LIBRARY Michigan State This is to certify that the thesis entitled A Transient Coherency Measure and Its Application to Transient Security Enhancement of Electric Energy Systems— A Systems and Operations Research Approach presented by Humayun Akhtar has been accepted towards fulfillment of the requirements for PH.d . Systems Science degree m n 1' Major professor QMW mafia/5‘, H72 0-7639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. .Iill‘t.l"'|“l‘l‘ “ 4 I‘I‘I A TRANSIENT COHERENCY MEASURE AND ITS APPLICATION TO TRANSIENT SECURITY-ENHANCEMENT OF ELECTRIC ENERGY SYSTEMS - A SYSTEMS AND OPERATIONS RESEARCH APPROACH BY Humayun Akhtar A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1979 ABSTRACT A TRANSIENT COHERENCY MEASURE AND ITS APPLICATION TO TRANSIENT SECURITY-ENHANCEMENT OF ELECTRIC ENERGY SYSTEMS - A SYSTEMS AND OPERATIONS RESEARCH APPROACH BY Humayun Akhtar A transient coherency measure is developed and shown to be an excellent tool for analyzing the transient response of an electric energy system in order to obtain insight into the dynamic structure of the interconnected system following a contingency. The transient response is disected into a sequence of events by showing the times at which even numbered terms of a Taylor~series approxima- tion are added to the transient coherency measure to in— dicate the instants that successive stages of generators further from the disturbance begin to accelerate. The measure is evaluated for a deterministic and probabilistic disturbance and indicates the lines and generators that are affected by the disturbance and the relative stiffness of the interconnection from the disturbed generator to the portion of the system affected during each time interval of propagation of the disturbance. The transient coherency measure is thus shown to assist in the identification of weaknesses in dynamic system structure and the changes in Humayun Akhtar the dynamic structure of an electric energy system as the length of the observation interval increases. The transient coherency measure is also shown to be an excellent measure of system security and reliability with the use of the equal-area criterion and the concept of an equivalent-line connected to an infinite bus. It is further shown to be a stability measure by showing that the transient coherency measure summed over all pairs of internal generator buses is proportional to the sum of the square of the eigenvalues of the system with zero damping. The transient coherency measure in its use as a security measure is shown to have potential for on-line application and thus provide the industry with a valuable tool for on-line transient security enhancement. In this direction, the off-line dispatch problem is reformulated as an on-line linearized tracking secure dispatch problem. By viewing the security problem in terms of five operating states, the associated sub-control problems are formulated and shown to have the familiar linear-programming or quadratic programming format from operations research, de- pending upon the addition of the transient security measure to the performance index. An algorithmic control structure is prOposed for solving the sub-control problems and it is shown that with the help of appropriate weightings, the reformulations can significantly enhance the security, reliability and stability of the system. Humayun Akhtar Thus the measure is shown to move the system in a direction to (l) stiffen weak lines, (2) reduce the vulnerability of any particular bus to a loss of syn- chronization or stability, (3) improve the overall stiff- ness of the interconnected network and (4) prepare the system for controlled islanding when it is moving toward total collapse. IN THE NAME OF ALLAH THE BENEFICIENT, THE COMPASSIONATE, THE MERCIFUL 6&2?" WEB To my Wife Yosria ii ACKNOWLEDGEMENTS The author would like to express his deep apprecia- tion to his major advisor, Dr. Robert A. SchJueter for his guidance, encouragement, suggestions and dedication during the course of his entire program. Dr. Herman E. Koenig is thanked for cultivating an initial interest in the energy area of systems-science and for his suggestions in improving the quality of this thesis. Dr. Gerald L. Park is thanked for his periodic guidance, suggestions, technical discussions and encourage- ment. Dr. Robert O. Barr is thanked for his guidance in planning and making a distinct contribution in the author's understanding of advanced topics in Systems-Science and Operations Research through his outstanding teaching ex- pertise. Dr. Ron C. Rosenberg is thanked for his useful dis- cussions, extreme encouragement and high moral support during the course of this thesis. The author is forever thankful to his parents, Aftab Akhtar and Amina Aftab, for their everlasting love and for providing the right foundations. iii Finally, and most importantly, this dissertation would not have been possible without the inspiration for attaining high professional recognition provided by the author's wife, Yosria, to whom he is eternally in- debted. iv TABLE OF CONTENTS Chapter Page I INTRODUCTION . . . . . . . . . . . . . . . 