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Application of catastrophe theory to voltage stability analysis of power systems Hjartarson, Thorhallur 1990

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APPLICATION of CATASTROPHE THEORY to VOLTAGE STABILITY ANALYSIS of POWER SYSTEMS  Thorhallur Hjartarson  B.A.Sc.  University of Iceland, 1988  A THESIS SUBMITTED IN PARTIAL FUMLLMENT 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  July 1990 © Thorhallur Hjartarson, 1990  In presenting degree  this  at the  thesis  in  partial fulfilment  of  University of  British Columbia,  I agree  freely available for reference copying  of  department publication  this or of  and study.  thesis for scholarly by  this  his  or  her  the  purposes  representatives.  may be It  thesis for financial gain shall not  u^l€c4ric&^  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  that the  I further agree  permission.  Department of  requirements  EL^Qi*\eeP*  that  advanced  Library shall make it  by the  understood be  an  permission for extensive  granted  is  for  that  allowed without  head  of  my  copying  or  my written  Abstract  In this thesis catastrophe theory is applied to the voltage stability problem in power systems. A general model for predicting voltage stability from the system conditions is presented and then applied to both a simple 2-bus explanatory power system and to a larger more realistic power system. The model is based on the swallowtail catastrophe which with its three control variables is able to determine the voltage stability of the system. The model is derived direcdy from the systems equations. The voltage stability of the system at each specified system bus is determined by comparing the values of the swallowtail catastrophe control variables with those of the unique region of voltage stability. The control variables are calculated from the system operating conditions. If the control variables specify a point inside the stability region, the system is voltage stable; otherwise it is voltage unstable.  ii  Table of Contents  Abstract  ii  List of Figures  v  List of Tables  vi  Acknowledgments  vii  1 Introduction  1  1.1  Literature Review on Voltage Stability  3  1.2  Literature Review on Applications of Catastrophe Theory  7  2 A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  9  2.1  Introduction  9  2.2  Developing the Model  9  2.3  Testing the Model  14  3 A General Catastrophe Model for Voltage Stability in Power Systems  19  3.1 General System Equations  19  3.2  Solving the System  20  3.3  General Swallowtail Catastrophe Modelling  26  4 Examination of the General Model on a Test System  29  4.1  Test System  29  4.2  Initial System Conditions  29  4.3  Voltage Variations  31  4.4  Load Variations  33  4.5  Load and Voltage Variations  37  5 Conclusions  39  Appendix A The Catastrophe Theory and the Swallowtail Catastrophe  41  A.1 Introduction  41  A.2 The Elementary Catastrophes  42  iii  44  A3 The Swallowtail Catastrophe Appendix B Program Code: General 6-Bus Case  48  References  69  iv  List of Figures  1.1  Steady-State Stability Region  7  1.2  Transient Stability Region  8  2.3  Simple Power System  9  2.4  Bifurcation Set For The Swallowtail Catastrophe  14  2.5  Voltage Versus MVA At Load Bus (pf = 0.9285)  15  2.6  The Voltage Stability Manifold of the Swallowtail Catastrophe  16  2.7  Manifold foru < 0  16  2.8  Manifold for u > 0  17  2.9  The Travel of the Voltage Curve through the Catastrophe  18  2.10  Voltage Curve near Critical Point  18  3.11 Non-Radial Power System  22  3.12 Bifurcation Set For The Swallowtail Catastrophe  28  4.13 A 6-Bus Power System  30  A.14 The Tail of the Swallow  46  A.15 Manifold for s = 0  47  A.16 Manifold for s > 0  47  v  List of Tables  2.1  The Travel of the Voltage Curve through the Catastrophe  17  4.2  Load Flow Results for 6-Bus Power System  30  43  Control Variables for Initial Conditions  31  4.4  Control Variables with 82.6% Initial Voltage Values  32  4.5  Control Variables with 58% of Voltage Values at Bus 1  33  4.6  Control Variables with 52% of Voltage Values at Bus 2  33  4.7  Control Variables with 29% of Voltage Values at Bus 3  34  4.8  Control Variables with 72% of Voltage Values at Bus 4  34  4.9  Control Variables with 70% of Voltage Values at Bus 5  35  4.10  Control Variables with 5.5timesthe Initial Reactive Loads  35  4.11  Control Variables with 240 times the Initial Reactive Load at Bus 1  36  4.12  Control Variables with 465 times the Initial Reactive Load (small) at Bus 2  36  4.13  Control Variables with 59 times the Initial Reactive Load at Bus 3  36  4.14  Control Variables with 11.3timesthe Initial Reactive Load at Bus 4  37  4.15  Control Variables with 8.5timesthe Initial Reactive Load at Bus 5. .  4.16  Effects of both Load and Voltage on Voltage Stability  vi  . 37 38  Acknowledgments  I would like to thank my supervisor Dr. M.D. Wvong for his invaluable insight and assistance with the work involved in this thesis. I would also like to thank all friends and family who have been supportive through the ordeal of my educations era. Special thanks to friends Ivar and Iris and of course their new bom daughter Alma, for constantly feeding me real home made meals and providing me with good company during lonely winter nights. Also special thanks to all the friends who provided many motivating discussions and interchanging of views over the odd beer.  vii  Chapter 1: Introduction  Chapter 1 Introduction  The blackout problem of electric power systems has traditionally been associated with the steady state and transient stability problems. Steady state and transient stability are the phenomena involved in connection with the loss of a major portion of a grid due to the inability of certain generators to maintain synchronism in the face of small and large disturbances respectively. These types of instability are, generally speaking, well understood today. System stability is being preserved to a greater extent than ever before by the advent of faster and more effective stabilizers, and more reliable protection systems. Inrecentyears a category of instability, usually termed voltage instability or collapse, and associated with the inability of a power system to maintain bus voltage magnitudes, has been responsible for several major blackouts world-wide. Earlier stability problems were concerned with the relationship between active power and phase angles of generators. The static voltage stability problem is, on the other hand, concerned with the relationship betweenreactivepower and voltage magnitudes of generators. As power systems become heavily loaded, there is a possibility that the power systems might suffer from a cascading voltage collapse due to lack of reactive power. Also there is the dynamic voltage stability problem, which involves frequency as a parameter as well. In this thesis, only the static voltage stability problem is considered since the dynamic case is more complex and the static problem is currently being researched in greater detail. As frequency is a critical parameter in the balance between real (MW) generation andreal(MW) load throughout the power system, so transmission voltage levels reflect the balance between the supply and demand ofreactivepower. While frequency is uniform throughout the power system, voltage levels can vary markedly across a transmission network, which is designed to operate at a particular voltage level. As aresult,it is generally accepted that the voltage stability problem,  1  Chapter J: Introduction  which is associated with the inability of a power system to maintain bus voltage magnitudes, is due to a deficit of reactive power at certain buses in the network. The actual process of collapse may therefore be triggered by some form of disturbance, resulting in significant changes in the reactive power balance in the system. The operating environment of many present-day power systems substantially increases the vulnerability of the system toreactivedeficit problems and therefore difficulties in maintaining system voltage profiles. Several factors have contributed to this situation. There is increasing difficulty in obtaining power plant sites in the vicinity of major power consumers. Also, the exploitation of hydro power resources has proceeded spectacularly to a point where remote, large generation plants have been developed. As a result, electrical power is often transported through high capacity lines over long distances from generators to consumer. Furthermore, the strengthening of transmission networks has been curtailed in general by high costs, and in particular cases by the difficulty of acquiring right-of-way. This has resulted in increased loading and exploitation of the older circuits thus resulting in increasing voltage stability problems. Other factors include the relative decrease in the reactive power outputs of generating units, and shifts in powerflowpatterns associated with changing fuel costs and generator availability. Numerous approaches to predict voltage collapse have been suggested (see following section). A fast alternative method is to apply catastrophe theory to the reactive power equations where an unique solution set for the stability exists. If the solution according to the catastrophe theory for any particular operating condition falls within this unique set then the system is voltage stable for that particular condition. In this thesis, the swallowtail catastrophe is applied to the voltage stability problem. In Chapter 2 a well known simple voltage stability example is modeled and examined. A more general model for any interconnected power system is derived in Chapter 3 which is then, in Chapter 4, applied  2  Chapter 1: Introduction  to a larger more realistic example. 1.1 Literature Review on Voltage Stability One of thefirstto address the problem of voltage collapse was W.R.Lachs [1]. There the phenomena is explained and an example is given which shows how a voltage collapse can occur. The importance of reactive compensation is discussed in detail since the reactive compensation on a EHV system must provide for both an overall, and a regional, balance and be able to withstand any feasible reactive disturbance on the system. In [2] a more recent review of the problem was done. Work about voltage stability conditions, proximity indicators, control strategies and planning network reinforcement is discussed. Some of the points made follow here. Some work has been reported on defining and establishing voltage stability criteria, i.e., criteria that may be used to determine whether or not an operating condition is stable from a voltage stability viewpoint It is suggested that an operating condition is stable from the voltage viewpoint if every load bus voltage increases when a source voltage increases or when a shunt capacitor is switched in at a load bus. Transformer taps are a major contributing factor in system voltage collapse and voltage instability is characterized in association with the slow tap-changing transformer dynamics. Stability conditions are derived in terms of allowable transformer taps settings using eigenvalue analysis. One criterion for voltage stability of a given operating condition states that for an operating point, voltage stability is ascertained when at mat operating point, an elementary increase of reactive demand is met by a finite increase in reactive power generation. Analytical computation of an index, which is defined on the basis of this criterion, for the simple two-bus system has been presented.  