 (2.16) and the transient stability region in terms of the power system parameters then takes the shape of the swallowtail bifurcation set. The region is defined by the above u, v and w parameters. 23 General Dynamic Equivalent Method for Multimachine Power Systems The general dynamic equivalent method [15] [16]will be discussed in this section. The equation of motion of machine i in a multimachine power system using classical model representation is given by 10 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Si = u>i i = 1, . . . . ,n MiSi = Pmi ~ Pei Pei = electrical power output of machines n = E [A j c o s % + Cij sin 6^ Pmi — mechanical power input Mi = inertia constant Ei = internal generator voltage Ui = rotor speed bij = transfer susceptance 5,j = transfer conductance = rotor angle = $ - *j Dij = EiEjQij Cij = EiEjbij (2.17) (2.18) (2.19) (2.20) The critical machines are those machines that tend to respond actively to the occurrence of the fault and may lose synchronism. Therefore, in order to determine the transient stability of the power system it would be sufficient to group these critical machines as one equivalent machine and study the response of this equivalent machine with respect to the undisturbed equivalent machine representing the rest of the system. Consider that A represents the critical machines for a specific three-phase fault. These machines are considered as one equivalent critical machine which oscillates against B, the rest of the power 11 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory system which is not significantly disturbed by the fault and also considered to be equivalent to one machine. Let k£A i€B where Mo and So are , respectively , the inertia constant and the angle of the centre of angle of the power system with the critical machines excluded. Let then But and similarly Vk = h - Sc Oi = 6i - #o V> = Sc- So M<kTA M»ttB 0 k£A " «€B «=EM-sE(«) keA ieB MkSk = Pmk - Pek n = Pmk - 2 [D'* c o s fa* + cki sin (nk - m) Mi Si = Pmi - Pei n = p™ - £ [ D i i c o s W ~ei) + casin(*< - 9i) (2.24) (2.25) (2.26) (2.27) J = 1 Substituting equation (2.26) and (2.27) in (2.25) and fol lowing some mathematical manipulation we obtain the swing equation of the critical machines against the rest of the system. 12 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory This is explained i n the following steps: Mci> = ]T \ Pmk - ^ \Dki cos (tj* -rfi) + Cki s in ( 7 7 * - 7 ? , ) ] keA K _ Mc Mo 1=1 E { P™ - E cos ~9i) + CH s i n ~ 9>)] ieB [ j=l separating the Dkk term from the rest o f the equation we obtain Mcxl> cos (j]k - rj,) + CM s in (% - rj,) keA K i=k n •> + ^2[Dkicos(rik-r],) + Cki sin(rjk-n,) \ Mc Mo E{P™ - E [DH c o s -*>) + C'i s i n ^ ~ °i) ieB jeA + E [Di> cos (*«' - 9i) + CH s i n _ 9i) grouping the appropriate terms together i.e. M, Mcip ieB ' Dki cos - r)i) + Cfc/ sin ( 7 7 * - 7 / , ) - EE[ - I F E E^cos - + a * s i n - M} 0 ieB jeB } -{EE [Dki cos fa* ~ *w) + ^ s i n fa* - rn) ^k£A l^k - w E E [ D ' i c o s ('•• -)+c«isin (ft - )]} Let P m ~ { E P M F C M „ E P m ' } KkeA " ieB J = \ E E \Dki cos fa* ~ + s i n fafc ~ _M Mo Y, E [^ o-cos (9i c a s i n ($i - ej)]} i€B j€B (2.28) (2.29) (2.30) (2.31) (2.32) 13 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Simplifying the Pc term K Un A P - ^ ^ - F E E [ A i cos (ft - Bi) (2.33) Therefore , Mcxi> = Pk - | ]jT E I"0*' c o s fa* ~ + Cw sin (f?jt - »») ^ E E [DH c o s (ft " ^) + ^ (ft - )] } (2.34) where we have defined Pk = Pm — Pc (2.35) Since ip = 6c- 60 rji = 6i - 6C therefore we can write Bk = Vk + i> Tji = ft - V Substituting equation (2.37) in equation (2.34) and rearranging : M, $ = PK~ { E E [DK c o s fa* ~ ft + VO + sin (7/fc - ft + V) /I . - X L kfc£.4 «yfc E E [ D k i c o s f a * - + Ckisinfa* - ft + vo]} Lr- A Ir- 13 ' k€A i^B(2.36) (2.37) (2.38) 14 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Expanding the sine and cosine terms of the above equation we obtain : ^ = flc-|EE D k i [COS (^  ~ v k ) c o s ^ + s i n ( f t - s i n ^ ^ E E A.' cos (ft - r}k) cos V> + sin (ft - sin tp Mo k€A ieB n - Y2 E ^ [SIN ~Vk)costp- cos ( f t - ?7fc) s in 0 ~ W E E ^ s i n ( f t ~ 7 ? f c) c o s ^ ~ c o s ( f t ~ Vk) s in V> } 0 k€A ieB ' (2.39) $ = P K" { { E E [AK s i n (*•' - + c o s (ft - Vk)] k£A i^k M. - y ^ E E iDki s i n (ft ~Vk)~ Cki cos (ft - r}k)) I sin V> fce-4 tgB {Wo E E [A.