1 II A GENERALIZED STATE MODEL AND DISTURBANCE MODEL 0 O O O O O O O O O O O I O O O O l 6 III A GENERALIZED MEAN SQUARE COHERENCY MASURE O O O O I O O O O O O O O O O O 2 8 IV A TAYLOR SERIES EXPANSION OF THE MEAN- SQUARE COHERENCY MEASURE . . . . . . . . 32 V THE TRANSIENT COHERENCY MEASURE AND POWER SYSTEM TRANSIENT RESPONSE . . . . . . . 44 VI AN OPTIMAL COHERENCY BASED SECURE DISPATCH FORMULATION . . . . . . . . . . . . . . 80 VII PERFORMANCE INDEX JUSTIFICATION FOR SECURITY DISPATCH . . . . . . . . . . . 132 VIII THE SECURITY CONTROL PROBLEM . . . . . . . 149 IX CONCLUSIONS AND FUTURE INVESTIGATION . . . 177 BIBLIOGRAPHY . . . . . . . . . . . . . . . 135 APPENDIX . . . . . . . . . . . . . . . . . 189 CHAPTER 1 INTRODUCTION Scenario: Increased and steadily increasing system inter- connections have created rather complex networks in the United States having load areas located far from the gen- erating plants. The NY blackout of 1965 initiated a serious interest in reliability and security studies of Electric Energy Systems. However, the recent 1977 NY outage provides sufficient evidence of the inadequacy of these efforts and their achievement, especially so far as on-line methods are concerned. To enable the systems planning division of the modern utility to meet existing requirements on system security, evaluate existing systems through transient stability studies and plan future development of the net- work, it is imperative that improved models be developed which adequately represent the real power system without including or deleting more than necessary from the model. For a better assessment of this situation, a review of some of the characteristics of an electric energy system is in order [1]: (i) Power plants will have to transmit power to distant load centers and economic growth will mean the in- crease in the size of these plants. The size of the plants is therefore dictated essentially by three factors which are . geographical . environmental . economic realities (ii) Due to increased demand coupled with delays in com- missioning of new units such as nuclear power plants, the stress on existing units is increasing and trans- mission corridors are-becoming more crowded with the increase in large amounts of power being transmitted over large distances. This will result in an in- creased probability of system instability as trans- mission lines carry nearly full rated power. (iii) As EHV interconnections increase, the effects of the disturbances will be felt in more and more re- mote parts of the system, thus increasing the severity of any disturbance. It is with these characteristics in perspective, then, that the modern utility spends a considerable amount of effort in terms of time and money to perform transient stability studies. The objective of their study is to aid the systems planners . to evaluate a power system's ability to with- stand large disturbances . to perform routine long range and short range planning of electric energy systems which could involve solution schemes for various combina- tions of proposed generator and transmission configurations, solutions for coordination of protection schemes, and design of controls for existing systems. Given this background, in any physical situation, most of the major disturbances can propagate through tie- lines to neighboring systems with the result that it be- comes important to represent in the power system model, not only the power system in question, hereafter referred to asifluainternal or study system, but also the neighbor- ing system, hereafter referred to as the external system. However, the extensive nature of the interconnection makes the representation of the external system a difficult chore. For example, to study the Michigan system, it becomes nec- essary to represent systems as far north as Ontario, Canada and as far east as New England. It is no wonder, then, that a single transient stability run requires large computers, a significant amount of computer time, and consequently high costs. In order to circumvent some of these difficulties, the concept of coherency in power system models comes into the picture. What is Coherency: Coherency is a relatively new area of interest in power systems studies. As befits the introduction of such a topic, the level of presentation is generally monotone, increasing from section to section and chapter to chapter. Qualitatively, coherency is a quality or state of systematic connectedness or interrelatedness, especially when governed by legical principles. It is, then, this state of connectedness and interrelatedness by logical principles which form the basis of this dissertation, by the integration of machines into congruent sets based on a consistent pattern. E Quantitatively, coherency is ideally defined to hold for any two buses for which the ratio of their com- plex bus voltages is constant over time. This definition of coherency for a particular fault or disturbance implies that for two buses to be coherent, the following two con- ditions are satisfied [4]: . the difference in voltage angles at the two buses, i and j, is constant . the ratio of voltage magnitudes is constant for buses i and j. Mathematically, therefore, this may be expressed as: . 