3  Chapter 1: Introduction  As noted in [3], a region-wise framework is presented which accounts for several of the mechanisms to predict voltage collapse and can serve as a basis for comparing the effectiveness of performance indices, to predict, on-line, voltage collapse problems in power systems. The basis of the framework is the voltage stability region, which accounts for both static and dynamic mechanisms of voltage collapse. Based on this region, static and dynamic performance indices are denned to predict the static mechanisms of voltage collapse in the input or injection space and the dynamic mechanism in the post-contingency state space, respectively. Sekine et al. also studied the dynamic phenomena of voltage collapse [4] by the method of dynamic simulation using induction motor models. From the viewpoint of dynamic phenomena, the voltage collapse starts locally at the weakest node and spreads out to the other weak nodes. In [5] the stability limit problem as a static divergence or bifurcation characterized by the disappearance of an equilibrium point was studied. Beyond this point, solutions to the load flow equations cannot be obtained. In [6] a suggestion on the use of the minimum singular value of the Jacobian matrix of the load flow equations as a security index was made, and static control strategies based on this index were derived. Mori et al. presented a method [7] for estimating critical points on static voltage instability in electric power systems. The difficulty of evaluating critical points in y (specified value) space although it is easy tofinda critical point in x (voltage) space with the conventional method is clarified. This is because there exists a set of specified values that provides a singular point in voltage space. Emphasis is put on evaluating the singular points in y space that is closest to operating conditions. A nonlinear programming technique is utilized to evaluate critical points in voltage and specified value space. In [8] a voltage stability index based on the feasibility of solution to the load flow equations at each bus was developed. 4  Chapter I: Introduction  In [9] a security measure to indicate vulnerability to voltage collapse based on an energy function for system models that include voltage variation and reactive loads is denned. The system dynamic model, the energy function and the security measure arefirstmotivated in a simple radial system. Application of the new measure and its computational aspects are then examined in a standard 30-bus example (New England System). The new measure captures nonlinear effects such as var limits on generators that can influence the vulnerability to collapse. The behavior of the index with respect to network load increases is nearly linear over a wide range of load variation, facilitating prediction of the onset of collapse. One common drawback of all these methods is that the operating constraints on system equipment (i.e., MW and MVAR limits of system generating units) are not taken into consideration. Production capabilities of generating units are important considerations, more so since voltage collapse is considered to be a reactive power problem. One suggestion is the use of repeated loadflowcomputations, as power injections are increased, to determine the voltage stability limit. Having determined the limit, the margin to collapse is then available. However, besides being computationally very demanding, this approach may be inadequate due to the unreliable behavior of the Newton-Raphson method of loadflowanalysis in the vicinity of the voltage stability limit. This behavior is linked to the singularity of the Jacobian at the voltage stability limit, and the existence of close multiple loadflowsolutions around that limit. Another suggestion is the use of a combination of loadflowanalysis and sensitivity parameters in order to reduce the computation time and circumvent the numerical ill-conditioning known to occur as the voltage stability limit is approached. Another possible short-cut to the repeated load flow calculations is a quadratic extrapolation method which Borresmans et. al. proposed. These methods are inherently approximate. The third suggestion is an approximate method to evaluate the condition at the voltage stability  5  Chapter I: Introduction  limit In [10] the problem of voltage collapse is approached by simulation of a power system using a slightly modified transient stability program. A sufficiently complex system with 39 buses and 10 generators is used in simulations. The system is stressed by progressively increasing the system load through a multiplier k. A very small change of k (order 1%) is used as collapse-inducing disturbance. Total system voltage collapse was observed after the disturbance. Significance of thesefindingsand directions for future research are presented and applicability of real-time control discussed. Several voltage collapses have had a period of slowly decreasing voltage followed by an accelerating collapse in voltage. In [11] this type of voltage collapse based on a centre manifold voltage collapse model is analyzed. The essence of this model is that the system dynamics after bifurcation are captured by the centre manifold trajectory and it is a computable model that allows prediction of voltage collapse. Both physical explanations and computational considerations of this model are presented. The use of static and dynamic models to explain voltage collapse are clarified. Voltage collapse dynamics are demonstrated on a simple power system model. In [12] a method of determining the voltage stability limit of a general multimachine power system was presented. In the method, the search for the voltage stabihty limit is formulated as an optimization problem of maximizing the system total MVA load. With this formulation, difficulties related to singularity of the loadflowequations Jacobian matrix, and convergence of the load flow solution around the voltage stability limit, are avoided. Voltage dependence of the MVA loads may be taken into account in determining the voltage stability limit. The method also accommodates device constraints or limitations in system controls (e.g. generator VAR limits and limits on transformer tap settings). A security margin is denned which serves as a measure of the security of the system as far as voltage collapse is concerned.  6  Chapter 1: Introduction  12 Literature Review on Applications of Catastrophe Theory In [13] M.D. Wvong and A.M. Mining introduce the application of catastrophe theory to transient stability assessment of power systems. A derivation is shown for a one-machine infinite-bus power system for which a swallowtail catastrophe manifold is derived from the swing equation of the system. From this a stability region is defined which is valid for changing load conditions and fault locations so that the stability of the system can be easily assessed. Figure 1.1 shows the steady-state stability region while Figure 1.2 shows the transient stability regioa  This method of transient stability assessment has several advantages: •  The regions of stability obtained by the method are well defined in terms of the control variables and the critical clearingtimeregardless of the state variables.  •  The computation required to define the stability are few and can be done in a very short time.  •  The method may be used for an on-line assessment of transient and steady-state (dynamic) stability.  7  Chapter I: Introduction  Figure 1.2:  Transient Stability Region  In [14] A.A. Sallam did similar studies on steady state stability assessment in power systems with similar results. Catastrophe theory has also been applied to otherfieldsin science with success. Some of the work done is described below: In [15] catastrophe theory is used to classify caustics and their associated traveltime diagrams. These traveltime diagrams are multivalued functions of the position coordinate when caustics are present. In [16] an experimental and theoretical investigation is presented for the forced vibration of a one-degree-of-freedom system with a non-linear restoring force. It is shown that the characteristics of these systems can be described by the cusp catastrophe model. And in [17] and [18] the rainbow effect in ion channeling is analyzed by the use of the catastrophe theory.  8  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  Chapter 2 A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  2.1 Introduction  In this chapter we are going to look at the application of catastrophe theory to voltage stability problems in a simple power system. This will give us insight into how the catastrophe model is developed. In later chapters we will do the same for a more realistic interconnected multi-bus power system. The simple 2-bus power system is shown in Figure 2.3. The problem is formulated with regard to voltage collapse at the load bus (bus 2). V1P1.Q1  V2P2.Q2  Source Load  Simple Power System  Figure 23:  22 Developing the Model  A catastrophe model for voltage stability is derived from the following system equations: Mioi + (ViV /Xi)sin(* - S ) = Pi  (2.1a)  r  2  1  2  -(ViV /Xi)cos(tf - Si) + (l/Xi)V? = 2  2  (V V /X )sin(^ - Si) = P 1  2  1  2  9  2  Qi  (2.1b) (2.1c)  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  (ViVj/XOcosto - *i) -  ((1/Xi) - B)V?. = Q  2  (2.1d)  Since we are only dealing with static voltage stability our angles do not change, hence 6j = 0. In our case bus 2 (the load bus) will be looked at from a voltage stability viewpoint. Therefore equation (2.Id) will be the equation that the catastrophe model will be built on. It will be the changes in V diat will be approximated. But since Vi is also present in equation (2.Id), we need to solve 2  for it in terms of V . This can be done from equation (2.1b) as follows: 2  V  -  2  V,V  2  QJXJ = 0  cos (<5 -6^2  (2.2)  This is a quadratic equation with the solution: Vi = \ (v cos(6 -60± 2  v  2  /(V cos2(^ -^))+4QiX ) 2  2  2  1  (2.3)  Substitution of equation (2.3) into equation (2.1d) will give [(Vl/(2X0)cos (* -^) 2  2  + ( V / ( 2 X ) ) c o s ( « - ^)^/(V cos2(^-^ )-r4Q X )] 2  1  2  2  2  1  -((1/X,) - B)V = Q  1  2  2  2  1  (2.4)  This can be rearranged as:  + V,( ° C  '>>) ^ V | co.' (h - «,) + 4Q.X,  S  (2.5)  where Q is a potential function of the variable V . 2  2  The catastrophe manifold Ms is the set  M = {V eR:/ (V )=0} s  s  3  a  10  (2.6)  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  where f  s  is the derivative of our potential function  / (V ) = 2V,[( S  C0S  2  '<*-'•>) - ( J - _ )]  2( ° ^ )vico '(feC  S(  y^«rf«i-  B  2  +  with respect to the state variable V :  <l)  2  S  ( " ^" €  <l)+  (  < l )  W  +  4Q X 1  1  )4Q,X,-0  (2.7)  Now let,  V  2Xi x  =  cos  2  (2.8)  (6 - Si) 2Xi 2  Equation (2.7) can then be rewritten as 2di V ^ c o s (* -* ) + 4Q Xi 2  2  2  +2C V^ cos (« -  1  + 4C Q X = 0  2  2  2  1  2  i  1  (2.9)  Now by applying Taylor's series expansion, the voltage at anytimecan be written as:  , , , AV? AVjj AV* V = V + AV + -7T + -jr0 + -KT 1 24 A 1  2  20  1  1  2  (2.10)  or with x = AV2 V =V 2  2 0+  x  x +  2  y  +  -  x  x  3  +  2  4  4  (2.11)  By substituting this into equation (2.9) and ignoring terms of higher than fourth degree we get the following:  11  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  2  V  1  x  20 +  +  f  f  +  ^]  +  + 2V x + (V + l)x + i + ^ V + ^ ± ^ x ]  2  2  0  20  x cos (c7 - 61) 2  2  V  C [v  4  20  + 4} / + 4C2Q1X! 1  2  + 2C cos  2  2  + 2V x + (V* + l)x +  2  2  +  2  2 0  (6 - «i)QiXi  20  =0  (2.12)  The square root in equation (2.12) is going to give us problems in the catastrophe formulation which we will need to take care of. One way is to eliminate it by a Taylor series approximation. The root can be rewritten as follows:  Jlfl ) 4Qi  cos(tf - ^)[(v 2  2 2 0  +c  6l)  + (2V o)x + (V + l)x 2  2  20  (2.13) The corresponding Taylor series would be: cos ^2\ 3b 5V +6V o+l 4b 2  K  + 1V 12b  where  2  4  2  20  2  4  b b / , 3V2 + V2, 5V|p\ 4 20 + 2 b 4 b* J +  0  +  x  6  4QiXi + V20 cos (£ - 6i)  b=  (2.14)  6  ....]  (2.15)  2  2  This can be simplified further as cos (6 - Si)[ k + kix + k x + k x + L,x ] 2  2  0  2  3  3  4  (2.16)  By substituting back into equation (2.12) and collecting like terms we have an equation of the general form:  12  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  EL4X  4  + aax + a x + aix + ao = 0 3  (2.17)  2  2  where ao =  + C V ko 2  20  ai = 2CiV -rC (ko + k V o) 20  2  a = Ci(V + 1) + C 2  20  a = C, ( ^ ) n  1  2  +h + k V )  2  2  20  (2.18)  + C ( ^ + I + k + knVao) 2  2  and Ci = 2 c o s ( « 2 - « i ) C i C = 2 cos (6 - £ i ) C  (2.19)  2  2  2  2  Now we let x = y + a , where a = - 4 ^ . Then our equation is  y + uy + vy + w = 0 4  2  (2.20)  where u,v and w are our control variables,  If  3a \ u= — a - - — 84 \ o 84/ 2  2  (2.21)  Hence we have developed a swallowtail catastrophe model for voltage stability for a simple 2 bus example. The bifurcation set for this catastrophe is shown in Figure 2.4.  13  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  W  A  Figure 2.4:  Bifurcation Set For The Swallowtail Catastrophe  23 Testing the Model Several studies have been done on voltage collapse for our simple 2-bus example. In [12] it was shown that for normal stable operation \ZR\ > \Zi\, where ZR is the load impedance and ZL is the line impedance. Figure 2.5 shows the variation of VR, the load bus voltage, against MVA demand SR, at constant power factor. Point A (where VR = V^ ) represents the critical system state. The upper LT  segment (VR > V^ ) is considered the stable operating region, i.e. ,{  < 1. It can be seen that in  the stable region, increasing the sending end voltage increases the receiving end voltage whereas, in the unstable region, increasing the sending end voltage actually reduces the receiving end voltage. The catastrophe model was compared with the above analysis in the following manner. Two voltage points were taken from the graph in Figure 2.5, where one point was in the voltage stable area and the other in the unstable area. By feeding the same data used in Figure 2.5 into the catastrophe  14  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  Figure 2.5:  Voltage Versus MVA At Load Bus (pf = 0.9285)  model it could be determined whether the model agreed with the analysis. All the stable points were found to lie above the catastrophe manifold shown highlighted in Figure 2.6, while all the unstable points were found to lie beneath it The critical point A in the voltage curve shown in Figure 2.5 was found to be close to S = 1.576397 p.u. The turning point in the curve according to the Catastrophe model was almost the same or close to 5 = 1.576292 p.u. This is the point at which our system shows stability as suggested by Figure 2.6. For the swallowtail catastrophe shown above there are basically three different cases, with only one being stable, that could result from our analysis. These are, above the manifold as shown in Figures 2.7 and 2.8, inside the tail as shown in Figure 2.7, and outside the manifold as shown in Figures 2.7 and 2.8.  15  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  V  Figure 2.6:  The Voltage Stability Manifold of the Swallowtail Catastrophe W  Outside Figure 2.7:  Manifold Manifold for u < 0  It was found that almost all the points of the voltage curve, stable or unstable, resulted in a control variable u > 0. Only the points close to the critical point of the curve resulted in a control variable u < 0. As the voltage curve went from unstable to stable (the critical point) it travelled from 'outside the manifold' to 'inside the tail', the two unstable regions, to 'above the manifold', the stable  16  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System. W  i ^^bove  Outside  1.  Figure 2.8:  Manifold  Manifold for u > 0  Regions of the Catastrophe  Intervals of S  Intervals of V?  (p.u.)  (p.u.)  Outside Manifold, u > 0 (Unstable)  0-1.569  0-0.598  Outside Manifold, u < 0 (Unstable)  1.569-1.573  0.598-0.609  Inside Tail (Unstable)  1.573-1.576  0.609-0.630  Above Manifold, u < 0 (Stable)  1.576-1.573  0.630-0.658  Above Manifold, u> 0 (Stable)  1.573-0  0.658-1.05  Table 2.1: The Travel of the Voltage Curve through the Catastrophe  region. Table 2.1 shows the points at which the voltage curve entered each region of the catastrophe. Similarly this can be shown as Figure 2.9 illustrates where the points shown are referred to Figure 2.10, which is a close up of the voltage curve of Figure 2.5 near the critical point. It should be noted that since the catastrophe is actually three-dimensional, the curve in Figure 2.9 is not exact It only serves for explanatory purposes and is not drawn to scale.  17  Chapter 2: A Catastrophe Model for Voltage Stability in a Simple 2-Bus System.  W A  Figure 2.9:  The Travel of the Voltage Curve through the Catastrophe  Figure 2.10:  Voltage Curve near Critical Point  18  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  Chapter 3 A General Catastrophe Model for Voltage Stability in Power Systems  In the previous chapter, voltage stability was studied for a simple 2-bus power system using a swallowtail catastrophe model. To apply the method to larger, more realistic, power systems it is useful to develop a general model for voltage stability of any power system. 3.1 General System Equations A general swallowtail catastrophe model can be developed for an n-bus power system directly from the systems equations. The most generalized way of stating the power flow equations of a power system [19] is n  Si = Pi+jQi =  ^{YsViVj + M^+jD^} j=i  D  =  E { l ' l l i K « " JBijJetoe-* + MjS] + jDj*j} V  V  G  j=i  = E {l ill il(G - JBiiXcos (Si - 6j) + j sin (Si -S )] + MjS] + j D ^ v  v  S  s  D  = E {l ill ilK » v  v  G  c o s  (ft ~ft)+ By sin (4 - Sj))  -rj(Gij sin (Si - Sj) - By cos (S - Sj))] + MjS] +}D S } (3.22) {  j  i  Since we are only dealing with static voltage stability our angles do not change, hence Sj Now by separating the real and imaginary terms:  Pi = E l ill jl[ ii v  v  G  c o s  (ft - ft) + « B  s i n  (4 - ft (3.23)  D  Qi = E l ill jl[ ii (ft - ft) - « v  v  G  sin  B  19  c o s  (ft - ft  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  or since  we have P i  irSi^  cos  = E  (ft - ft) - * x  s i n  (ft - ft)]  j=i ij + « K  A  irSit^  ^ = E  (3.25) si  »(ft - ft) + xy cos (ft - 6^)]  Again since static voltage stability is being modelled, all phase angles remain constant. It is only in dynamic voltage stability that changes in phase angles occur. The following constants can therefore be denned: Rij cos (Si — 6}) — Xij sin (Si - ft)\  <8 =  u  y  7  , _ / Rij sin (ft - ft) + Xij cos (Sj - 6$)  * Voiiagc stability  =  I  Ri + X i  (3.26)  ,  is synonymous with reactive power stability where voltage is the dynamic  variable (likewise, the phase angle is the variable in real power stability studies). For this reason only the reactive power equations are of interest in this study. Therefore our set of equations is:  Qi=  £ I  V  M I  V  ^  (3.27)  j=i 3.2 Solving the System  We now define any bus k as the one to be examined from a voltage stability viewpoint Its equation on which the catastrophe model will be built is a single case of the set of equations above or D  Qk=  ElVkHVjIVkj  20  (3.28)  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  where Qk is a potential function of the variables VJt,Vj (j = l,2,..,n). But these variables are related through the other system equations. Therefore by solving the system equations the variables Vj (j = 1,2, ..,n) could be stated in terms of only V*.  Essentially what we have is a set of n functions of the form:  Qi = fi(Vi,Vj)  (3.29)  V = g (V )  (3.30)  which can be rearranged as  i  i  J  if Qi is considered to be constant. However, Q^ is not considered a constant since it is our potential function. All the reactive powers effectively vary, but for buses other than the bus under study the variation is small enough to be neglected. Thus by going through the system and solving at each node i for the corresponding voltage Vi and substituting into the next equation and so on, our potential function can be written in terms of V% only. For a strictly radial system this can be done in a straightforward manner and does not need further details. On the other hand for a non-radial system (loop or circular connections) problems will arise because of the nonlinearities of the equations. Figure 3.11 shows an example of this. Here we have four buses which are all interconnected, and have two connections to the rest of the system at buses u and z.  To be able to solve the whole system we therefore need to be able to solve implicitly  for the voltages at buses u and z.  21  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  bus U  other buses Figure 3.11:  Non-Radial Power System  The functions at the four buses upc,y,z will be  Qx = f (v ,v ,v ,v ) x  x  u  y  z  Qy = f y ( V , V , V , V ) y  u  x  z  (3.31) Qu  =  Qz  =  fu(V ? V , V y , V , Vj) u  x  z  fz(V , V , V » V y , V j ) z  u  x  where K and V, are voltages at other buses connected to buses u and z. This can be rewritten as V ^ g ^ V v . V , ) Vy = g y ( V , V „ V ) u  x  (3.32) V\l  =  gu(V , Vy , V , Vj) x  z  • V = g (V ,V ,Vy,Vj) z  x  u  x  This cannot be solved explicidy for V or V and therefore voltages V{ and Vj can't be solved either u  z  and so on. 22  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  Note, however, if V has an explicit solution in terms of V , i.e., X  u  (3.33)  V = gx(V ) x  u  then Vu can be solved as V = g ( V , V ) = g (V ,g (V )) u  u  i  x  u  i  x  u  = h (V0  (3.34)  u  where bus i precedes bus u in sequence. The same then goes for Vj etc. Therefore we need to write V and V in the same form as shown in equation (3.33), that is U  Z  with an explicit solution of only one variable. Since we are dealing with small changes in voltage and since these changes are going to be smaller the further away we are from the specific bus of study they can be dismissed as shown below. When solving for Vu, then V and V can be written as X  Y  V x « g x i ( V , V , V ) = h (V ) u  ¥O  l0  x  u  (3.35) Vy «gyl(V ,V ,V ) = M V u ) u  x0  z0  where V Q, V o and V o are the initial voltage values at buses x,y and z. Now we rewrite the equation X  v  2  for Vu as V = g„(V ,V ,V„Vi)wg (g i(V ,V o,V o),g -i(V ,V ,V,o),V ,Vi) u  x  y  u  x  u  y  x  >  u  x0  l0  (3.36) or  V = h (Vi,V o,V o,V o)=n (Vi) u  u  x  y  z  u  (3.37)  since Vro, V o and V o are constants. y  z  Similarly when solving for V , V or V^ can be written as Z  X  V » 6 « 2 ( V „ V o , V o ) = h (V,) y  x  u  x  (3.38) V «gy (V ,V o,V o) = h (V ). y  2  z  x  u  23  y  z  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  Again we rewrite the equation for V as Z  V, = gz(V ,V ,V ,Vj) * gz(V o,gx2(V ,V o,V o),g (V ,V o,V ),V ) u  x  u  y  z  u  y  y2  u  u  x0  j  (3.39)  or V, = h,(Vj, V , Vxo.Vyo) = h,(Vj).  (3.40)  u0  Each of the solutions of equation (3.30) will be a quadratic solution of equation (3.29) which can be rewritten as V j V ^ + v? £  Qi = Vi J2  V  (3.41)  2ij  j=lj#  j=lj#i  where Vj = VJQ when systems are inteconnected as described above. VjVio  E  Qi  Vf + V i - ^ J  E  E  V>2ij  = 0  (3.42)  V^ij  which will give the solution:  , 2  E Vi^ia  *5  D  E  1  E VjVV D  E V>2« 2va«  \  <h»  N \ i=ij#  / /  +  4Q;  (3.43)  D  E  V^ij  i=to#i  This will, however, become unsolvable as we go through the system since the degree of the solution will increase by one at each step. But this problem can be overcome by approximating all voltages at buses far away from the 1  specific bus as Vi = V 1  i0  + AVi  far away = all buses but the ones next to the specific bus.  24  (3.44)  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  by disregarding higher orders of AVj. Now equation (3.41) can be solved simply for AV, or even better it can be solved in the same way for Vi = V;o + A V ; . Qi^CVio + AVO  J2 VjVis  + (Vio + A V ; )  E  2  V>2ij  = (V + AVi) £ VjVia + (V + 2V AV ) E V*ii 2  i0  i0  (Vio + AVO E = (V + AVi) i0  i0  i  VjViij +(2V (V + AV )-V ) E  ^  2  i0  E  V  J^S  i0  i  + io E  ^  2 V  0  -v  2 0  E  (3.45)  ^  Therefore,  Qi + v o E V>ij 2  2  Vi = V  i 0  + A V i = —s E  " VjViij + 2V J  ~  1 J ?  i0  „ E  (3.46) ^  However, the solutions of Vj which have previously been solved in terms of V,- could have been substituted into equation (3.46) leaving only one unknown voltage variable. We shall call this variable, Vj. Now equation (3.46) can be written as Vi = r - ^ - v li + mjVi  (3.47)  where fc;,/,,m, are constants. This solution of Vj is then substituted into the remaining system equations to solve for V/ in the same way. All buses, except those near bus k, are solved this way. Since the voltage changes are larger the closer to the specific bus (the voltage collapse bus) more accurate voltage solutions are desired. This time the precise solutions of equation (3.43) are used for solving these buses next to the specific bus. The format of that equation will change somewhat though since all known Vj solutions of equation  25  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  (3.47) have been substituted in. The solution will be of the following form:  V i = -^(ai + AVO + ^ ' + A V O ' + l i = ~ ( a i + A V ) + \ JtfVl  + 2aiAV + a? + i  k  k  (3.48)  7  where £ is the specific bus. The potential equation at bus it is similar to equation (3.41) or, D  Qk = V  II  £  k  ^2kj  VjVikj + Vl Y,  (3.49)  where all the Vj's have previously been solved so that only one variable is left in the equation, namely V . k  33 General Swallowtail Catastrophe Modelling Since our potential function Q is of the same form as previously shown for a 2-bus example, the k  steps involved in developing the catastrophe model are basically the same from here on and there is no reason for going through them again in detail. As said previously, the catastrophe manifold Ms is the set Ms = {V 6 R : f(V ) k  where f  s  = 0}  k  (3.50)  is the derivative of our potential function Qf with respect to the state variable V  k  By applying Taylor's approximations of functions for the changes in voltage, the voltage at any time can be written as:  V =V 2  20  + AV +  AV|  2  A V | AVJ  2  6  +  24  +  (3.51)  or with x = A V as 2  __  ..  V = V „4-x _ 2  2  +  26  X  +  -  X +  -  (3.52)  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  After approximating the squarerootswith a 4th order Taylor series and collecting like terms we get:  a4X + aax + a x + aix + ao = 0 4  3  (3.53)  2  2  where  52 TJVikj + V  ao = V k 0 D a  i=  E  E  jV'ikj+2V o 52 ^2kj k  j=lj*k  52 V>2kj  T/Vikj + (V k 0 + 1)  j=lj#k  *> = \ E  V2kj  D T  j=lj#k ^2=2  £  2 k 0  (3.54)  j=lj/k  T J W ^  J=U#k  +  I) E  ^  ' j=lj#k  v  £r!(! T j j? the r-th constant in the corresponding taylor series. Now by letting, x = y + a , where, a =  our equation can be written as:  y + uy + vy + w = 0 4  (3.55)  2  where u,v and w are our control variables:  If 3 a?A u = —I a ) a4 \ 8 a4 / a — +, -1^ai!| \ v = —If a i - — &4\ 2 an 8 a| / _ ao aj an a a 3 a ~a7~Ta|l6ai~256ai 2  _ (3.56)  2  2  2  W  27  4  Chapter 3: A General Catastrophe Model for Voltage Stability in Power Systems  This is precisely the same swallowtail catastrophe model for voltage stability as derived earlier for our simple 2-bus example. The bifurcation set for this catastrophe is shown here again in Figure 3.12.  28  Chapter 4: Examination of the General Model on a Test System  Chapter 4 Examination or the General Model on a Test System  4.1 Test System  Even though the catastrophe model had proven to work for a simple 2-bus system, studies needed to be done on larger systems for realistic purposes. This realistic system was needed to have the following four characteristics and conditions for our purposes: •  Non-radial system.  •  Losses accounted for.  •  Load flow results available.  •  Generator and load buses.  The system chosen [20] was a 6-bus single loop system as shown in Figure 4.13. This system has line-losses parameters available, a load flow has been computed and it has two generator buses and four load buses. Table 4.2 shows the loadflowresults which are needed for the catastrophe analysis. Bus 6 of the system is a slack bus which means that the voltage remains constant and the demand of power required would always be met. Thus bus 6 was excluded from our cases of voltage stability studies (No voltage change, no voltage collapse). All the otherfivebuses were of course examined though as the following sections show. This system was then formulated according to the general model described in Chapter 3. 4.2 Initial System Conditions  First the system was analyzed using the initial loadflowconditions given in Table 4.2. Table 4.3 shows the results from the catastrophe program at each of the five buses. The stability and the unstability regions are identical to the ones shown and discussed in Chapter 2 since we are still dealing with the swallowtail catastrophe.  29  Chapter 4: Examination of the Genera! Model on a Test System  •J34.1 p.u. _l 0.08+J0.37 p.u. jO.133 p.u. t=0.909 0.123+j0.518p.u. 0.723+j 1.050 p.u. t=0.974  jO.3 p.u. 0.282+J0.64 p.u.  -J28.5 p.u.  Figure 4.13:  Bus  Voltage Magnitude, V  Number  (p.u.)  1.  A 6-Bus Power System  Voltage Angle, 6 (°)  Reactive Power, Q (p.u.)  0.932061  -9.841571  0.218091  2.  1.003495  -12.770030  0.016298  3.  1.103648  -3.389008  0.370004  4.  0.924158  -12.282888  0.179988  5.  0.922237  -12.182555  0.153299  6.  1.05  0  0.497890  Table 4.2: Load Flow Results for 6-Bus Power System  As expected the system is voltage stable at all buses for the initial conditions but of interest  30  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9888  7.9888  11.7420  Stable  2.  1.1586  2.5105  3.9941  Stable  3.  5.3547  7.1235  17.1505  Stable  4.  1.0936  3.3229  3.1817  Stable  5.  5.1193  6.9659  7.1436  Stable  Stability Census  Table 4.3: Control Variables for Initial Conditions is to find out which bus is the least or most stable and so on. In later sections we will vary some of the state variables and then see clearly which buses are more sensitive to voltage collapse. But it is possible to predict from the above control variables which buses would be most susceptible to voltage instability although such a prediction might not be very accurate. This kind of prediction would be based on the proximity to the catastrophe manifold. Each of low control variable u, high v and low w by themselves or especially in combination could indicate this proximity. By looking at Table 4.3 it can be seen that both buses 2 and 4 have arelativelylow control variable u. Similarly buses 1 and 3 have arelativelyhigh control variable v and buses 2 and 4 have a low control variable w. From this it could be concluded that again buses 2 and 4 would be more vulnerable to voltage collapse than buses 1, 3 and 5. This would make sense since both these buses (2 and 4) are the load buses furthest away from the two generators. 43  Voltage Variations  As previously stated the two state variables for static voltage collapse are voltage and reactive power. Lower voltage and/or higher reactive loads would make a power system more vulnerable to voltage instability. In this section the effects of strictly voltage variations on voltage collapse are examined using the catastrophe model. Voltage values were lowered until the model predicted a voltage collapse  31  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9887  7.9887  7.6945  Stable  2.  0.7667  1.8239  3.9941  Stable  3.  5.3014  6.9908  12.3260  Stable  4.  -661.50  6940.07  -19200.91  Unstable  5.  5.2232  7.1251  9.6979  Stable  Table 4.4: Control  Stability Census  Variables with 82.6% Initial Voltage Values.  at one of the buses. However since a new loadflowwould effectively be needed for each new voltage profile the exact voltage value at which the collapse occurs may not be accurate. But the purpose of this test is not tofindthe exact voltage collapse points for the power system in Figure 4.13 as much as it is to prove that the catstrophe model works for voltage collapse predictions. Thefirsttest was done by varying all the voltage profiles equally. Voltage collapse was found to  occur at bus 4 at 82.6% of the value of the initial voltages as shown in Table 4.4. The other buses are still stable at this point but the whole system would collapse though due to the instability at bus 4. Individual voltage variations are also of interest although there are of course infinite combinations of the different bus voltages. The tests done here involve varying only one bus voltage value while the other voltages remain constant at the initial values. Since bus 4 proved to be the weakest one in the uniform voltage variation test above it was of particular interest for this examination and as can be seen from the following tables it still has the highest bus voltage for which it goes unstable (the weakest bus). Studies done on bus 3, a generator bus, found that voltage collapse occurs at a very low voltage 32  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9888  7.9888  2.3813  Unstable  2.  1.1586  2.5105  3.9941  Stable  3.  5.4484  ' 7.2298  19.5364  Stable  4.  0.9080  2.8032  2.7308  Stable  5.  5.1121  6.9491  6.8755  Stable  Stability Census  Table 4.5: Control Variables with 58% of Voltage Values at Bus 1.  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9886  7.9886  11.7386  Stable  2.  0.2643  2.7717  1.4320  Unstable  3.  5.2575  7.0099  15.5127  Stable  4.  0.8394  2.6126  2.5673  Stable  5.  5.1188  6.9471  6.7781  Stable  Stability Census  Table 4.6: Control Variables with 52% of Voltage Values at Bus 2.  (29%) and the model cannot be used for some of the other buses at this voltage (buses 1 and 5). This is not at all unreasonable since there is no way the system would support these voltage conditions. Also voltage collapse rarely occurs at generator buses because of greater voltage control. 4.4 Load Variations  Now we will examine the effects of reactive load variations with the voltage constant at the initial values. This testing was done in a similar way as before, only in this case by increasing the loads  33  Chapter 4: Examination of the General Model on a Test System  Stability Census  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  -  -  -  -  2.  1.1381  2.4769  3.9709  Stable  3.  5.0389  6.8004  2.0884  Unstable  4.  1.0709  3.2673  3.1439  Stable  5.  -  -  -  -  Table 4.7: Control Variables with 29% of Voltage Values at Bus 3.  Bus  Control  Control Variable,  Control  Stability Census  Number  Variable, u  V  Variable, w  1.  5.9887  7.9887  10.6383  Stable  2.  1.1536  2.5235  4.0283  Stable  3.  5.3788  7.1301  17.0745  Stable  4.  28.7936  131.5285  131.0456  Unstable  5.  5.1146  6.9520  6.9110  Stable  Table 4.8: Control Variables with 72% of Voltage Values at Bus 4 .  until voltage collapse occurred at one of the buses. Thefirsttest involved increasing all the bus loads uniformly until voltage instability was reached. This happened at 5.5timesthe initial loads as shown in Table 4.10. Again it was bus 4 which failed first. The tests for the individual changes of loads at each bus with the other loads constant at the initial values provided some interesting results as the following tables show. The model does not 34  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Stability Census  Number  Variable, u  V  Variable, w  1.  