-COS (*«• -»/*)+SIN (*«• - %)] /te.4 «'eB n <. + 53 E S l n (ft ~ Vk) ~ Dki COS (ft - T)k)] | COS V> | Therefore, the swing equation of the single machine representing the group of critical machines has the form : Mci> = Pk-Tk s in (if> - ak) (2.41) where •k£A ieB - { E ^ - ^ E E ^ - S ^ - ^ ) ] } (2.42) Tk = y/al + b\ (2.43) 15 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory -I ak ak = tan — h ak = Mc Mo ]C YI [Dki cos " +^sin ~ + 2 E S i Q ^ ' ~ ~ A, C O S ~ (2.45) MO + YI E S l n (ft ~rlk) + Cki COS (ft - Tjk) k£A i?k 2.4 Application of Catastrophe Theory to Multimachine Power Systems (2.46) During the transient period an exchange of energy takes place between the rotor of the critical machines and the post-fault network [18] . The kinetic energy generated by the accelerating power during the fault-on period must be ful ly absorbed by the post-fault network in order to maintain stability. Us ing equation (2.41 ) from the previous section, namely Mk^k = Pk-Tk sin (V>* - ak) (2.47) which represents the motion of the group of critical machines represented as a single machine with respect to the rest of the system, also represented as a single machine, for a certain three-phase fault. Since we have assumed that the rest of the system is not responding to the disturbance, it is reasonable to use the pre-disturbance angles Oo and 770 to calculate the parameters Pk , Tk and a * . B y solving equation (2.47) for ipk, the stable and unstable points are computed i.e. Pk-Tksin(xli-ak) = 0 (2.48) 16 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory fa = stable equlibrium point (2.49) and the unstable equil ibrium point ( UEP ) is rk = * - rk (2.50) Mul t ip ly equation (2.47) by fa and integrate between ip^ and fa with respect to time we, obtain the kinetic energy generated by the fault : ^Mkfa K . E . (2.51) = Pi {it ~ - Tl [cos (fk - a() - cos {fa - a°k) where P / , 7 1 / and a { are the fault-on parameters and fa is the clearing angle. The potential energy of the post-fault system is derived in the same fashion but is integrated between fa and fa using the post-fault parameters -\Mkfak = P.E. = p m - n ) - n c o s (rk - <%) - c o s -(2.52) The L .H .S . of equation (2.52) represents the kinetic energy produced during the fault and the R .H .S . represents the potential energy of the post-fault network. In order for the system to be stable the kinetic energy should be equal to or less than the potential energy. Therefore, ±Mkfak - Plfak - Tl cos (fa -al)+ku = 0 (2.53) where we define ku = Ppkfa + Tpk cos (fa - apk) (2.54) 17 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Expanding xpi by a Taylor series and using the first two terms only (2.55) and V'fc = Iktc (2.56) where lk = Mk Pk - J*fc(«0+) (2.57) Replacing the cosine term by cosine series expansion up to the fourth order and defining (2.58) then after some mathematical manipulation, we get the catastrophe manifold equation as shown in the fo l lowing steps : \Mk{lktcf - PPK (V2 + x) - Tpk cos (V>2 + x - al) + ku = 0 Mklkx - PI +*) - n cos^ ( x + ^ ~ ^ 2! (2.59) 4! + ku = 0 M f c 7 f c x - p f c p ( ^ ° + x ) - ^ 1 -(x + /?)2 , (* + /?)' 2! + 4! + ku = 0 (2.60) Therefore, expanding and simplifying equation (2.60) we obtain .2 rpp rpp .tJL x4 _ ik_Q 3 + 24 6 P rpp rpp 2 4 ^ x 24~' + *" = 0 (2.61) 18 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory dividing equation (2.61) by - y r to give x 4 + (4/3) x a - 12 1 -24 n 24 Mklk + Tvkf3-Ppk-^-f (2.62) = 0 In order to obtain the swallowtail catastrophe manifold , we have to eliminate the third order term in equation (2.62), by letting : x = y-P (2.63) Therefore, ( y _ / ? ) 4 + 4 / 3 ( y - / 3 ) 3 - 1 2 24 1 - 0 2 -, (y - P? Mk7k + Tpp - Ppk - ^p*] (y - P) 24 rrP 1k and expanding we get ( y 4 - 4y 3 /? + 6 y 2 / ? 2 - 4y /? 3 + /? 4 ) + 4/? ( y 3 - 3y 2 /? + 3y/? 2 - 03) + (6/? 2 - 12) (y2-2yP + P*) '24 r Tp -\ lMklk + Tlt5-Pl--±p\{y-(S) = o = o rearranging the terms and simplifying the above equation y4 - 12y2+U {PI - Mklk ) y Tv v- k 24 + ^(PPK^K -ku- PIP + Mklkp) + 24j = 0 equation (2.66) is in the form of the swallowtail catastrophe manifold i.e : (2.64) (2.65) (2.66) y4 + uy3 + vy + w = 0 (2.67) 19 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory where u -12 w v Mklk -ku -Ppkf3 + Mklkp +24 (2.68) The control variables u , v , w for the swallowtail catastrophe obtained for the multimachine power system can be compared to the control variables of the single machine infinite bus system namely equation (2.15 ) . It is seen that the control variables have the same form and the equations derived for the multimachine power system reduces to that of the single machine infinite bus system when the number of the critical machines is one and the rest of the system is also one. The bifurcation manifold is reduced from three dimensions to only two dimensions in v and w as u = —12. The boundaries of the bifurcation set of Figure 2.3 represents the degenerate transient stability limits of the power system. It should be noted that for a generator the stable points are in the region of a positive v and w and for a motor the stable points lie between a negative v and a positive w. Several comments are in order here : • During a three-phase short-circuit of a generation bus the transfer admittances between machine k and other machines are zero i.e. gkj = bkj = 0 so Tk = 0 and the electric power output during fault-on period is found using equation 2.18 • When combining critical machines in an equivalent machine if the fault duration is short, the machine angle offsets will not change ; thus we may use the prefault steady-state values along with fault-on values for the b's and the g's to compute the fault-on parameters. 