5. '1: vi IViIeJI—i Cij = 5; = 'I6' = constant A ' J _1 ts [CIT] 3 ‘lee Ideal coherency is sufficient for a mathematically rigorous procedure for replacing a group of coherent buses by a single equivalent bus. However, the model which results cannot be represented in terms of normal power system com- ponents and thus could not be used in conventional transient stability programs. It has been shown that accurate dynamic equivalents can be deveIOped using the above pro- cedure for the case where buses that are to be equivalenced are determined based on a relaxed definition of coherency which depends only on differences in voltage angles and is independent of voltage magnitude. Relaxed Coherency, then, refers to two buses as being co- herent if the difference of their voltage angles remains constant to a certain prespecified tolerance, s, over time. CijA = 6i(t) - 6j(t) < 8 te[0,T] Thus groups of generators that "swing-together" when dis- turbed are called coherent and knowledge of coherent be- havior allows a simpler and cheaper representation of machines in large stability studies. Coherency Based Modelling: Several methods for developing Coherency Based Dynamic Equivalents have been proposed [3,7]. They use various definitions of electrical distance and various clustering altorithms for determining coherent groups. These electrical distance methods were heuristically based, requiring tuning of parameters for each system studied, and required validation of the equivalent against the results of the step by step integration of the system nonlinear differential equations during fault and post fault conditions, in a base case transient stability. In an effort to overcome the above difficulties, Systems Control Incorporated (SCI) in Palo Alto, California has developed a method of producing coherency based dynamic equivalents which accurately reproduces the effects of the external system when the internal or study system is subjected to a specific disturbance. This method com- prises of the following steps: 1) Classify groups of generators to be equivalenced into coherent groups by use of the relaxed co- herency approach for processing swing curves from base case transient stability studies 2) Aggregate the generators in each coherent group to form one or more equivalent generators. The result is a reduced equivalent network in the form of normal power system components which can be used without modifying present transient stability programs. This pro- cedure, however, is beset by severe limitations which are [5]: . The base case transient stability studies are expensive in terms of computer requirements. Consequently a large initial effort is required to form the equivalent. . The advantageous applications of the method are limited to studies in which multiple transient stability cases are to be simulated with faults concentrated in a local area. The coherency-based equivalencing method is not directly usable by utilities with small computers and with transient stability programs which cannot handle the full system representa- tion. The two major limitations of the coherency based dynamic equivalents method are: (l) the equivalent is determined based on a single contingency in a particular location. The equivalent can thus only be used to study similar contingencies occurring in approximately the same local area. (2) there is no theoretical justification for the coherency based method of producing dynamic equivalents when coherency is defined based strictly on the difference in voltage angles. Two coherency measures considered by SCI were the max-min measure C1 (T) = max {5 k2 k(t) - 6£(tl} - min {6k(t) - 5£(t)} t6 [O'T] t6 [OIT] and the rms measure _ 1 T 2 k2(T) - SQRT f f0{[6k(t) - 6k(0)] - [52(t) - 52(o>1} dt. The max-min measure was chosen by SCI over the rms measure. The following discussion attempts to establish the restric- tive conditions under which the max-min measure is a good choice and to show the conditions under which an rms or mean square coherency measure is a better choice for (l) analyzing the propagation of disturbances (2) analyzing changes in power system dynamic structure (3) use as a measure of power system security. The max-min measure in its present form is a transient coherency measure as it selects the maximum change in the angular difference which will occur in the transient in- terval [0,T] and depends upon T, the disturbance, and that portion of the system dynamics which is affected by the disturbance when the angular difference achieves its maximum or minimum values. For this coherency measure to give an accurate indication of the coherent groups and dynamic structure, it must be assumed that the coherent groups do not change during the short interval in which the angular differences on every pair of buses achieve their maximum and minimum values. If the observation interval is short enough, this assumption that the coherent groups do not change is cer- tainly valid, but the time-interval over which the co- herency measure and any dynamic equivalent derived based on it are confined to the short interval. On the other hand, an rms or mean square coherency measure averages the angular difference in the interval and thus the rms coherent equivalent may not be as accurate as an appropriate min-max measure over a short interval but would likely be a better equivalent over a longer