5.9888  7.9888  12.0828  Stable  2.  1.1018  2.3576  3.8180  Stable  *>  J).  5.4118  7.1429  17.0061  Stable  4.  1.0936  3.3229  3.1817  Stable  5.  5.0856  6.9298  2.1775  Unstable  Table 4.9: Control Variables with 70% of Voltage Values at Bus 5 .  Bus  Control  Control Variable,  Control  Stability Census  Number  Variable, u  V  Variable, w  1.  5.9893  7.9893  12.2171  Stable  2.  3.9361  8.6089  10.4715  Stable  3.  5.6998  7.4385  17.3363  Stable  4.  15.3072  12.5569  2.3516  Unstable  5.  5.1128  6.9538  6.9585  Stable  Table 4.10: Control Variables with 53 times the Initial Reactive Loads.  provide a stability census at the load change bus simply because the reactive power at that bus is not a variable in the catastrophe model. This is because the potential function of the catastrophe model is the reactive power equation at the bus in question. This is fully acceptable since for the value of the function to change then the variables of the function must change as well. As before only the initial loadflowdata was used, so the data is not accurate, but this is sufficient to show that the catastrophe model is valid. This inaccurate data is probably the reason for the high  35  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  2.  5.6703  12.0631  5.4013  Unstable  3.  5.6220  7.4188  31.6700  Stable  4.  0.7265  2.4308  2.5738  Stable  5.  5.1047  6.9267  6.5458  Stable  Stability Census  Table 4.11: Control Variables with 240 times the Initial Reactive Load at Bus 1.  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.7234  8.2060  9.6336  Stable  3.  -5.9235  -14.7004  -16.2437  Unstable  4.  0.9568  2.8519  2.6492  Stable  5.  5.1190  6.9724  7.2764  Stable  Stability Census  Table 4.12: Control Variables with 465 times the Initial Reactive Load (small) at Bus 2.  Bus  Control  Control Variable,  Control  Stability Census  Number  Variable, u  V  Variable, w  1.  5.9901  7.9901  14.5094  Stable  2.  1.7603  3.1642  1.5576  Stable  4.  1.2349  3.2494  1.4174  Unstable  5.  5.1954  7.0921  10.6266  Stable  Table 4.13: Control Variables with 59 times the Initial Reactive Load at Bus 3.  36  Chapter 4: Examination of the General Model on a Test System  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9887  7.9886  11.5493  Stable  2.  0.9214  1.9520  3.3876  Stable  3.  -  -  -  -  5.  2.6522  .10.5990  -0.4502  Unstable  Stability Census  Table 4.14: Control Variables with 11.3 times the Initial Reactive Load at Bus 4.  Bus  Control  Control Variable,  Control  Number  Variable, u  V  Variable, w  1.  5.9888  7.9888  11.6117  Stable  2.  1.1765  2.5387  4.0117  Stable  3.  5.2966  7.1363  17.7766  Stable  4.  16.0800  6.8469  -0.2222  Unstable  Stability Census  Table 4.15: Control Variables with 8.5 times the Initial Reactive Load at Bus 5.  load before bus 1 collapses and why in one case a solution is not found for bus 3, a generator bus (Table 4.14). 4.5 Load and Voltage Variations Now since the effects of both the state variables have been individually looked at it would be interesting to examine how variations of both of them simultaneously would affect the voltage stability and the catastrophe model. It is an obvious conclusion from the proceeding sections that lower voltage and higher reactive loads bring a system closer to voltage instability. Table 4.16 shows how the two state variables can bring the system quickly to collapse if both are changing. The higher the load becomes the higher the critical voltage point (the voltage value for which the system  37  Chapter 4: Examination of the General Model on a Test System  Reactive Load Factor  Critical Voltage Factor for Collapse (Bus 4 in All Cases)  1.0  0.826  2.0  0.823  3.0  0.873  4.0  0.929  5.0  0.979  5.5  1.0  6.0  1.022  7.0  1.06  Table 4.16: Effects of both Load and Voltage on Voltage Stability.  becomes unstable). It should be noted that the factors in Table 4.16 are uniform over all buses and collapse occurs at bus 4 (the weakest bus) in all cases.  38  Chapter 5: Conclusions  Chapter 5 Conclusions  It has been shown i n this thesis that catastrophe theory can be used to predict voltage collapse i n power systems. A general swallowtail catastrophe model has been developed and applied to a realistic power system. F r o m this model the voltage stability status for any given system condition can be predicted.  The model is based on finding the control variables of a swallowtail catastrophe based on the system equations for a specific bus and determining i f they lie within a unique stability region i n the control space. This is determined for one bus at a time where the single state variable of the swallowtail catastrophe is the voltage at the particular bus. are solved i n terms of this voltage.  Prior to this all other bus voltages  However because o f the nonlinearity and the possibility of  interconnections i n the system some approximations need to be done to obtain this single state variable.  In some cases the approximations involve using initial voltage values for some of the  bus voltages far away from the particular bus. But since during voltage collapse, voltage changes obviously take place, and although those changes might be minor at buses far away from the point o f collapse the effects o f these approximations need to be studied further.  The model was first tested on a simple 2-bus power system which had some previous voltage stability results available. The results from the catastrophe model were compared w i t h the k n o w n results and when proved to be accurate the model was extended to a more realistic interconnected power system. F o r a six-bus test system the model predicted voltage collapse at high loads and/or l o w voltages as would be expected.  N o general methods are available to predict voltage collapse quickly and easily i n this way, so the use o f catastrophe theory is novel and very promising.  39  Chapter 5: Conclusions  These results are very encouraging for further work using catastrophe theory for voltage stability studies especially as a possible online stability tool. More research lies ahead before this will be realized because the problem of voltage stability is very complex. Studies need to be done on very large power systems where voltage stability data exists. One such system is the so-called New England power system which has been designated as a voltage stability study system. Also it is of interest to try to formulate dynamic voltage stability as well, but that would include the power angles as a state variable and therefore increase the complexity of the system. This would bring us closer to a general voltage and power stability solution for power systems.  40  Appendix A The Catastrophe Theory and the Swallowtail Catastrophe  A.1 Introduction Catastrophe theory [21] grows where algebra, calculus, and topology meet each other, and is concerned with the study of real-valued functions of several real variables. As a part of mathematics, catastrophe theory is a theory about singularities. When applied to scientific problems, therefore, it deals with the properties of discontinuities directly, without reference to any specific underlying mechanism. The theory attempts to study how the qualitative nature of the solutions of equations depends on the parameters that appear in the equations. Elementary catastrophe theory is the study of how the equilibria ^j{C ) of V($j\ C ) change as Q  a  the control parameters C change. Catastrophe theory shows that the number of qualitatively different a  configurations of discontinuities that can occur depends upon the number of control variables, which are generally few, and not upon the number of state variables, which may generally be many. If wc have a family of functions V:F x C ^ R  (A.56)  where J is a manifold, usually R , and C is another manifold, usually R . R is the state space and 1  n  r  n  R is the control space. The state space has dimension n and the control space has dimension r. A T  manifold is a term to indicate a high-order surface (hypersurface), e.g., a 1-dimensional (lst-order) manifold is a curve, a 2nd-order manifold is a contour, etc. The catastrophe manifold, N, is the subset R x R denned by n  T  V,V (x) = 0 c  (A.57)  where V (x) = V(x,c) and x is the state variable. This is the set of all critical points of all the c  potentials V in the family V and V is the partial derivative with respect to x. c  x  41  N o w we find the singularity set, S, which is the subset o f the manifold, N, that consists o f all singular points o f V. These are the points at which  V V (x) = 0  (A.58)  V V (x) = 0  (A.59)  x  c  and 2  c  T h e singularity set 5 is then projected d o w n onto the control space R to obtain the bifurcation set B. T  T h e bifurcation set is the image o f the catastrophe manifold N i n the control space C. The bifurcation set B provides the projection o f the stability region o f all possible stable points o f V i n terms o f the control variables, which usually represent the system parameters.  A.2 The Elementary Catastrophes F o r systems with less than five control variables, there are seven distinct types o f catastrophes [21] and no more than two state variables are involved i n any o f these. These catastrophes are called the elementary catastrophes. •  r = l , rt=l. The fold catastrophe.  V (u,x) = U X + i  •  (A.60)  3  F  x  r=2, n=l. The cusp catastrophe. V ( u , x ) = u i x + ^u x + i x 2  c  (A.61)  4  2  •  r=3, n = l . The swallowtail catastrophe.  V (u,x) = m x + ^u x + ^ x 2  s  I  42  2  o  3  + Ix* o  (A.62)  r=3, n=2. The hyperbolic umbilic catastrophe.  (A.63)  V ( U , X ) = U i X i + U X + U3X1X2+X1 + X j D  2  .  2  r=3, n=2. The elliptic umbilic catastrophe.  V (u,x) = uixi + u x + un(x + x )-t-Xj - 3xix E  2  2  •  2  2  2  r=4, n=l. The butterfly catastrophe. V (u,x) = uix + l x + \x* + ±x + ^x z o 4 0 D  2  4  6  U 2  •  (A.64)  (A.65)  r=4, «=2. The parabolic umbilic catastrophe.  V (u,x) = uixj + u x -(n  2  2  U3X  2  +  U4X +x x + x 2 2  2  2  (A.66)  The set in R x R where the differential of V is zero turns out to be a differential manifold n  T  u  for each one of the elementary catastrophes. It is sometimes called the catastrophe manifold, and in other contexts the equilibrium set. This set is then projected onto R . The collection of the critical T  points of the projection is of particular interest. It is sometimes called the catastrophe set. The image of the catastrophe set under the projection is the set of all critical values of the projection. It is sometimes called the bifurcation set. In this thesis, catastrophe theory has been applied to voltage stability analysis in power systems for the purpose of finding stable and unstable regions directly from the system equations. The elementary catastrophe chosen for the study was the swallowtail. Some reasoning was behind this choice. Firstly, our system equations only have one state variable in terms of voltage stability, namely 43  voltage. Therefore the hyperbolic, elliptic and the parabolic umbilics were eliminated since they all have two state variables. This left us with a choice of anything from a one-dimensional to a fourdimensional control space. The fold was immediately thought to be too simple to produce accurate enough results. For similar reasons the swallowtail prevailed over the cusp, i.e., without getting to complicated it would be more accurate. Finally, the butterfly was considered to be complicated, simply for the reason that the control space would be 4-dimensional and hard to visualize. Also some previous studies on power systems had been done using the swallowtail catastrophe [13],[14]. A3 The Swallowtail Catastrophe The derivative of the potential function V (u,x) = uix + ^u x + ^u x + ^x z o o s  2  n  2  (A.67)  5  3  with respect to the state variable x is = 11!