20 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory Figure 23: The transient stability limits given by the swallowtail catastrophe for a multimachine power system. 2 1 Chapter 2: Transient Stability Analysis of Multimachine Power Systems Using Catastrophe Theory 2.5 Identification of the Critical Machines The method presented depends upon the accurate identification of the crit ical machines for a specified disturbance. Correct identification could be achieved by calculating the unstable equilibrium points for al l machines in the power system; the machine having the highest unstable equil ibrium point would be identified as the crit ical machine [19] , but its drawback is the calculation of the unstable equil ibrium points, which is time consuming. In this thesis for a certain three-phase fault sequence occurring at either generator or non-generator buses, the crit ical machine (s) are identified as fol lows : • Calculate the init ial acceleration for each machine using Pmi ~ Pzi {IQ ) (2.69) where Pei ( t j ) is the electrical power output during fault at the instant of fault occurrence. The machines which have high and positive initial accelerations are injecting kinetic energy to the system; therefore, they al l contribute to the system instability and should be combined to form a single critical machine. In practice, only two, at times three, machines having the largest init ial acceleration w i l l be declared as candidates [15]. 22 Chapter 3: Numerical Examples Chapter 3 Numerical Examples In this chapter two test systems are presented to demonstrate the validity and advantages of the application of catastrophe theory to transient stabihty assessment of power systems. Three and seven machine power systems are used, where three-phase short circuits are considered at different locations. For each test system there w i l l be a one line diagram, the steady state loadflow and the systems data. Transient stabihty regions in terms of systems catastrophe control parameters are given for each example used and they are compared with the time solution. Each three-phase short-circuit case considered is evaluated by the fol lowing steps : • Construct the systems reduced prefault, during-fault and the post-fault matrices. • Identify the crit ical machine or machines for each case. • Calculate the general dynamic equivalent parameters i.e. Pk, Tk, cxk, i>k and M * . • Calculate the bifurcation set parameters for the swallowtail catastrophe. • Each case is then compared with the time solution. 3.1 The Three-Machine System This system has nine buses, three machines and three loads [20] . It is widely referred to in the literature as the Western Systems Coordinating Counci l ( W S C C ) test system. A one-line diagram for the system is given in Figure 3.4 . The prefault normal load flow is given in Figure 3.5. Transmission l ine parameters and loads impedances are given in per unit on a 100 M V A base in Table 3.2. Generator data and initial operating conditions are given in Table 3.3. Three-phase short circuits are considered at different locations. The transient stability of each fault location is evaluated by the use of the swallowtail catastrophe. 23 Chapter 3: Numerical Examples Bus No . Admittances (pu) G B Generators 1 1 • • 4 0.0 -8.446 2 2 • • 7 0.0 -5.485 3 3 • - 9 0.0 -4.168 Transmission Lines 4 - 5 1.365 -11.604 4 • 6 1.942 -10.511 5 • 7 1.188 -5.975 6 - 9 1.282 -5.588 7 • 8 1.617 -13.698 8 - 9 1.155 -9.784 Shunt Admittances Load A 5 - 0 1.261 -0.263 Load B 6 - 0 0.878 -0.035 Load C 8 - 0 0.969 -0.160 4 - 0 0.167 7 - 0 0.227 9 - 0 0.283 Table 3.2: Network parameters of the three-machine system 24 Chapter 3: Numerical Examples Generator Data Initial Operating Conditions Gen. No. H (Mw/MVA) < E (pu) <$o(deg) 1 23.64 0.0608 7.16 1.056 2.272 2 6.40 0.1198 1.63 1.050 19.732 3 3.01 0.1813 0.85 1.017 13.175 Table 33: Three-machine generator data and operating conditions 2 Load C ;i Figure 3.4: Nine-bus three-machine power system 25 Chapter 3: Numerical Examples i o 8 o Is > o - * r s . O 2 2 00 — so© 8 O ~~s. ( C ' l H © 0 0 s i S e*re-3* (Z-8C-) ror- 6'Or I , 0 K6TZ-)>(0-ZZ^ 9'lZ-<;91z| v-l so 0 ^ 1 -< 8 0 . 