in- terval where the coherent groups and dynamic structure have changed. The mean square measure is capable of pro- viding a probabilistic description of the disturbance in terms of system parameters and is not dependent upon the location of the disturbance. It can also provide a deterministic description for a specific disturbance in a specific location. The mean square measure has potential for applica- tion in the comparison of modal and coherent equivalents or comparison based on coherent equivalents derived on different observation intervals. In addition, it has potential for use as a security measure similar to the measure used by Byerly et al. [28, 29] max (6i - ej) which is similar to the max-min coherency measure and is dependent upon the fault and its location. The transient coherency measure is proposed as a security measure in this research and is shown to have potential for on-line application because it . provides a better measure of security . provides a better measure of reliability . provides a better measure of stability . is a closed form analytical expression 10 . stiffens the network adaptively . enables the analysis of the propagation of disturbances in the system . permits the analysis of changes in power system dynamic structure . does not depend upon type and location of the disturbance and can permit the representation of different disturbances in different loca- tions both deterministically and proba- bilistically. The Specific topics in the development of a transient coherency measure and its application to transient security enhancement are now introduced. A linearized power system state model and dis- turbance model are developed to permit an analysis of the spread of a contingency, deterministic or probabilistic, in the power system. The linearized models developed are used to represent dynamic fluctuations in both the in- ternal system and in the external system which is far from the internal system where the disturbances occur and the models developed are accurate. Contingencies such as load shedding, generator dropping, electrical faults, and line switching can be handled easily by these models. In order to determine coherent groups using these models, a generalized mean square coherency measure is developed. This measure of coherency between internal generator buses is capable of handling the set of deter- ministic and probabilistic disturbances already described. W‘_-' 11 The mean square coherency measure is expressed as a Taylor series expansion and its coefficients are analyzed. Computer programs are written to obtain the coherency matrix for a system based on these coefficients which are found to be recursive. This Taylor series expansion is then used to define a transient coherency measure and thereby analyze the transient response of a power system for obtaining better insight into dynamic structure and associated changes in the interval following the disturbance. The analysis of the power system transient response is made possible by defining the aforementioned transient coherency measureixx which the number of terms in the Taylor series increases with the observation interval in order to keep the approximation error bounded. The transient response of a power system is sub- divided into a finite sequence of events by showing the time instants at which even numbered terms are added to the measure, and signifying the time instants at which successive sets of generators or stages further from the disturbance location begin to accelerate. The propagation of the disturbance in the power system with the help of this transient coherency measure is thus used to indicate the lines and generators that are affected by the disturbance and the relative stiff- ness of the interconnection from the disturbed generator 12 to the portion of the system affected by the disturbance in any interval, which begins and ends when successive even numbered terms are added in the series as the dis- turbance propagates to successive stages of generators. The chief advantage of this analysis of the transient coherency measure for a Specific disturbance is to aid in determining the cause of major fluctuations at a particular location and at a particular time which could lead to system instability. Consequently, the usefulness of the analysis is found in (a) determining the effects on transient stability of adding generation or transmis- sion capacity at a particular location and (2) performing contingency analysis for systems operations and planning. It is hypothesized and later supported by com- putational results on example systems that the transient coherency measure, evaluated for either a deterministic or probabilistic disturbance, can be used to determine coherent groups, dynamic system structure, and weaknesses in the associated structure. Changes in these are shown to occur due to changes in the stiffness of the inter- connection of that portion of the system which is affected by the disturbance caused by the Spread of the initial contingency. When the transient coherency measure is evaluated for a deterministic contingency, the changes in associated coherent groups and structural weaknesses can be analyzed. 