+ u x + u x + x  / (u,x) = 5  2  2  (A.68)  4  a  The catastrophe manifold Ms is the set Ms = {(u,x) € R x R : / (u,x) = 0} n  (A.69)  5  The catastrophe set Cs is the set of all points in Ms for which d e t  ef^x) ox  _  d  e  t  ,  +  2  ^  +  ^  =  u 2 +  2 u  ^  x+  _  0  ( A > 7 0 )  '  v  Cs is therefore the solution set of the system ui + u x 2  + U.-JX + x 2  4  =  0  (A.71) u + 2u x + 4x = 0 n  2  3  Eliminating the second term in the first equation yields an equivalent system of equations: ui - unx - 3x = 0 2  4  (A.72) u + 2u x + 4x = 0 3  2  3  44  From this we obtain C = {(st + 3t , -2st - 4t , s, t) : (s, t) e R } 2  4  3  (A.73)  2  s  The bifurcation set Bs is the projection of Cs onto it*: 3  B = {(st + 3t , -2st - 4t , s, t) : (s, t) G R } 2  4  3  (A.74)  2  s  We can have a closer look at Bs by studying its cross sections with the planes E? x {s}. The projection onto B? of the cross section is the image of the real line under the mapping A : R —• R ,A (t) = (st + 3t , -2st - 4t )  (A.75)  B n (R x s) = A (R) x {s}  (A.76)  s  2  8  2  4  3  that is, 2  6  B  Since X"(—t) = (X\(t), — A | ( t ) ) , the image X"(R) is symmetric in theui axis. \"(R) is a curve in R?. Its slope can be computed from (X )'(t) = 2(s + 6 t s  2  )(t, - 1 ) .  We obtain  t^O  The curve turns left. The direction of movement along  m  m  .  A*(.ft)  (A 77)  is given by  W i - 5 f c * !(t! l l),  ~wm'^mw '45  11  (A7S1 <A 78>  -  Figure A.14:  The Tail of the Swallow  If s < 0, s + 6t changes sign from positive to negative at t\ = --^/(-s/6) and again from 2  negative to positive at t = y/(-s/6). 2  At all other points 6 is continuous. We have  S(t) = /  lim  t->tr  (ti,-l)=  S  2  4s  -lim«(t)  /-S  (A.79) lim  *(t) =  .  ( t , - l ) = - lim S(t)  1  2  s  2  t-*t^  4s  f-s  The points A ( t i ) and A ( t ) are cusp points. The tail of the swallow can now be seen in Figure A.M. s  s  2  If 5 = 0, s + 6t = 6f , hence 6(t) is continuous, except for t = 0. Even there it has a limit: 2  2  lim^(t) = (0,-1). The curve is shown in Figure A.15. If s > 0, s + 6t > 6t , hence £(<) is continuous. X is an embedding. The curve is a 2  2  3  one-dimensional submanifold. It is shown in Figure A. 16.  46  s =o Figure A.15:  Manifold for s  **(R)____  S>0 Figure A.16:  Manifold for s  47  Appendix B Program Code: General 6-Bus Case  The swallowtail catastrophe model for voltage stability was programmed using the C programming language [22], T h e program was mostly built o n function prototyping were each parameter and variables were calculated i n their o w n function. Data was read from a file and then the program found the control variables and determined voltage stability from them. The program is a general program for the 6-bus example where any or all o f the buses could be examined for voltage stability. A general program for any power system would need to use arrays o r pointers to greater extent and subprograms would be needed for the system solving prior to the catastrophe manipulation.  A n example o f the datafile read i n to the program follows the program listing.  /*  GENERAL SWALLOWTAIL CATASTROPHE PROGRAM FOR A N A L Y S I N G VOLTAGE S T A B I L I T Y FOR A S P E C I F I C 6-BUS EXAMPLE * /  •include •include /* double  <math.h> <stdio.h>  Global X[6]  Variables  Declarations  * /  , R [ 6] , X c l , Xc5, Q [6] , V O [ 6 ] , t l , t5, d [ 6] , a l [ 6] , b e [ 6] , o m [ 6] , k s i [ 6] ,  j[7],k[7],l[7],m[7];  double Xcl,Xc5,tl,t5; int chosebus,cbus,bus,nbus,cas;  /*  Main  Program  * /  m a i n () < /*  Function  Declarations  * /  48  d o u b l e V t () , V t l o w ( ) ,T() , Tlow ( ) , c o n t r o l U () , c o n t r o l V ( ) , c o n t r o l W ( ) , paramAO () ,paramAl(),paramA2(),paramA3(),paramA4() , c o n s t a n t C l () , c o n s t a n t C 2 ( ) , c o n s t a n t C 3 () ,constantC4(),constantC5(),constantC6() ,S0<),S1(),S2(),S3(),S4(),T0(),T1(),T2() ,T3<),T4(),P0(),P1(),P2(),P3(),P4(),R0() ,Rl(),R2(),R3(),R4(),plO(),pll(),pl2() ,pl3O,pl4(),rl0<>,rll(),rl2(),rl3() , r l 4 0 , k 4 ( ) , 1 4 0,ni4(),k3<),13(),m3 0 , j2 0 , k 2 0,12 0 m 2 ( ) , k l O , l i t ) , ml () , a l p h a 5 {) ,beta5 O , omega5 ( ) , k s i S O ,alpha4 O , b e t a 4 O,omega4 0 , k s i 4 ( ) ,alpha3(),beta3(),omega3(), ksi3{) ,alpha2 0 ,beta2 0 , omega2 0 , k s i 2 () ,alphal(),betal(),omegal0,ksil0; f  /* V a r i a b l e s */  double contv.x; int i , c ; /*  F u n c t i o n f o r Input o f Data C a l l e d */  d a t a f (); /*  F u n c t i o n f o r C h o o s i n g V o l t a g e C o l l a p s e Bus */  buschoice(); for  (c = 1; c <= c a s ; C + + ) ( i f (chosebus == 6) { cbus = c; ) else { cbus = chosebus;  /*  Loop f o r Each Bus  /*  Case o f A l l Buses Examined  */  Case o f S i n g l e Bus Examined  */  /*  */  i  /*  Initializing  Arrays  */  m[0] = 0.0;m[l] = 0.0;m[2] = 0.0;m[3] = 0.0;m[4] = 0.0;m[5] •= 0.0;m[6]=0.0 1[0] = 0.0;1[1] = 0.0;1[2] •= 0.0;1[3] = 0.0;1[4] = 0.0;1[5] = 0.0;1[6]=0.0 k[0] = 0 . 0 ; k [ l ] = 0.0;k[2] = 0.0;k[3] = 0.0;k[4] = 0.0;k[5J = 0.0;k[6]=0.0 j [ 0 ] = 0 . 0 ; j [ l ] = 0.0;j[2] = 0 . 0 ; j [ 3 ] = 0.0;j[4] = 0.0;j[5] = 0.0;j[6)=0.0 k[0] = - V 0 [ 0 ] ; k [ 6 ] •= - V 0 [ 0 ] ; a l [ l ] = a l p h a ! ( ) ; a l [ 2 ) •= a l p h a 2 () ; a l [3] = a l p h a 3 () ; a l [4] •= alpha4 0 ; al[5] = alpha5{);al[0]=0.0; b e [ l ] - b e t a l ().-be [2J = beta2 O ;be [3] = b e t a 3 ( ) ; b e [ 4 ] = beta-4 O ; be[5] = b e t a 5 ( ) ; b e [ 0 ] = 0 . 0 ; om[l] = omegal () ; om[2] «= omega2 () ;om[3] = omega3 () ;om[4] = omega4(); om[5] = omega5();om[0]=0. 0; k s i [ l ] = k s i l ( ) ; k s i [ 2 ] = k s i 2 ( ) ; k s i [ 3 ] = k s i 3 () ; k s i [4] = k s i 4 () ; k s i [ 5 ] = k s i 5 ( ) ; k s i [ 0 ] = 0.0; /* for  Control  f o r Bus C o n s t a n t s C a l c u l a t i o n s  */  ( i = 5; i >= cbus + 1; i ~ ) < bus = i ; i f (bus == 5) { i f (bus == cbus + 1) { k[bus] = k 4 ( ) ;  49  IK);  Constants  of  Type  Four  k l Oi 110; ml();  Constants  of  Type  One  Constants  of  Type  Three  Constants  of  Type  Two  k 4 () 14 0 ml 0  Constants  of  Type  Four  k l O; 110; ml();  Constants  of  Type  One  Constants  of  Type  Three  Constants  of  Type  1[bus] mfbus] ) else f k[bus] 1 [bus] m[bus] }  ) else  { n b u s = b u s + 1; If ( b u s == c b u s + 1 ) ( k3(), k[bus] 130; 1[bus] m3 O ; m[bus] ) else  (  j [bus] k[bus] 1[bus] m [bus] )  for  (1=1; bus i f  i  <=  j2() k2() 120 m2()  cbus-1;  i++)  = i ; (bus • { i f  (bus  1) ==  k[bus] 1[bus] m[bus]  cbus  -  1)  )  el se <  k[bus] 1[bus] m[bus] > else  nbus = bus 1; i f ( b u s == c b u s (  - 1)  k[bus] 1[bus] m[bus] )  = k3(); = 1 3 0 , = n>3(),  j[bus)  j2(); k2(); 12 0 ; m2 () ;  else  k[bus] 1[bus] m[bus] )  ) Type Type  One Two  Bus beside Slack General Bus  Type  Three  Bus  beside  Bus  Voltage  Study  50  Bus  Two  Type F o u r /*  Bus b e s i d e V o l t a g e  Output o f R e s u l t s  Study Bus and b e s i d e  S l a c k Bus  */  p r i n t f ("\n") ; p r i n t f ( " C o l l a p s e a t Bus Number % i \ n " , c b u s ) ; p r i n t f ( " C o n t r o l V a r i a b l e U: %4. 4 f " , c o n t r o l U ( ) ) ; p r i n t f ( " ,V: %4 . 4 f " , c o n t r o l V () ) ; p r i n t f ( " ,W: %4.4fcontrolW()); p r i n t f ("\n") ; /* C h e c k i n g f o r P o s i t i o n i n g With Respect t o M a n i f o l d s if  (fabs(Vt()) > fabs(controlV())). { i f (controlUO > 0.0) < p r i n t f ( " S y s t e m P o i n t Above M a n i f o l d , ) else  */  U > 0");  ( if  ( c o n t r o l W O > 0.0 ii c o n t r o l W O < 5. 0/4 . 0*pow ( c o n t r o l U () , 2. < printf("System Point Inside Manifold"); } else  ( if  ( c o n t r o l W O > 5.0/4 . 0*pow ( c o n t r o l U () , 2. 0) ) < p r i n t f ( " S y s t e m P o i n t Above M a n i f o l d , U < 0 " ) ;  else  } {  if  (fabs(Vtlowl) ) < fabs(controlV() ) ) { printf("System Point Inside Manifold");  else  ) {  else  printf("System )  )}})  P o i n t Below  Manifold");  <  printf("System )  Point  Outside  Manifold");  printf("\n\n");  )  ) /*  Function  f o r Input o f Data  */  d a t a f () {  I*  Variable Declarations  F I L E *Y, * fopen () ; char junk[8], t ; d o u b l e mV[6],mQ[6], float temp; int i,a;  if  */  test;  ( (Y « fopen ( " i n p u t . d a t " , V ) | == NULL)  /*  51  Opening D a t a F i l e */  printf("Error, )  Data F i l e  NOT  Found!");  else < for  ( i = 0; i <= 5; ++i) < f s c a n f (Y,"%s %s % f " , j u n k , junk,stemp); d [ i l = (double) temp;  /* Phase A n g l e s */  ) for  ( i = 0; i <= 5; ++i) { f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; V C [ i ] = (double) temp;  /* V o l t a g e s  */  }  ( i = 0; i <= 5; ++i)  for  (  f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , s t e m p ) ; Q [ i ] = (double) temp;  /* R e a c t i v e Loads */  ) for  ( i = 0; i <= 5; ++i) ( f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; R [ i ] = (double) temp;  /* L i n e R e s i s t a n c e s */  ) for  ( i = 0; i <= 5; ++i) ( f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; X [ i ] = (double) temp;  /* L i n e R e a c t a n c e s  ) f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; X c l = (double) temp; f s c a n f (Y,"%s %s % f " , j u n k , j u n k , S t e m p ) ; Xc5 = (double) temp; f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; t l = (double) temp; f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , s t e m p ) ; t 5 = (double) temp; f o r ( i = 0; i <= 5; ++i) ( f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; m V [ i ) = (double) temp; V0[i) = V0[i]*mV[i]; ) f o r ( i = 0; i <= 5; ++i) < f s c a n f ( Y , " % s %s % f " , j u n k , j u n k , S t e m p ) ; mQ[i] = (double) temp; Q[i] = Q[i]*mQ[i]; } f c l o s e (Y) ; ); a = 1; r e t u r n (a) ; } /* F u n c t i o n f o r C h o o s i n g V o l t a g e C o l l a p s e Bus  /*  A l l Buses  Capacitor  /*  Reactances.  /*  Transformer  /*  Ratio.  /*  Voltage  /*  b u s c h o i c e () < i n t a; p r i n t f ( " W h i c h Bus (1-5), 6 i f a l l b u s e s : " ) ; scanf("%i",Schosebus); i f (chosebus •== 6) < c a s = 5;  /*  */  ) else  52  Load  */  */ */ */ */  Factors  Factors  */  <  cas = 1 ;  /*  Only One  Bus  */  }  a = 1; r e t u r n (a) ; ) /*  F u n c t i o n f o r F i n d i n g C a t a s t r o p h e P o i n t Vt  /*  F o r the M a n i f o l d C o m p a r i s i n g ,  */  a S w a l l o w t a i l M a n i f o l d P o i n t */  d o u b l e Vt () { d o u b l e v,pow(); v = 2 . 0 * c o n t r o l U ( ) * T ( ) + A . 0*pow (T () , 3. 0) ; r e t u r n (v) ;  )  /*  F u n c t i o n f o r F i n d i n g the C a t a s t r o p h e P o i n t Vtlow  /*  F o r the M a n i f o l d C o m p a r i s i n g ,  */  a S w a l l o w t a i l M a n i f o l d P o i n t */  double Vtlow() { d o u b l e v, pow () ; v = 2.0*controlU()*Tlow() + r e t u r n (v) ;  A.0*pow(Tlow(),3.0);  )  /* /*  F u n c t i o n f o r F i n d i n g the C a t a s t r o p h e V a r i a b l e t To F i n d the S w a l l o w t a i l M a n i f o l d P o i n t Vt */  double  (  */  TO  d o u b l e t , s q r t ( ) , pow ( ) ; if  (controlWO ( t = 0.0; )  < 0.0  ss c o n t r o l U O  >  0.0)  e l se ( t = s q r t ( - c o n t r o l U O / 6 . 0 + s q r t (pow ( c o n t r o l U () , 2 . 0)/36. 0 + c o n t r o l W () /3. 0) ) ;  )  r e t u r n (t) ; ) /* /*  F u n c t i o n f o r F i n d i n g the C a t a s t r o p h e V a r i a b l e tlow To F i n d the S w a l l o w t a i l M a n i f o l d P o i n t V t l o w */  double Tlowl) ( d o u b l e t , s q r t () ,pow () ; if  (controlWO { t =  < 0.0  it c o n t r o l U O  >  0.0)  0.0;  ) else {  53  */  t  -  sqrt (-controlU ()/6.0 + controlW()/3.0));  s q r t (pow ( c o n t r o l U (), 2. 