0 soQi O o so 26 Chapter 3: Numerical Examples Three-phase faults applied at the generator buses, the lines that were opened, the number of critical generators that were involved, the electrical power produced during the fault as well as the catastrophe control parameters v and w are shown in Table 3.4. Those buses which are not generator buses are shown in Table 3.5. The transient stability region using the general dynamic method is shown for generation buses in Figure 3.6 and for non-generating buses in Figure3.7 . All stable cases are shown inside the region in terms of the catastrophe control parameters. Node Grounded Line Opened No. Critical Generators Elec. Power DuringFault V W 7 7 - 8 1 0.0 1.9 13.5 7' 7-5 1 0.0 1.05 16.1 9 ' 9 - 6 1 0.0 0.7 28.5 9 9-8 1 0.0 1.02 27.1 4 4 - 5 3 0.304 1.02 31.7 4 4 -6 3 0.304 1.16 31.9 Table 3.4: Cases of faulted generating buses 27 Chapter 3: Numerical Examples Node Grounded L ine Opened No . Cri t ical Generators Elec. Power DuringFault V W 5 4 - 5 1 0.652 2.0 27.0 5 5 - 7 1 0.652 7.2 14.3 6 6 - 4 2 0.720 1.8 29.0 6 6 - 9 2 0.720 3.8 23.1 8 8 - 9 2 0.487 1.0 27.2 8 8 - 7 2 0.487 1.5 26.0 Table 3.5: Cases of non-generating buses 28 Chapter 3: Numerical Examples Chapter 3: Numerical Examples Chapter 3: Numerical Examples 3.2 The Seven-Machine System The C I G R E 225 K V test system is shown in Figure 3.8. This system has 10 buses and 13 unique branches. Buses 1 through 7 are generating buses while loads are located at buses 2, 4, 6, 7, 8, 9, and 10. The base values used are 225 K V and 100 M V A [21] . The systems bus data, branch data and the systems loadflow summary is given in Table 3.6, Table 3.7and Table 3.8 respectively. Bus P gen. X P l o a d Q load # ( M W ) ( % ) ( M W ) ( M V A R ) 1 217.00 7.4 0.00 0.00 2 120.00 11.8 200.00 120.00 3 256.00 6.2 0.00 0.00 4 300.00 4.9 650.00 405.00 5 230.00 7.4 0.00 0.00 6 160.00 7.1 80.00 30.00 7 174.00 8.7 90.00 40.00 8 0.00 0.0 100.00 50.00 9 0.00 0.0 230.00 140.00 10 0.00 0.0 90.00 45.00 Table 3.6: C IGRE 7-machine bus data 31 Chapter 3: Numerical Examples Bus Bus R (pu) X (pu) CHARG ( MVAR ) 1 3 0.0099 0.0484 20.250 1 4 0.0099 0.0484 10.125 2 3 0.0450 0.1237 20.250 2 10 0.0164 0.0638 30.375 3 4 0.0119 0.0780 30.375 3 9 0.0114 0.0553 20.250 4 5 0.0040 0.0198 20.250 4 6 0.0075 0.0198 121.50 4 9 0.0488 0.1916 20.250 4 10 0.0164 0.0652 30.375 6 8 0.0188 0.0628 20.250 7 8 0.0119 0.0780 30.375 8 9 0.0488 0.1916 20.250 Table 3.7: CIGRE 7 - machine branch data 32 Chapter 3: Numerical Examples Figure 3.8: CIGRE 7 - machine test system 33 Chapter 3: Numerical Examples Three-phase faults are applied and the transient stability is evaluated for each fault. The bus which the fault is applied, the number of critical generators involved, the values of the catastrophe control parameters and weather the system is stable or not is shown in Table 3.9. The transient stability regions in terms of the swallowtail catastrophe control parameters are shown in Figure 3.9which show good agreement with the time solution. Bus V m a g Vang Pgen Qgen # ( p u ) ( d e g ) ( M W ) ( M V A R ) 1 1.106 7.9 227.83 -49.54 2 1.156 0.35 120.00 232.96 3 1.098 6.42 256.00 -59.687 4 1.110 4.07 300.00 746.462 5 1.118 6.17 230.00 -9.748 6 1.039 5.89 160.00 -434.255 7 1.054 7.84 174.00 39.866 8 1.034 4.50 0.00 0.00 9 1.032 1.95 0.00 0.00 10 1.124 0.88 0.00 0.00 Table 3.8: CIGRE 7 - machine voltage and power summary 34 Chapter 3: Numerical Examples Node Grounded No . Cri t ical Generators Elec. Power DuringFault V W Stable 1 2 1.0404 2.9 25.1 yes 2 2 1.057 0.9 33.5 yes 3 2 0.8045 3.9 19.3 yes 4 2 0.317 5.4 21.04 yes 5 1 0.00 12.76 -26.75 no 6 1 0.00 9.4 5.5 yes 7 1 0.0 5.24 21.4 yes 8 1 0.7152 12.53 11.27 yes 9 3 1.098 0.33 44.0 no 10 2 0.1507 0.245 40.0 no Table 3.9: Cases of CIGRE 7-machine grounded buses 35 Chapter 3: Numerical Examples -24 Figure 3.7: Results for the 3-machine power system with faulted load buses. All cases are stable. 30 Chapter 4: Conclusion Chapter 4 Conclusion Catastrophe theory has been applied to the study of stability o f various dynamic systems such as aircraft stability [22] , and in recent years to the steady state stability problem of power systems [23]. However, that application was l imited only to salient-pole type synchronous generators. Then after swal low tail catastrophe was applied to transient stability of single-machine infinite bus system [17] and also to transient stability of multimachine power systems with the worst case approach i.e. only one generator being crit ical [18]. This thesis suggests a method to solve the transient stability problem of multimachine power systems with the system having more than one crit ical machine for a specified disturbance. Here the crit ical machines during a three-phase fault are identified, singled out and combined to be one equivalent machine and also the rest of the system as another single equivalent machine using the general dynamic equivalent approach. Then the energy balance equation is derived from the equation of motion of the equivalent critical machine against the rest o f the system. The energy balance equation is then used to form the equil ibrium surface of the swallowtail catastrophe manifold from which the transient stability region is derived by the bifurcation technique. The results obtained by this general swallowtail catastrophe approach is in good agreement with those obtained by the time solution method. It should be noted that the application of swallowtail catastrophe to transient stability of multimachine power systems has the fol lowing advantages : • The regions of stability are wel l denned in terms of the swallowtail catastrophe control parameters u, v and w. • The computations required to define the stability regions are few and done in a very short time. • The generator swing equations need not be solved. 