13 In order to analyze changes in coherent groups and weak- nesses in dynamic structure that are independent of the location of the disturbance or its magnitude, a zero mean independent identically distributed (ZMIID) step change in shaft acceleration on all generators is used. When the transient coherency measure is evaluated for the ZMIID probabilistic disturbance, it is shown that it is an excellent transient security measure. Following a comparison between the structural information contained in the transient and dynamic measures, these measures are shown to provide the information needed by system planning divisions and operators to obtain a realistic assessment of system security caused by weaknesses in dynamic structure which could be corrected by a change in generation scheduling and network configuration. The transient coherency measure is then used to reformulate the off-line dispatch problem as an on-line linearized tracking secure dispatch problem with a transient security measure which augments the-formulation in the form of an approximated quadratic performance index. The transient security measure is justified as a measure of security, reliability and stability to enable its usage in the performance index of the reformulated on- line secure dispatch problem. This is done by using the concept of an equivalent line connecting a specific gen— erator to an infinite bus representing the rest of the 14 system. The equal-area criterion is used as a tool in the analysis for providing the justification of security and reliability. This measure is shown to be a stability measure by expressing the transient coherency measure as a function of the eigenvalues of the system matrix. By using the anaIOgy of a harmonic oscillator the stability property of the index is then confirmed. The power system operation is subdivided into five operating states and the sub-control problems are formulated. An algorithmic structure to solve these sub-problems is proposed and the sub-problems are then reformulated with the addition of the transient security measure. Time frames for controls are discussed. By assigning appropriate weighting coefficients it is shown that the transient security measure along with the reformulations can assist the operator or automatic control system to move the system in a direction whioh improves system security by stiffening weak parts of the system in the alert insecure mode when postulated next contingency tests are being done. In addition, it enables the system to prevent cascading by continually stiffening the network in the face of an actual transient emergency, provides the system operator with a tool for lowering generation and load to assist in controlled islanding when the system is under extremsis and system tearing is a strong likelihood and assists the operator in resynchronization of areas in the optimal load 15 restoration process. In addition, it protects those parts of the system which have already been restored from the effects of any further contingencies. Thus the transient coherency measure in its use as a security measure is seen to provide a very useful measure of security in each power system operating state -- a feature which was not available to the industry before. CHAPTER 2 A GENERALIZED STATE MODEL AND DISTURBANCE MODEL The principal objective of this chapter is to develop a generalized power system state model and a generalized disturbance model which can be used to obtain an analytical expression for the mean square coherency measure [6]. This objective will be met in the following development which (i) defines the power system (ii) briefly reviews the need for the model (iii) describes and develops the detailed state space model of interest (iv) describes and develops the disturbance model appropriate for a mean square coherency measure used to describe the spread of a disturbance. The power system being modeled consists of (1) an electrical network consisting of trans- mission lines and transformers that connect generation and load. The equations that describe power flows in this network are the network equations of Kirchoff (2) dynamic models of the generators that produce the power injections into the network. The model would generally include models of the generator, voltage regulator, and governor turbine energy system. The system model and disturbance model developed in this chapter are not intended to accurately describe l6 17 the effects of a particular contingency but rather to permit an analysis of the spread of this contingency through a power system. The system and disturbance models de- veloped here are similar to ones developed for an rms coherency measure used to obtain equivalents for an ex- ternal system which is distant from the location of the disturbance. These system and disturbance models will be slightly modified and used to represent the power system dynamic fluctuations in both the internal system and in the external system far from the disturbance where these models [6] are known to be accurate. The linearization of the classical transient stability model, the elimination of Q-V dynamics, the elimination of load dynamics, and the approximations used for disturbance are all justified based on the need to analyze the dependence on system structure and time sequence of the spread of the contin- gency in both the internal and external system. The accuracy of the model for the portion of the system close to the disturbance is therefore sacrificed. Having established the raison d'etre of obtaining a simplified power system model, the next step is to specify the assumptions needed to meet this objective and finally to quantify the model in terms of state space notation. 