0 ) / 3 6 . 0  }  r e t u r n (t) ; ) /*  Function for Finding Control Variable U  */  double controlU() { double u; u = 1.0/paramA4 () * (paramA2 O - 3.0 paramA3 () *paramA3 () / (8.0*paramA4 () ) ) ; r e t u r n (u); 4  ) /*  Function for Finding Control Variable V  */  double c o n t r o l V O double v; v = 1.0/paramA4 ()* (paramAl <) - paramA2 () *paramA3 () / (2. 0*paramA4 ()) paramA3 () *paramA3 () *paramA3 () / (8.0*paramA4 () *paramA4 () ) )S 0 ; r e t u r n (v); ) /* Function for Finding Control Variable W * /  +  double controlWO ( d o u b l e w; w = paramAO ()/paramA4 () - paramAl () *paramA3 ( ) / ( 4 . 0*paramA4 () * paramA4()) + paramA2()*paramA3()*paramA3()/(16.0*paramA4()* paramA4()*paramA4()) - 3.0*paramA3()*paramA3()*paramA3()* paramA3 () / (256. 0*paramA4 () *paramA4 () *paramA4 () *paramA4 () ) S () ; r e t u r n (w) ;  ) /*  Function  f o r F i n d i n g C a t a s t r o p h e Parameter Ao  */  d o u b l e paramAO () ( d o u b l e aO; aO = c o n s t a n t C l f ) + c o n s t a n t C 2 () *V0 [cbus] + 0.5* (POO + R O O ) + c o n s t a n t C 3 ( ) *V0[cbus]*V0[cbus]*S0() + constantC40*V0[cbus]*S00 + constantC5() *V0[cbus]*V0[cbus]*T0() + constantC6()*V0[cbus]*T0O; r e t u r n (aO) ; ) /*  F u n c t i o n f o r F i n d i n g C a t a s t r o p h e Parameter A l  */  double paramAl0 ( double a l ; a l = c o n s t a n t C 2 ( ) + 0.5*(P1() + R l () ) + const ant C3 () * (VO [cbus] *V0 [cbus ] *S1 () + 2*V0 [cbus] *S0 () ) + constantC4 ()* ( V 0 [ c b u s ] * S l () + SO () ) + c o n s t a n t C 5 () * (VO [cbus] *V0 [cbus] *T1 () + 2*V0 [cbus] *T0 () ) + c o n s t a n t C 6 ( ) * ( V 0 [ c b u s ] * T l () + T O O ) ; return(al); ) /*  F u n c t i o n f o r F i n d i n g C a t a s t r o p h e Parameter A2  d o u b l e paramA2()  54  */  d o u b l e a2; a2 = c o n s t a n t C 2 ()/2.0 + 0 . 5 M P 2 O + R2{)) + c o n s t a n t C 3 {) * (VO [cbus] * V 0 [ c b u s ] * S 2 ( ) + 2 * V 0 [ c b u s ] * S l () + (V0[cbus] + 1.0)*S0()) + c o n s t a n t C 4 () * (VO [cbus] *S2 () + S l ( ) + SOO/2.0) + c o n s t a n t C 5 () * (VO [cbus] *V0 [cbus] *T2() + 2 * V 0 [ c b u s ] * T l () + (VOtcbus] + 1.0)*T0()) + c o n s t a n t C 6 ( ) * ( V 0 [ c b u s ] * T 2 ( ) + T l () + T O O / 2 . 0 ) ; r e t u r n (a2) ; ) /*  F u n c t i o n f o r F i n d i n g C a t a s t r o p h e Parameter A3  */  double paramA3 () < d o u b l e a3; a3 •= c o n s t a n t C 2 O/6.0 + 0.5* (P3() + R3()) + c o n s t a n t C 3 () * (VO [cbus] * V 0 [ c b u s ] * S 3 ( ) + 2*V0[cbus]*S2() + (V0[cbus] + 1.0)*S1() + (V0[cbus]/3.0 + 1 . 0 ) * S 0 O ) + c o n s t a n t C 4 ( ) * ( V 0 [ c b u s ] * S 3 ( ) + S2 0 + S 1 O / 2 . 0 + SOO/6.0) + c o n s t a n t C 5 ( ) * ( V 0 [cbus] *V0 [cbus] *T3 0 + 2*V0 [cbus] *T2 () + (V0[cbus] + 1.0)*T1() + (VOtcbus] /3.0 + 1 . 0 ) * T 0 O ) + c o n s t a n t C 6 ( ) * ( V 0 [ c b u s ] * T 3 ( ) + T2 0 + T 1 O / 2 . 0 + T O O / 6 . 0 ) ; return(a3); 1 /*  F u n c t i o n f o r F i n d i n g C a t a s t r o p h e Parameter A4  */  d o u b l e paramA4() { d o u b l e a4; a4 = c o n s t a n t C 2 0 / 2 4 . 0 + O.S*(P4() + R4 () ) + c o n s t a n t C 3 () * (VO [cbus] * V 0 [ c b u s ] * S 4 ( ) + 2*V0[cbus]*S3() + (V0[cbus] + 1.0)*S2() + (V0(cbus]/3.0 + 1.0)*S1() + ((VOtcbus] + 1. 0) /12 . 0) *S0 () ) + c o n s t a n t C 4 () * (VOtcbus) *S4 () + S3 0 + S 2 O / 2 . 0 + S 1 O / 6 . 0 + S 0 O / 2 4 . 0 ) + c o n s t a n t C 5 () * (VO [cbus ] *V0 [cbus] *T4 () + 2*V0[cbus]*T3() + (VOtcbus] + 1.0)*T2() + (VO [cbus]/3.0 + 1.0)«T1() + ((V0[cbus] + 7 . 0 ) / 1 2 . 0 ) * T 0 O ) 4 c o n s t a n t C 6 () * (VO [cbus] *T4 () + T3() + T 2 O / 2 . 0 + T 1 O / 6 . 0 + T O O / 2 4 . 0 ) ; r e t u r n (a4) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t CI */ A C o n s t a n t F o r C o l l e c t i o n o f L i k e 'Terms  */  double c o n s t a n t C l ( ) ( double CI; CI = - 1 . 0 / 2 . 0 * ( k [ c b u s - l ) * a l [ c b u s ] + k [ c b u s + 1 ] * k s i [ c b u s ] ) ; r e t u r n (CI); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t C2 */ A C o n s t a n t F o r C o l l e c t i o n o f L i k e Terms  */  double constantC2() { d o u b l e C2; C2 *= 2.0*om[cbus] + 2.0*be(cbus] - 1 [cbus-1] * a l [cbus] - 1 [cbus+l] * k s i [cbus] r e t u r n (C2) ; ) /*  F u n c t i o n f o r F i n d i n g C o n s t a n t C3  */  55  /*  A Constant For C o l l e c t i o n of  L i k e Terms  */  double constantC3() { double C3; C3 = ( 1 . 0 / 2 . 0 ) * l [ c b u s - l ] * l [ c b u s - 1 ] ; r e t u r n (C3) ;  ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t C4 */ A C o n s t a n t F o r C o l l e c t i o n o f L i k e Terms  */  d o u b l e c o n s t a n t C 4 () < d o u b l e C4; C4 = ( 1 . 0 / 2 . 0 ) * 1 [ c b u s - 1 ] * k [ c b u s - l ] ; r e t u r n (C4); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t C5 * / A C o n s t a n t F o r C o l l e c t i o n o f L i k e Terms  */  double constantC5() ( d o u b l e C5; C5 = ( 1 . 0 / 2 . 0 ) * l [ c b u s + l ] * l [ c b u s + U ; r e t u r n (C5) ;  )  /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t C6 * / A C o n s t a n t F o r C o l l e c t i o n o f L i k e Terms  double  */  constantC6()  {  double C6; C6 = ( 1 . 0 / 2 . 0 ) * k [ c b u s + l ] * l [ c b u s + l ] ; r e t u r n (C6) ; j  /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t SO * / C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e S0() < d o u b l e SO; 50 = 1 . 0 / s q r t ( p l O ( ) ) ; return(SO); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t SI */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e SI () { double SI; 51 = - ( 1 . 0 / 2 . 0 ) * p l l ( ) / p o w ( p l 0 ( ) , 1 . 5 ) ; r e t u r n (SI) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t S2 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  56  */  d o u b l e S2() < d o u b l e S2; 52 •= (1.0/ (2.0*pow(pl0 ( ) , 0 . 5 ) ) ) * ( p o w ( p l l () ,2.0) / (4.0*pow(pl0 0 , 2 . 0 ) ) - pl20/plO(>); return(S2); } /* /*  Function Constant  f o r F i n d i n g C o n s t a n t S3 */ from T a y l o r S e r i e s F o r Square  Root  */  d o u b l e S3 () ( d o u b l e S3; 53 = (1.0/ (2.0*pow (plO () ,0.5) ) ) * ( 3 . 0 * p l l ( ) * p l 2 0 /(2.0*pow (plO () ,2.0) ) - p l 3 ( ) / p l 0 ( ) - 5.0*pow(pllO,3.0)/(8.0*pow(plO(),3.0))); return(S3); > /* /*  Function Constant  f o r F i n d i n g C o n s t a n t S4 */ from T a y l o r S e r i e s F o r Square  Root  */  d o u b l e SI () < d o u b l e S4,pow(); 54 = (1.0/ (2.0*pow(pl0 () , 0.5) ) ) * ( 3 . 0 * p l l () * p l 3 () / (2 . 0*pow (plO (), 2 . 0) ) - p l 4 ( ) / p l 0 ( ) + 3.0*pow(pl2 () ,2.0) / (4.0*pow(pl0 () ,2.0) ) - 1 5 . 0 * p o w ( p l l () ,2.0) * p l 2 0 / (8.0*pow(pl0 (), 3.0)) + 35.0*pow ( p l l () , 4.0) / (64.0*pow(pl0() , 4.0) ) ) ; r e t u r n (S4) ; ) /* /*  Function Constant  for Finding from T a y l o r  C o n s t a n t TO */ S e r i e s F o r Square  Root  */  f o r F i n d i n g C o n s t a n t TI */ from T a y l o r S e r i e s F o r Square  Root  */  Root  */  d o u b l e TOO i d o u b l e TO; TO = 1 . 0 / s q r t ( r l O () ) ; r e t u r n (TO); ) /* /*  Function Constant  double T I 0 ( double T I ; TI •= - ( 1 . 0 / 2 . 0 ) * r l l ( ) / p o w ( r l 0 ( ) , 1 . 5 ) ; return(TI); } /* /*  Function Constant  for Finding from T a y l o r  C o n s t a n t T2 */ S e r i e s F o r Square  d o u b l e T2() ( d o u b l e T2; T2 = (1.0/ (2.0*pow ( r l O () , 0.5) ) ) * (pow ( r l l 0 , 2. 0) / (4 . 0*pow ( r l O () ,2.0) ) - rl20/rlO()); r e t u r n (T2); )  57  /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t T3 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e T3 () ( d o u b l e T3; T3 = (1.0/ (2.0*pow ( r l O () , 0.5) ) ) * < 3 . 0 * r l l () * r l 2 () / (2. 0*pow ( r l O () , 2 . 0) ) - r l 3 ( ) / r l 0 ( ) - 5 . 0 * p o w ( r l l () , 3.0) / (8.0*pow(rl0 () ,3.0) ) ) ; r e t u r n (T3); } /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t T4 . */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e T4 ()  (  d o u b l e T4,pow () ; T4 = (1.0/ <2.0*pow ( r l O () , 0.5) ) ) * (3. 0 * r l l () * r l 3 () / (2. 0*pow ( r l O <) , 2. 0) ) - r l 4 ( ) / r l 0 ( ) + 3.0*pow(rl2 () , 2.0) / (4.0*pow ( r l O () ,2.0) ) - 1 5 . 0 * p o w ( r l l ( ) , 2 . 0 ) * r l 2 ( ) / (8. 0*pow ( r l O ( ) , 3.0) ) + 3 5 . 0 * p o w ( r l l 0,4.0) / (64 . 0*pow (rlO () , 4 . 0) ) ) ; r e t u r n (T4) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t P0 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e P0() < double P0,sqrt 0 ; P0 = s q r t (plO () ) ; r e t u r n (P0) ; } /* /*  F u n c t i o n f o r F i n d i n g Constant PI */ C o n s t a n t f r o m T a y l o r S e r i e s F o r Square Root  */  double PIO  <  double P I , s q r t ( ) ; PI = ( 1 . 0 / 2 . 0 ) * p l l ( ) / s q r t ( p l 0 ( ) ) ; r e t u r n (PI) ;  ) /* /*  F u n c t i o n f o r F i n d i n g Constant P2 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e P2 () ( d o u b l e P2,pow(); P2 = ( 1 . 0 / 2 . 0 ) * p o w ( p l 0 ( ) , 0 . 5 ) * ( p l 2 ( ) / p l 0 ( ) - p o w t p l l ( ) , 2 . 0 ) / (4.0*pow(pl0 () , 2.0) ) ) ; r e t u r n (P2) ; } /* /*  F u n c t i o n f o r F i n d i n g Constant P3 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e P3 () { d o u b l e P3,pow(); P3 = (1.0/2.0) *pow(pl0 (), 0.5) * ( p l 3 O / p l 0 () - p l l () * p l 2 () / (2. 0*pow (pi 0 () , 2. 0) ) + pow ( p l l (),3.0) / (8. 0*pow(pl0 (),3.0) ) ) ;  58  r e t u r n (P3) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t P3 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e P4() { d o u b l e P4, pow () ; P4 = (1.0/2.0) *pow(plO (), 0.5) * (pl4 0 / p l O () - p l l 0 * p l 3 ( ) / ( 2 . 0 * p o w ( p i 0 0 , 2 - pow (pl2 () , 2. 0) / (4 . 0*pow (plO () , 2. 0) ) + 3.0*pow ( p l l () ,2.0) * p l 2 () / (8.0*pow(pl0 () , 3.0) ) - 5 . 0 * p o w ( p l l () , 4.0) / (64.0*pow (plO () , 4 . 0) ) ) ; r e t u r n (P 4 ) ; ) /* /*  F u n c t i o n f o r F i n d i n g Constant R0 */ C o n s t a n t from T a y l o r S e r i e s F o r Square  Root  */  Root  */  F u n c t i o n f o r F i n d i n g Constant R2 */ C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e R0 () < d o u b l e R0,pow(); R0 = p o w ( r l 0 () , 0.5) ; r e t u r n (R0) ; ) /* /*  F u n c t i o n f o r F i n d i n g Constant R l */ C o n s t a n t from T a y l o r S e r i e s F o r Square  d o u b l e R l () ( double Rl,pow(); R l = (1.0/2.0) * r l l ( ) / p o w ( r l 0 () , 0.5) ; return(Rl);  )  /* /*  d o u b l e R2() ( d o u b l e R2, pow () ; R2 = (1.0/2.0) * p o w ( r l 0 () , 0.5) * ( r l 2 O / r l O () - p o w ( r l l ( ) , 2 . 0 ) / (4.0*pow (rlO () , 2. 0) ) ) ; r e t u r n (R2) ; } /* ' F u n c t i o n f o r F i n d i n g C o n s t a n t R3 */ /* C o n s t a n t from T a y l o r S e r i e s F o r Square Root  */  d o u b l e R3 () ( d o u b l e R3,pow () , s q r t () ; R3 = ( 1 . 0 / 2 . 0 ) * s q r t ( r l O O )* ( r l 3 ( ) / r l O () - r l l 0 * r l 2 O / (2 . 0*pow (rlO () , 2 . 0) + pow ( r l l () ,3.0) / (8.0*pow(rl0 () ,3.0) )) ; r e t u r n (R3) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t R3 */ C o n s t a n t from T a y l o r S e r i e s F o r Square  Root  d o u b l e R4 () f  59  */  d o u b l e R4,pow () , s q r t () ; R4 = (1.0/2.0) * s q r t ( r l D ( ) ) * ( r l 4 ( ) / r l O () - r l 1 ( ) * r l 3 ( ) / ( 2 . 0 * p o w ( r l O < ) , 2 . 0 ) ) - p o w ( r l 2 () ,2.0) / ( 4 . 0 * p o w ( r l 0 () ,2.0) ) + 3 . 0 * p o w ( r l l 0 , 2 . 0 ) * r l 2 ( ) / ( 8 . 0 * p o w ( r l O ( ) , 3.0)) - 5.0*pow ( r l l () , 4.0) / (64.0*pow ( r l O () , 4.0) ) ); return(R4); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l O */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  */  d o u b l e plO () < d o u b l e p0,pow(); pO = pow(1[cbus+1],2.0)*pow(V0[cbus] ,2.0) + 2 . 