37 Chapter 4: Conclusion New areas of research need to be explored in order to reach the goal of an efficient on-line direct method of transient stability analysis. The future research should include the fol lowing : • Stability controls such as fast valving, braking resistors, single pole switching, series capac-itors and generator trippings are usually applied in practice to restore transient stability of power systems. The inclusion of these controls in the transient stability using the swallowtail catastrophe approach would be of great interest to power utility companies. • In this thesis we only considered three-phase faults. However single-phase faults as wel l as multiple disturbances also occur. 38 r References [I] E. Kimbark and R. Byerly, "Stability of Large Power Systems,". A Volume in the IEEE Press Selected Reprint Series, pp. 221-231, 1974. [2] E. W. Kimbark, "Power System Stability," Vol. 1. John Wiley and Sons Inc., New York, 1948. [3] H. W. Dommel and N. Sato, "Fast transient stability solutions," IEEE Transactions on Power Apparatus System, July / August 1972. [4] H. W. Dommel, "Digital computer solution of electromagnetic transients in single and multiphase networks," IEEE Transactions Power Apparatus and Systems, Vol. PAS-88, pp. 388-399, April 1969. [5] F. S. Prabhakara and A . H. El-Abiad, " A simplified determination of transient stability regions for lyapunov methods," IEEE Transaction Power Apparatus Systems, Vol-94, No.2, pp. 672-689, March / Apri l 1975. [6] A . A . Fouad, "Application of transient energy functions to practical system problems," IEEE Special Publication on Rapid Analysis of Transient Stability, Winter meeting 1987. [7] M . Ribbens-Pavella and F. J . Evans, "Direct methods for studying dynamics of large-scale electric power systems," A Survey Automica, Vol. 21, No. 1, pp. 1-21, 1985. [8] P. Saunders, "An Introduction to Catastrophe Theory,". Cambridge University Press, 1980. [9] C. Panati, "Catastrophe theory," Newsweek, January 19, 1976. [10] M . Davis and A . Woodcock, "Catastrophe Theory,". Clarke, Irwin and Company Limited, Toronto, 1978. [II] I. Stewart and T.Poston, "Catastrophe Theory and its Applications,". Pitman Publishing, London, 1979. [12] A . Mihir ig, "Transient Stability of Multi-Machine Power Systems by Catastrophe Theory,". PhD thesis, Faculty of Applied Science, University of British Columbia, December 1987. [13] A . A . Sallam and J. L. Dineley, "Catastrophe theory as a tool for determining synchronous power system stability," IEEE Transactions on Power Apparatus and System, Vol. PAS-102, No. 3, pp. 622-630, March 1983. [14] M . D. Wvong and A . Mihirig, "Application of catastrophe theory to transient stability analysis of power systems," Proc. IASTED International Conference on High Technology in Power Industry, Bozmann, Montana, pp. 101-105, August 1986. [15] Y. Xue and T. V . Cutsen, " A simple direct method for fast transient stability assessments of large power systems," IEEE Transactions on Power Systems, Vol. 3, No. 2, pp. 400-412, May 1988. 39 [16] A. Rahimi and G. Schaffer, "Power system transient stability indexes for on line analysis of worst case dynamic contingencies," IEEE Transactions on Power Systems, Vol. PWRS-2, No. 3, pp. 660-665, August 1987. [17] M. Wvong and A. Mihirig, "Catastrophe theory applied to transient stability assessment of power systems," Proc. IEEE, 133, part C, 314-318, September 1986. [18] M. D. Wvong and A. Mihirig, "Transient stability regions of multi-machine power systems by catastrophe theory," Proc. IASTED International Conference on High Technology in the Power Industry, Phoenix, Arizona, U.SA., March 1988. [19] A. Michel and A. Fouad, "Power system transient stabihty using individual machine energy functions," IEEE Transactions, CAS-30, pp. 266-276, May 1983. [20] Anderson and Fouad, "Power System Control and Stability," Vol. 1. Iowa State University Press, Ames, Iowa, 1977. [21] M. Pai, "Power System Stability," Vol. 3. North Holland Publishing Co., 1980. [22] R.K.Mehra, "Catastrophe theory, non-linear system identification and bifurcation control," Proc. IEE of 1st Automatic Control Conference, SanFrancisco, California, June 1977. [23] A. Sallam, "Power system stability using catastrophe theory," IEE Proceedings, Vol. 135, pt C, No. 3, pp. 189-195, May 1988. [24] W. Stevenson, "Elements of Power System Analysis,". McGraw-Hill Book Company, 1982. [25] Stagg and El-Abiad, "Computer Methods in Power System Analysis,". McGraw-Hill Kogakusha, Ltd, 1968. [26] A. Fouad, "Application of transient energy functions to practical power system problem,," IEEE special publication on Rapid Analysis of Transient Stability, Power Winter Meeting,, 1987. 