0 * k [ c b u s + l ] * l [ c b u s + l ] * V 0 [ c b u s ] + m[cbus+l]  + pow(k[cbus+1],2.0);  return(pO); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l l */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  */  d o u b l e p l l () ( d o u b l e p i , pow ( ) ; p i = 2 . 0 * ( p o w ( l [ c b u s + l ] , 2 . 0 ) * V 0 [ c b u s ] + k[cbus+1]*1[cbus+1]) ; r e t u r n (pi) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l 2 */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  double pl2() < d o u b l e p2,pow(); p2 = pow(1[cbus+1],2.0)*(V0[cbus]  */  + 1.0) + k [ c b u s + 1 ] * 1 [ c b u s + 1 ] ;  r e t u r n (p2) ,) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l 3 */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  */  double p l 3 ( ) { d o u b l e p3,pow () ; p3 = p o w ( l [ c b u s + l ] , 2 . 0 ) * ( V 0 [ c b u s ] / 3 . 0 + 1.0) + k[cbus+1]*1[cbus+1]/3.0; r e t u r n (p3) ; ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l 4 */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  */  d o u b l e p i 4 () {  d o u b l e p4,pow(); p4 = pow(1[cbus+1],2.0)*(V0[cbus] + 7.0)/12.0 + k[cbus+1]*1[cbus+1]/12. 0; return(p4); } /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t p l O */ Parameter o f C o l l e c t e d Terms I n s i d e Square Root  60  */  double r l O ( ) < d o u b l e rO,pow(); rO = pow(l [cbus-1] ,2.0)*pow(V0[cbus],2.0) + 2.0*k[cbus-1]*1[cbus-1]*V0[cbus] + m[cbus-l]  +  pow(k[cbus-1],2.0);  r e t u r n (rO); } /* /*  F u n c t i o n f o r F i n d i n g Constant r l l V P a r a m e t e r o f C o l l e c t e d Terms I n s i d e Square  double  i  Root  */  r l l ()  double rl,pow(); r l = 2 . 0 * ( p o w ( l [ c b u s - l ] , 2 . 0 ) * V 0 [ c b u s ] + lc [cbus-1 J *1 [cbus-1] ); return (rl); )  /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t r l 2 */ P a r a m e t e r o f C o l l e c t e d Terms I n s i d e Square Root  */  d o u b l e r l 2 () { d o u b l e r2,pow ( ) ; r 2 = p o w ( l [ c b u s - l ] , 2 . 0 ) * ( V 0 [ c b u s ] + 1.0) + k [cbus-1 ] *1 [cbus-1 ] ; return(r2); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t r l 3 */ P a r a m e t e r o f C o l l e c t e d Terms I n s i d e Square  Root  */  double r l 3 ( ) {  d o u b l e r3,pow(); r 3 = p o w ( l [ c b u s - l ] , 2 . 0 ) * ( V 0 [ c b u s ] / 3 . 0 + 1.0) + k [ c b u s - 1 ] * 1 [ c b u s - 1 ] / 3 . 0 ; return (r3); ) /* /*  F u n c t i o n f o r F i n d i n g C o n s t a n t r l 4 */ P a r a m e t e r o f C o l l e c t e d Terms I n s i d e Square Root  double  i  r l 4 () 1  d o u b l e r4,pow(); r4 = p o w ( 1 [ c b u s - 1 ] , 2 . 0 ) * ( V 0 [ c b u s ]  + 7.0J/12.0 +  r e t u r n (r 4) ; ) /*  Constants  f o r Type 4  */  d o u b l e k4 () I d o u b l e k4; k4 = a l [ b u s ] * V 0 [ 0 ] / ( b e [ b u s ] +  om[bus]);  r e t u r n (k4) ; ) double  (  */  14 0  d o u b l e 14; 14 = k s i [bus] / (befbus] +  om[bus]);  61  k[cbus-1]*1[cbus-1]/12.0;  r e t u r n (14) ; ) d o u b l e m4() { d o u b l e m4; m4 = 4.0*Q[bus]/(be[bus]  + omfbus]);  return(m4); )  ,'* C o n s t a n t s f o r Type 3 */ d o u b l e k3 () < d o u b l e k3; k3 = ( a l [ b u s ] * j [ n b u s ] / ( l [ n b u s ]  + m [ n b u s ] * V 0 [ b u s ] ) ) / ( b e [ b u s ] + om[bus]);  r e t u r n (k3) ;  )  d o u b l e 13 0 < d o u b l e 13; 13 = k s i [ b u s ] / ( b e [ b u s ] + om[bus]); return(13); } d o u b l e m3() { d o u b l e m3; m3 = (4.0*(Q[bus] - a l [bus] *k [nbus] / (1 [nbus] + m[nbus] *V0 [bus] ) ) ) / (be [bus + om[bu s]) ; return(m3);  )  /* C o n s t a n t s f o r Type 2 */  double j2()  (  double j2; j2 = k s i [bus] *m[nbus]*VO [bus] *V0 [bushr e t u r n (j2) ;  ) d o u b l e k2 () { d o u b l e k2; k2 = C [ b u s ] * 1 [ n b u s ] + (2.0*m[nbus)*V0[bus] * ( b e [ b u s ] + om[bus]);  +  1[nbus])*pow(VO[bus],2.0)  r e t u r n (k2) ; ) d o u b l e 12() { d o u b l e 13; 13 - V O [ b u s ] * ( b e [ b u s ] + om[bus])*(3.0*m[nbus]*V0[bus] + al[bus]*k[nbus] - m[nbus] *Q[bus] ; r e t u r n (13) ; )  62  + 2.0*l[nbus))  double  m2 ()  { d o u b l e m2; m2 = 2 . O ' k s i [ b u s j * m [ n b u s ] * V 0 [ b u s ] +  ksi[bus]*1[nbus];  return(m2); > /*  Constants  double  f o r Type  1 */  k l ()  { double k l ; k l = Q[bus]  +  (om[bus)  +  be[bus])*pow(VO[bus],2.0);  return(kl); ) double  11()  { double 11; 11 = a l [bus]*V0[0]  + 2.0*(om[bus]  + b e [ b u s ] ) *V0 [ b u s ] ;  r e t u r n (11) ; ) double ml() { double ml; ml = k s i [bus] ; return(ml); ) /*  Constants  double  from R e a c t i v e  Power  Equation  at Bus 5 * /  a l p h a 5 ()  { drvjVle alph =  alph; (X[5]*cos(d[5)-d[0]) + pow(X[5J ,2.0) ) ;  + R [5 ] * s i n (d [ 5 ] - d [0] ) ) / (pow (R [ 5 ] , 2 . 0)  r e t u r n (alph) ; ) double  beta5()  ( double bet; bet = -X[5]/(pow(R[5],2.0)  + pow (X [ 5 ] , 2 . 0) ) -  return(bet); ) double  omega50  ( d o u b l e omeg; o m e g = 1 . 0 / ( p o w ( t 5 , 2 . 0) * X [ 4 ] ) ; return(omeg);  1 double  k s i 5 ()  ( double  alph;  alph  -(cos(d[5]-d[4]))/(2.0't5*X[4));  =  return(alph); )  63  1.0/Xc5;  /* C o n s t a n t s f r o m R e a c t i v e Power E q u a t i o n a t Bus 4 */ double alpha4() ( double alph; a l p h = < X [ 3 ] * c o s ( d [ 4 ] - d [ 3 ] ) + R [ 3] * s i n (d {4 ) -d [3] ) ) / (pow (R [3], 2 . 0) + pow(X[3],2.0)); return(alph); } double beta4() ( double bet; bet = -X[3] / (pow (R[3) ,2.0) + pow(X[3],2 . 0) ); return(bet); } d o u b l e omega4() ( d o u b l e omeg; omeg = - 1 . 0 / X [ 4 ) ; return(omeg); ) double  k s i 4 ()  [  double k s i ; ksi = (cos(d[4]-d[5]))/(2.0*t5*X[4]); r e t u r n (ksi) ; } /* C o n s t a n t s double  (  from R e a c t i v e Power E q u a t i o n a t Bus 3 */  alpha3()  double alph; alph = -<X[2]*cos(d[3]-d[2]) + pow (X[2] ,2.0) ) ;  + R [2] * s i n (d [3]-d [2] ) ) / (pow (R [2] , 2. 0)  r e t u r n (alph) ; ) d o u b l e beta3 ()  (  double bet; bet = X ( 2 ] / ( p o w ( R [ 2 ] , 2 . 0 ) + pow (X [2] , 2. 0) ) ; return(bet); }  d o u b l e omega3() { d o u b l e omeg; omeg = X [3] / (pow (R [ 3] , 2. 0) + pow (X [ 3) , 2 . 0) ) ; return(omeg); } d o u b l e k s i 3 () < double k s i ; k s i = - ( X [ 3 ] * c o s ( d [ 3 ] - d [ 4 J ) + R [ 3 ) * s i n ( d [ 3 ] - d [ 4 ] ) ) / (pow (R [3] ,2.0) + pow(X[3],2.0));  64  r e t u r n (ksi) ; ) /* C o n s t a n t s from R e a c t i v e Power E q u a t i o n a t Bus 2 */ double alpha2() ( double k s i ; k s i = c o s ( d [ 2 ] - d [ l ] ) / <tl*X[l] ) ; r e t u r n (ksi) ;  )  double  (  beta2()  double bet; bet = -1.0/XU); return(bet); }  double  omega2()  d o u b l e omeg; omeg = -X[2)/(pow(R[2),2.0) + pow(X[2],2 . 0) ) ; return(omeg); ) d o u b l e k s i 2 () { double alph; a l p h = { X [ 2 ] * c o s ( d [ 2 ] - d [ 3 ] ) + R [2] * s i n (d [ 2 ] - d [ 3] ) ) / (pow (R [ 2) , 2 . 0) + pow(X[2],2.0)); return(alph); ) '*  r o n r t a n t s f r o m R e a c t i v e Power E q u a t i o n a t Bus 1 */  d o u b l e a l p h a l () I double alph; alph = (X[0)*cos(d[l)-d[0)) + pow(X[0],2.0));  + R[0]*sin(d[1]-d[0]))/(pow(R[0],2.0)  return(alph); }  double  (  betal()  double bet; b e t = -X[0]/(pow(R[0],2.0) + p o w ( X [ 0 ] , 2 . 0 ) ) - 1 . 0 / X c l ; r e t u r n (bet) ; ) double  i  omegalO  d o u b l e omeg; omeg = 1 . 0 / ( p o w ( t l , 2 . 0 ) * X [ 1 ) ) ; r e t u r n (omeg) ; )  d o u b l e k s i l () ( double k s i ;  65  ksi = -cos(d[l)-d[2])/(tl*X[l]) return(ksi); )  F I L E FOR SYSTEM DATA  PHASE  ANGLES  d[0] d[l) d[2) d[3] d[4) d[5)  = = = = =  0.0 -0.17176782 -0.22287907 -0.05914935 -0.21262570 0.21437684  I N I T I A L VOLTAGE VALUES V0[0] V0[1] V0[2] V0[3] V0[4] V0[5]  = = = = = =  1.05 0.932061 1.003495 1.103648 0.922237 0.924158  REACTIVE POWERS Q[0] Q[l] Q[2] Q[3] Q[4] Q[5]  = = = = = =  0.497890 0.218091 0.016298 0.370004 0.153299 0.179988  LINE RESISTANCES R[0] R[l] R[2] R[3] R[4] R[5] LINE X[0] X[l] X[2] X[3] X[4) X[5)  = = = = = =  0.08 0.0 10.723 0.282 0.0 0.123  REACTANCES = = = = = =  0.37 0.133 10.050 0.64 0.3 0.518  CAPACITOR Xcl Xc5  REACTANCES  = 34.1 = 28.5  TRANSFORMER tl t5  RATIOS  = 0.909 = 0.974  VOLTAGE MULTIPLIERS mV[0) mV[l) mV[2) mV[3] mV[4] mV[5]  = = = = = =  1.0 1.0 1.0 1.0 1.0 1.0  REACTIVE LOAD MULTIPLIERS  mQ[0] mQ[l] mQ[2] mQ[3] mQ[4] mQ[5]  = = = = = =  1.0 1.0 1.0 1.0 1.0 1.0  68  References  [I]  W.R. Lachs. Voltage collapse in ehv power systems. IEEE/PES Winter Meeting, New York, NY, (057), January 1978.  [2]  O.O. Obadina and G.J. Berg. Voltage collapse in power systems. Canadian Conference on Electrical and Computer Engineering, pages 196-199, November 1988.  [3]  F. Mercede, J-C. Chow, H. Yan, and R. Fischl. A framework to predict voltage collapse in power systems. IEEE Transactions on Power Systems, 3(4):1807-1813, November 1988.  [4]  Y. Sekine and H. Ohtsuki. Cascaded voltage collapse. IEEE/PES 1989 Summer Meeting, Long Beach, California, (710), July 1989.  [5]  A.K. Pasrija H.G. Kwatny and L . Y . Bahar. Static bifurcations in electric power networks: Loss of steady-state stability and voltage collapse. IEEE Transactions on Circuits and Systems, CAS33(10):981-991, October 1986.  [6]  A. Tiranuchit and R J . Thomas. A posturing strategy against voltage instabilities in electric power systems. IEEE Transactions on Power Systems, 3(l):87-93, February 1988.  [7]  H. Mori and S. Tsuzuki. Estimation of critical points on static voltage stability in electric power systems. IFAC Symposium on Power Systems and Power Plant Control, pages 550-554, 1989.  [8]  P. Kessel and H. Glavitsch. Estimating the voltage stability of a power system. IEEE Transactions on Power Delivery, PWRD-l(3):346-354, July 1986.  [9]  CJ~. DeMarco and T J . Overbye. An energy based security measure for assessing vulnerability to voltage collapse. IEEE/PES 1989 Summer Meeting, Long Beach, California, (712), July 1989.  [10]  M.M. Begovic and A.G. Phadke. Dynamic simulation of voltage collapse. IEEE Power Industry Computer Application Conference, Seattle, Washington, pages 336-341, May 1989.  [II]  J.S. Thorp H-D. Chiang, R J . Thomas and L . Fekih-Ahmed. On voltage collapse in electric power systems. IEEE Power Industry Computer Application Conference, Seattle, Washington, pages 3 May 1989.  [12]  O.O. Obadina and G J . Berg. Determination of voltage stability limit in multimachine power systems. IEEE Transactions on Power Systems, 3(4):1545-1554, November 1988.  [13]  M.D. Wvong and A . M . Mihirig. Catastrophe theory applied to transient stability assessment of power systems. IEE Proceedings, 133(6):314-318, September 1986.  [14]  A.A.Sallam. Power system steady state stability assessment using catastrophe theory. IEE Proceedings, 135(3), May 1988.  69  [15]  M.G.Brown and F.D.Tappert. Catastrophe theory, caustics and traveltime diagrams in seismology. Geophys. J. R. astr. Soc, 88:217-229, 1987.  [16]  Y.Kume A.Murata and F.Hashimoto. Application of catastrophe theory to forced vibration of a diaphragm air spring. Journal of Sound and Vibration, 112(l):31-44, 1987.  [17]  N.Neskovic and BPerovic. Ion channeling and catastrophe theory. Physical Review Letters, 59(3):308310, July 1987.  [18]  N.Neskovic and BPerovic. The analysis of the rainbow effect in ion channeling by catastrophe theory. Nuclear Instruments and Methods in Physics Research, pages 66-68, 1988.  [19]  Olle I. Elgerd. Electric Energy Systems Theory. McGraw-Hill Book Company, second edition, 1982.  [20]  B.M. Weedy. Electric Power Systems. John Wiley and Sons, third edition, 1979.  [21]  Antal Majthay. Foundations of Catastrophe Theory. Pitman Publishing Inc., 1020 Plain Street, Marshfield, Massachusetts 02050, first edition, 1985.  [22]  Brian W. Kernighan and Dennis M. Ritchie. The C Programming Language. Prentice-Hall,firstedition, 1978.  70  

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