40 ) Appendix Fortran Program Listing of Swallowtail Catastrophe Applied to Multimachine Power Systems INTEGER N L , B U S ( 8 0 ) , N G E N , L N ( 8 0 ) , N L L L INTEGER F R ( 8 0 ) , T 0 ( 8 0 ) . N L L , N B B , N O D E , C A N C E L INTEGER N . R . N N . N B . N N F . R F , I I , J J . K K C NB: NUMBER OF BUSES C N: NUMBER OF GENERATORS IN THE SYSTEM C NN:MAXIMUM DIMENSION OF THE PREFAULT AND POST FAULT MATRIX C NNF:MAXIMUM DIMENSION OF THE DURING FAULT MATRIX C R: NN-N C R F : NNF-N C PARAMETER(NB=9,N=3) PARAMETER(NN=9, R=6) PARAMETER(NNF=8.RF=5) INTEGER N P R S T A ( N ) , B A S E ( N ) , K G ( 5 0 ) INTEGER C L I M ( 5 0 ) , N C L I M , C R I ( 8 0 ) , N C R I ( 8 0 ) REAL V ( 8 0 ) , A N G ( 8 0 ) , P G E N ( 8 0 ) , 0 G E N ( 8 0 ) , P L 0 A D ( 8 0 ) REAL 0 L 0 A D ( 8 0 ) . M A G V ( 8 0 ) , R V ( 8 0 ) , I V ( 8 0 ) ,A REAL M A G E ( 8 0 ) . G ( N . N ) , M A G D F ( N . N ) . T E T A F ( N . N ) . D E L T A ( 8 0 ) REAL M A G I ( N B ) . A L F A ( N B ) , M A G 0 P ( N , N ) , D E L TA t (80) REAL TETAP(N.N) REAL P M ( 8 0 ) , P M 1 ( 5 0 ) , P E F ( 5 0 ) . G A M M A ( 5 0 ) . P E P ( 5 0 ) REAL R A T I N G ( N ) , H ( N ) , M ( 5 0 ) REAL T O L . D E L T . T C REAL MNOT.CM,NCM.CMECH,NCMECH,CD,COF.COP REAL CDEL.NCDEL.CANG.NCANG,RATIO ,PMECH REAL ATEMP,A1 TEMP,A2TEMP,A3TEMP,BTEMP.B1TEMP,B2TEMP REAL 83TEMP,CTEMP,C1 TEMP,C2TEMP,C3TEMP REAL P R E D ( N , N ) , P R E C ( N , N ) , P R E P K ( 8 0 ) , P R E A K , P R E 8 K , P R E A L F REAL P R E T K , P R E C Y 1 ( 5 0 ) . P R E C Y 2 . P R E P C REAL DURD(N.N) ,DURC(N.N) ,DURPK.DURAK.0UR8K,DURALF REAL 0URTK.DURPC.DURPE(5O) REAL P O S D ( N . N ) . P O S C ( N . N ) , P O S P K , P O S A K , P O S B K . P O S A L F REAL POSTK.POSPC.POSPE REAL E T A X ( 8 0 ) , T E T A I ( 8 0 ) . B E T A . S P O S C Y , U P O S C Y . M K G A M K ( 5 0 ) REAL UK.UCAT.VCAT.WCAT COMPLEX SGEN(80) ' ,V1(80) ,CUR(60) ,SL0AD(80) COMPLEX YLOAD(BO) ,EPRIM(80) COMPLEX X P R I M D ( 8 0 ) , T E M P . Y ( 8 0 ) , Y S H ( 8 0 ) COMPLEX YBUS(NB.NB) . Y B U S K N B . N B ) , YBUS2(N8- 1 .NB- 1) COMPLEX YBUS3(NB.NB) COMPLEX Y N N ( N . N ) , Y N R ( N . R ) , Y R N ( R . N ) 41 COMPLEX YRR(R.R),YRRR(R.R).AA(R.R),DET,COND COMPLEX B(N,R).C(N.N),D(N.N) ,TEM COMPLEX YNNF(N.N),YNRF(N,RF).YRNF(RF,N) COMPLEX YRRF(RF,RF),YRRRF(RF,RF),AAF(RF,RF) COMPLEX BF(N,RF),CF(N,N),OF(N,N) COMPLEX YNNP(N.N),YNRP(N,R).YRNP(R.N) COMPLEX YRRP(R.R).YRRRP(R,R).AAP(R,R) COMPLEX BP(N,R).CP(N.N),DP(N,N) OPEN(UNIT=5>FILE=•FOUAD•,STATUS='OLD') OPEN(UNIT=6.FILE='OUT',STATUS='UNKNOWN') READ STATEMENTS READ(5,*)NL,NGEN,NODE,CANCEL,KK DO 228 1=1,NL READ(5,*)BUS(I ) ,V(I),ANG(I),PGEN(I),QGEN(I) , PLOAD(I).OLOAD(I).XPRIMD(I) CONTINUE READ(5,*)NLL DO 111 1=1,NLL READ15,*)LN(I),FR(1),TO(I),Y(I).YSH(I) CONTINUE DO 199 1=1,N READ(5,*)NPRSTA(I),RATING!I),H(I),BASE(I) CONTINUE CALCULATE TRANSIENT VOLTAGE,INITIAL OPERATING ANGLE,LOAD IMPEDANCE,CURRENT AT GENERATING NODE. A=3.14159/180.00 00 191 1=1,NL EPRIM(I)=(0.0,0.0) YLOAO(I)=(0.0,0.0) DELTA(I)=0.0 DELTA1(I)=0.0 CUR(I)=(0.0.0.0) ALFA(I)=0.0 MAGI(I)=0.0 PM(I)=0.0 CONTINUE DO 100 1=1,NL RV(I)=V(I)*COS(ANG(I)'A) IV(I)=V(I)'SIN(ANG(I) ,A) 42 V 1 ( I ) = C M P L X ( R V ( I ) , I V ( I ) ) M A G V ( I ) = S Q R T ( ( R V ( I ) * R V ( I ) ) • ( I V ( I ) • I V ( I ) ) ) I F ( ( P G E N t I ) .GT.O.O).OR.(0GEN(I).GT.0.0))THEN SGEN(I)=CMPLX(PGEN(I).OGEN(I)) CUR(I)=(CONJG(SGEN(I)))/(CONJG(V1(I))) EPRIMtI)=V1 (I) + (CUR(I)*XPRIMD(I)) ELSE E P R IMd ) = (0.0,0.0) ENDIF SL0A0(I)=CMPLX(PL0AD(I),QL0AD(I)) YL0AD(I)=(C0NJG(SL0AD(I)))/(MAGV(I)'MAGV(I)) 100 CONTINUE CALCULATE ABOVE VALUES IN POLAR FORM DO 192 1=1.NL MAGE(I)=SQRT((REAL(EPRIM(I))* REAL(EPRIMtI))) & +(AIMAGtEPRIMtI))* AIMAG ( EPRIM( I ) ) ) ) MAGI(I)=SORT((REAL(CUR(I))•REAL(CUR(I))) & •(AIMAGtCURtI))'AIMAGtCURtI)))) IF((AIMAG(EPRIMtI)).NE.O.O).OR.(REAL(EPRIMtI)).NE.O.O))THEN DELTA(I)=ATAN2(AIMAGtEPRIMtI)).REAL(EPRIMtI))) ELSE DELTA(I)=0.0 ENDIF IFt(AIMAGtCUR(I)).NE.0.0).OR.(REAL(CUR(I)).NE.0.0))THEN ALFAtI)=ATAN2(AIMAGtCURtI)),REAL(CUR(I))) ELSE CURtI)=0.0 ENDIF DELTA(I)=DELTA(I)/A JELTA1(I)=DELTA(I)'A ALFA(I)=ALFA(I)/A 192 CONTINUE INITIALISE ALL THREE MATRICES DO 112 1=1,NB DO 112 J=1,NB YBUS(I,J)=(0.0.0.0) YBUSKI .J) = (0.0,0.0) YBUS3(I,J)=(0.0,0.0) 112 CONTINUE DO 131 I=1.(NB-1) DO 131 J=1,(NB-1) YBUS2(I.J)=(0.0,0.0) 131 CONTINUE FORMULATE THE PREFAULT MATRIX 43 c 00 113 I=1,NB DO 114 J=1.NLL IF( (FR(J) .EQ.LN(I)).OR. (T0( J).EQ.LN( I )))THEN YBUS(1,1)=YBUS(I,I) + ( Y ( J)•YSH ( J ) ) ENDIF 114 CONTINUE YBUS(I,I)=YBUS(I,I )+YLOAD<I) 113 CONTINUE DO 115 K=1 ,NLL IF((FR(K).NE.LN(K)).OR.<TO(K).NE.LN(K)))THEN I=FR(K) J=TO(K) YBUS(I,J)=YBUS(I,J)-Y(K) YBUS(J.I)=YBUS(I,J) ENDIF 115 CONTINUE C REDUCE THE PREFAULT MATRIX C DO 146 I = (N +1),NN DO 132 J=(N+1),NN TEMP=YBUS(I,J) II=I-N JJ=J-N YRR(II,JJ)=TEMP YRRR(II,JJ)=YRR(II,JJ) 132 CONTINUE 146 CONTINUE DO 134 1=1,N DO 134 J=1.N YNN(I,J)=YBUS(I.J) 134 CONTINUE DO 136 1=1.N DO 136 J=(N +1),NN TEMP=YBUS(I.J) 11 = 1 JJ=J-N YNR(11,JJ)=TEMP 136 CONTINUE DO 138 I=(N+1),NN DO 138 J=1,N TEMP=YBUS(I.J) II=I-N JJ=J YRN(11,JJ)=TEMP 138 CONTINUE CALL CINVRKYRR,R,R,DET,COND) 44 CALL CMULT(YRRR,YRR,AA,R.R,R.R,R.R) CALL CMULT(YNR,YRR,B,N,R,R,N,R,N) CALL CMULT(B,YRN,C,N,R,N,N,R,N) CALL CSUB(YNN,C,D.N.N,N,N,N) FORMULATE THE DURING FAULT MATRIX 125 DO 125 1=1,NB DO 125 J=1,NB YBUS1(I.J)=YBUS(I,J) CONTINUE 117 DO 117 1=1.NB DO 117 J=1,NB IF((I.EQ.NODE).OR.(J.EQ.NODE))YBUS1(I.J)=(0.0,0.0) CONTINUE 120 1 19 DO 119 I=1.NB DO 120 J=1,NB K=J+1 IF((K.LE.NB).AND.{J.GE.NODE))THEN YBUS1(I,J)=YBUS1(I,K) YBUS1(I,K)=(0.0,0.0) ELSE ENDIF CONTINUE CONTINUE 121 DO 121 1=1,NB DO 121 J=1,NB K = I • 1 IF((K.LE.NB).ANO.(I.GE.NODE)(THEN Y B U S K I , J ) =YBUS 1 (K , J ) Y8USKK, J) = (0. 0,0.0) ELSE ENDIF CONTINUE 123 DO 123 1=1,NB-1 DO 123 J=1.NB-1 YBUS2U, J)=YBUS1(I . J) CONTINUE REDUCE THE FAULTED MATRIX DO 160 I=(N+1),NNF DO 161 J=(N+1).NNF TEMP=YBUS2(I,J) II=I-N JJ=J-N YRRF(II.JJ)=TEMP YRRRF(11,JJ)=YRRF(II.JJ) 45 161 160 CONTINUE CONTINUE 00 163 1=1,N DO 163 J=1,N YNNF(I.J)=YBUS2(I,J) 163 CONTINUE DO 165 1=1,N DO 165 J=(N+1),NNF TEMP=YBUS2(I,J) 11 = 1 JJ=J-N YNRF(II,JJ)=TEMP 165 CONTINUE DO 167 I=(N+1),NNF DO 167 J=1,N TEMP=YBUS2(I,J) II=I-N JJ=J YRNF(11,JJ)=TEMP 167 CONTINUE CALL CINVRT(YRRF,RF,RF,DET,COND) CALL CMULT(YRRRF,YRRF.AAF,RF,RF,RF,RF.RF,RF) CALL CMULT(YNRF.YRRF,BF,N,RF,RF,N,RF,N) CALL CMULT(BF.YRNF,CF,N,RF,N,N,RF.N) CALL CSUB(YNNF,CF,DF,N,N,N,N,N) DO 126 1=1,NB DO 126 J=1,NB YBUS3U , J)=YBUS(I . J) 126 CONTINUE C FORMULATE THE AFTER FAULT MATRIX C • DO 127 1=1.NB DO 127 J=1.NB IF((I.EQ.FR(CANCEL)).AND.(J.EO.TO(CANCEL)))THEN YBUS3(I.J)=(0.0.0.0) YBUS3(J,I)=(0.0,0.0) ENDIF 127 CONTINUE DO 128 1=1.NB 00 128 J=1 ,NB ' IF((I.EQ.FR(CANCEL)) .AND.(J.EO.FR(CANCEL)))THEN YBUS3I1,1)=YBUS3(I,I)-(Y(CANCEL)•YSH(CANCEL) ) C YBUS3(I,I)=YBUS3(I . I)-Y(CANCEL) ENDIF 128 CONTINUE DO 129 1=1,NB DO 129 J=1,NB 46 IF({I.EQ.T0(CANCEL)).AND.(J.EQ.T0(CANCEL>))THEN YBUS31I,I)=YBUS3(1,1)-(Y(CANCEL)+ YSH(CANCEL)) YBUS3(1,1)=YBUS3(I,I)-Y(CANCEL) ENDIF 129 CONTINUE C REDUCE THE AFTER FAULT MATRIX C DO 172 I=(N+1),NN DO 173 J=(N+ 1 ),NN TEMP=YBUS3(I,J) II=I-N JJ=J-N Y R R P d l , JJ)=TEMP YRRRP(II.JJ)=YRRP(II,JJ) 173 CONTINUE 172 CONTINUE DO 175 1=1,N DO 175 J=1,N YNNP(I,J)=YBUS3(I,J) 175 CONTINUE DO 177 1=1,N DO 177 J=(N + 1 ),NN TEMP=YBUS3(I.J) 11 = 1 JJ=J-N YNRP (II,JJ)=TEMP 177 CONTINUE DO 179 I = (N+1 ),NN DO 179 J=1,N TEMP=YBUS3(I,J) 11 = 1 N JJ=J YRNP(II,JJ)=TEMP 179 CONTINUE CALL CINVRT(YRRP,R.R,DET,CONO) CALL CMULT(YRRRP,YRRP,AAP,R,R,R,R,R,R) CALL CMULT(YNRP.YRRP,BP.N,R,R,N.R.N) CALL CMULT(BP,YRNP.CP,N,R.N.N,R.N) CALL CSUB(YNNP.CP.DP,N,N,N,N,N) C FAULTED AND AFTER FAULT MATRIX IN POLAR FORM C DO 193 1=1,N DO 194 J=1.N MAGDF(I.J)=SQRT((REAL(DF(I.J))•REAL(DF(I,J))) + (AIMAG(DF(I,J))* AIMAG(DF(I,J)))) MAGDP(I.J)=SQRT((REAL(DP(I,J))•REAL(DP(I,J))) 47 + (AIMAG(OP(I,J))* AIMAG(DP(I,J)))) I F ( R E A L ( D F ( I , J ) ).NE.0.0.OR.AIMAG(DF(I.J)).NE.0.0)THEN TETAF(I,J)=ATAN2(AIMAGIDF(I,J)),REAL(DF(I,J))) ELSE TETAF(I,J)=0.0 ENDIF IF(REAL(DP(I.J)).NE.0.0.OR.AIMAG(DP(I,J)).NE.0.0)THEN TETAP(I,J)=ATAN2(AIMAG(DP(I,J)),REAL(DP(I,J))) ELSE TETAP(I , J)=0.0 ENDIF T E T A F ( I , J ) = T E T A F ( I , J ) / A T E T A P ( I , J ) = T E T A P ( I . J ) / A CONTINUE CONTINUE D i j . C i j USING PRE,DURING AND POST REDUCED MATRIX DO 222 1=1,N DO 223 J=1,N PRED (I , J) =MAGE (I) * MAGE (J)•(REAL(D <I.J))) PREC(I,J)=MAGE(I)"MAGE(J)*(AIMAG(D(I , J ) ) ) DURD(I,J)=MAGE(I)'MAGE(J)*(REAL(DF(I,J))) DURC(I >J)=MAGE(I) ,MAGE<J)MAIMAG(DF(I,J))) POSD(I,J)=MAGE(I)* MAGE(J)*(REAL(DP (I , J ) ) ) POSC(I.J)=MAGE(I)* MAGE(J)•(AIMAG(DPI I , J ) ) ) CONTINUE CONTINUE MECHANICAL INPUT=GENERATED POWER - LOACAL LOAD DO 198 1=1,NL IF(PGEN(I) .GT.0.0)PM(I)=PGEN(I) -PLOAD( I) CONTINUE DO 221 1=1,N PEF(I)=0.0 PM1(I)=0.0 00 220 J=1 ,N ATEMP=DELTA1(I)-DELTA 1(J) PM1(I)=PM1(I)+((PRED(1,J)* COS(ATEMP)) +(PREC(I,J)*SIN(ATEMP))) P E F ( I ) = P E F ( I ) + ((DURD(I,J)* COS(ATEMP)) •(DURCU.JJ'SINUTEMP))) CONTINUE CONTINUE 48 C CALCULATE THE INERTIA CONSTANTS OF EACH MACHINE C • DO 200 1=1,N C TEMP3=NPRSTA(I)'RATING(I)'H(I) TEMP3=NPRSTA(1) ,H(I) M(I)=TEMP3/(60.0'3.14159'BASE(I)) IF(M(I).NE.0)THEN GAMMA(I)=(PM1(I)-PEF(I))/M(I) KG(I)=I ENDIF 200 CONTINUE C SORTING C DO 610 1=1,(N-1) DO 611 J=(I+1),N IF(GAMMA(I).LT.GAMMA(J))THEN ATEMP=GAMMA(I) BTEMP=KG(I) CTEMP=PEF(I) GAMMA(I)=GAMMA ( J) KG(I)=KG(J) P E F ( I ) = P E F ( J ) GAMMA(J)=ATEMP KG(J)=BTEMP PEF(J)=CTEMP ENDIF 611 CONTINUE 610 CONTINUE C WRITE(6,700) C 700 FORMAT)////) C WRITE(6.701) C 701 FORMAT{4X,'GAMMA' ,3X,'GEN' ,5X,'DUR FAULT PE ' ) C DO 999 1=1,N C WRITE(6,702)GAMMA(I),KG(I),PEF(I) C 702 F0RMAT(/4X,F11 5.5X,I3,4X.F11.5) C 999 CONTINUE C CM=CRITICAL MOMENT OF INERTIA C CANG=CRITICAL EQUIVALENT ANGLE C NCANG=NON CRITICAL EQUIVALENT ANGLE C CMECH=CRITICAL MECHANICAL POWER C NCMECH=NON CRITICAL MECHANICAL POWER C PMECH=EQUIVALENT MECHANICAL POWER C RATIO=CRITICAL MOMENT OF INERTIA/NON CRITICAL C ============================================== 49 CLIM(KK)=0.0 CLIM(KK)=KK DO 612 1=1,CLIM(KK) CRI(I)=KG<I) 612 CONTINUE NCLIM=NGEN-CLIM(KK) IF(NCLIM.EQ.O)THEN NCLIM=1 DO 637 1=1,NCLIM NCRI(I)=0 637 CONTINUE ELSE DO 613 1=1,NCLIM J=I+CLIM(KK) NCRI(I)=KG(J) 613 CONTINUE ENDIF CMECH=0.0 CDEL=0.0 MNOT=0.0 CM=0.0 CD=0.0 CDF=0.0 CDP=0.0 NCM=0.0 NCDEL=0.0 NCMECH=0.0 DO 600 1=1.CLIM(KK) CM=CM+M(CRI(I)) CDEL=CDEL+(M(CRI(I))'DELTA 1 ( C R I ( I ) ) ) CMECH=CMECH+PM1(CRI(I)) CD=CD+PRED(CRI(I).CRI(I)) CDF=CDF+DURD(CRI(I),CRI(I)> CDP=CDP+POSD(CRI(I),CRI(I)) 600 CONTINUE CANG=CDEL/CM DO 601 1=1.NCLIM IF(NCRI(I).NE.0)THEN NCM=NCM+M(NCRI(I>) NCDEL=NCDEL*(M(NCRI(I))'DEL TA1(NCRI(I))) NCMECH=NCMECH+PM1(NCRI ( I ) ) NCANG=NCDEL/NCM RATIO=CM/NCM ELSE NCANG=0.0 50 RATIO=0.0 ENDIF CONTINUE PMECH=CMECH-(RATIO*NCMECH) DO 602 1=1,N ETAK(I)=DELTA1(I)-CANG TETAI(I)=DELTA1(I)-NCANG CONTINUE ATEMP=0.0 A1TEMP=0.0 A2TEMP=0.0 A3TEMP=0.0 BTEMP=0.0 B1TEMP=0.0 B2TEMP=0.0 B3TEMP=0.0 CTEMP=0.0 C1TEMP=0.0 C2TEMP=0.0 C3TEMP=0.0 DO 622 1=1,NCLIM DO 623 J=1.NCLIM IF((NCRI(I).NE.0).OR.(NCRI(J).NE.0))THEN ATEMP=ATEMP+(PRED(NCRI(I).NCRI(J))' COS (TETAI ( N C R K I ) ) - TETAI (NCRI ( J) ) ) ) CTEMP = CTEMP+(POSD( N C R K I ) . N C R I ( J ) ) * COStTETAI ( N C R K I ) ) - TETAI (NCRI ( J) ) ) ) ENDIF CONTINUE CONTINUE PREPK(CLIM(KK))=PMECH-(CD -(RAT 10*ATEMP)) POSPK=PMECH-(CDP-(RAT 10"CTEMP)) PRECY1(CLIM(KK))=CANG-NCANG IF(CLIM ( K ).EQ.1)THEN DURPE(CLIM(K K))=PEF(CLIM(K K ) ) DO 641 1=1.CLIM (KK) DO 638 J=1,NCLIM IF(NCRI(J).NE.O.OJTHEN DTEMP=TETAI(NCRI(J))-ETAK ( CRI ( I ) ) CTEMP=CTEMP+( (POSD(CRKI) .NCRI(J))*COS(DTEMP))• ( P O S C ( C R I ( I ) , N C R I ( J ) ) * SIN(DTEMP))) C1TEMP=C1TEMP+((POSD(CRI(I).NCRI(J))'SIN(DTEMP))-(POSC(CRI(I),NCRI(J))"COS(DTEMP))) 51 638 641 ENDIF CONTINUE CONTINUE DO 639 I=1,CLIM(KK) DO 640 J=1,N I F ( J . N E . C R I ( I ) ) T H E N DTEMP=TETAI(J)-ETAK(CRI(I)) C2TEMP=C2TEMP+( (POSD(CRKI) , J) *SIN(DTEMP) ) + ( P O S C ( C R K I ) , J) 'COS (DTEMP) ) ) C3TEMP=C3TEMP+< (POS C ( C R K I ) , J) • SIN (DTEMP)) -(POSD(CRKI) , J)'COS(DTEMP))) ENDIF 640 CONTINUE 639 CONTINUE ELSE DO 616 I=1.CLIM(KK) DO 617 J=1 .NCLIM IF(NCRI(J).NE.0.0)THEN DTEMP=TETAI(NCRI(J))-ETAK(CRI(I)) BTEMP=BTEMP+((DURD(CRI(I),NCRI(J))* COS(DTEMP)) + (DURC(CRKI) ,NCRI(J))'SIN(DTEMP))) CTEMP=CTEMP+((POSD(CRKI),NCRI(J)>'COS(DTEMP))+ (POSCtCRI(I).NCRI(J))'SIN(OTEMP))) B1TEMP=B1TEMP+((DURD(CRI(I),NCRI(J))•SIN(DTEMP)) (DURC(CRKI) ,NCRI ( J ) ) 'COS(DTEMP)) ) C1TEMP=C1TEMP+((POSD(CRKI),NCRI(J))'SIN(DTEMP)) (POSC(CRI(I).NCRI(J))'COS(DTEMP))) ENDIF 617 CONTINUE 616 CONTINUE DO 618 1=1.CLIM(KK) DO 619 J=1,N IF(J.NE.CRI(I))THEN DTEMP=TETAI(J)-ETAK(CRI(I)) B2TEMP=B2TEMP+((DURD(CRI(I),J)'SIN(DTEMP))• (DURC(CRKI) .J)'COS(DTEMP))) C2TEMP=C2TEMP*((POSD(CRKI),J)'SIN(DTEMP)) + (POSC(CRI(I).J)'COS(DTEMP))) 52 & B3TEMP=B3TEMP+((DURC(CRI(I),J)"SIN(DTEMP))-(DURD(CRI(I).J)* COS(DTEMP))) C3TEMP=C3TEMP+((POSC(CRI(I),J)•SIN(DTEMP))-& (P0SD(CRI(I),J)'C0S(DTEMP))) ENDIF 619 CONTINUE 618 CONTINUE DURAK=B3TEMP+(RATIO'BTEMP) DURBK=B2TEMP-(RATIO*B1 TEMP) DURALF=ATAN2(DURAK,DURBK) DURTK=SQRT((DURAK*DURAK)+(DURBK*DURBK)) DURPE(CLIM(K))=DURTK'SIN(PRECY1(CLIM(K))-DURALF) ENDIF P0SAK=C3TEMP+(RATI0'CTEMP) POSBK=C2TEMP-(RAT 10'C1 TEMP) P0SALF=ATAN2(POSAK,POSBK) POSTK=SORT((POSAK'POSAK)•(POSBK•POSBK)) SPOSCY=POSALF + ASIN< POSPK/POSTK) UP0SCY=3.14159-SPOSCY P0SPE=P0STK'SIN(SPOSCY-POSALF) MKGAMKtCLIM(KK)}=PREPK(CLIM(KK))-DURPE(CLIM(KK)) BETA=PRECY1(CLIM(KK))-POSALF UK=(POSPK'UPOSCY) + (POSTK * COS(UPOSCY-POSALF)) UCAT=-12.000 VCAT=(24.0/POSTK)*(POSPK-MKGAMK (CLIM(KK ) ) ) WCAT=(24.0/POSTK) *((POSPK•PRECY1(CLIM(KK)))-UK-& (POSPK'BETA) + (MKGAMK(KK)* BETA))+24 . 00 C WRITE(6,700) C WRITE(6,703) C 703 FORMAT(5X, 'CRIT GENS' , 4X , ' DURPE' ,5X, 'MKGAMK') C WRITE(6,704)CLIM(KK),DURPE(CLIM(KK)),MKGAMK(CLIM(KK)) C 704 FORMAT(/6X,I3,5X.F11.5,5X.F11.5) C WRITE(6,700) C WRITE(6,705) C 705 FORMAT(5X,'CRIT GENS',4X,'POST CYE',4X,'UK') C WRITE(6.706)CLIM(KK),SPOSCY , UK C 706 FORMAT(/5X.I3.5X,F11.5,5X.F11.5) C WRITE(6.700) WRITE(6,707) 707 FORMAT(5X,'UCAT' ,4X,'VCAT' ,6X, 'WCAT') WRITE(6.708)UCAT,VCAT.WCAT 708 F0RMAT(/4X.F11.5.4X.F11.5.6X.F11.5) STOP END 53 

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