18 State Model Development The recent SCI work on coherency based dynamic equivalents has shown that the coherency can be evaluated using a model based on the following assumptions: The coherent groups of generators are inde- pendent of the size of the disturbance. Therefore, coherency can be determined by considering a linearized system model. The coherent groups are independent of the amount of detail in the generating unit models. Therefore, a classical synchronous machine model is considered and the excita- tion and turbine-governor systems are ignored. The effect of a fault may be reproduced by considering the unfaulted network and pulsing the mechanical powers to achieve the same accelerating powers which would have existed in the faulted network. The first assumption may be confirmed by considering a fault on a certain bus, and observing that the coherency behavior of the generators is not significantly changed as the fault clearing time is increased. The second assumption is based upon the observation that although the amount of detail in the generating unit models has a significant effect upon the swing curves, particularly the damping, it does not radically affect the more basic characteristics such as the natural frequencies and mode shapes. The third assumption recognizes that the gener- ator accelerating powers are approximately constant during faults with typical clearing times. These above assump- tions and their justification are quoted from [5]. These l9 assumptions were made and justified in a study that used the coherency measure as a basis for developing dynamic equivalents for an external system which is far from the location of the contingency. However, since the assump- tions are reasonable in most cases even in the internal system close to the location of the contingency and since a linear simple model is required to perform the desired analysis of the mean square coherency measure, the above assumptions are also made in this study. It is assumed that the power system includes N generator buses, K load buses and an infinite bus used as the synchronous reference. The mechanical equations of motion for each synchronous generator are do). 1 M. l i = l,2,...,N (2,1) where i subscript for generator i indicates that this variable represents a deviation from a specified steady-state operating point M. inertia constant - p.u. w. speed of generator rad/sec 6. generator rotor angle - radians D. damping constant - p.u. 20 ms synchronous frequency - radians/sec PMi mechanical input power - p.u. PGi electrical output power - p.u. Writing a deviational model for these generators around an operating point 5:, w PME, and PG: where 8! A6- = 6- - 5: l l Ami = mi - ms APMi = PMi - PM; APGi = PGi - PG; the equations (1) become dAwi Mi a-t——- : APMi - APGi " DiAwi dAdi (2.2) dt = Awl i = 1,2, . ,N The network equations in polar form are linearized with the real power equations decoupled from the reactive power equations to obtain: A_P_G_ agg/ag aPc/ag A5; = (2.3) A2}: agg/ag 333/39 AG where PGT = [PG PG PG 1 —— 1' 2'“" N PLT = [PL PL PL ] _ 1' 2,..., K T _ . i [5]., (52!. I ON] T - - g - [Oll ©2I°°°I GK] 21 APLi - deviation in real power at load bus i - p.u. AOL - deviation in voltage angle at load bus i - radians This model can be expressed in state space form by using the network equations in (2.3) to obtain 1 329 _§ BBQ 329 ABE ' 3E£ AA —APG="§'§ “2: ate;— a The state model becomes 3(t) 5 x(t) + a u(t) (2.4) where APM 5 = gg g = (2.5) A2 APL .9 .I. 9 : 9 l l A: ' g: ————— : —————— -M'1T -M-1D M'1 l M-lL _ _ _ _ _ ' _ _ g = Diag{Ml M2 ... Mn} 9 = Diag{Dl D2 ... DN} 3253 see 892* 822 g = —§§ — —§§ ~3§ —§§ (2.6) are aea’l V‘s—g ‘a—g 22 Disturbance Model The following disturbance model is chosen so that a deterministic as well as probabilistic contingency set could be handled. The initial conditions are assumed random with B{>_<_(O)} = Q E{§(0) §T(o>} = yx(0) (2.7) since the expected deviations from any operating state is zero but the variance of such deviations is nonzero. The coherency measure to be developed will be shown to depend on this VX(0). The initial conditions are in- cluded not to reflect any specific type of disturbance but rather the effects on the state from some hypotheti— cal disturbance whose statistics (2.7) may be inferred from internal and external operating conditions. The input, composed of the deviations in the mechanical input power ARM and the deviation in load power APE, can be used to model i) loss of generation due to generator dropping ii) loss of load due to load shedding iii) line switching iv) electrical faults These contingencies can be modeled by an input 3(t) that has the following form gm = u (t) (2.8) 1”) + 9—2 23 The vector function 31 t Z 0 51(t) = (2.9) Q t < 0 represents (i) the loss of generation due to generator dropping (ii) the loss of load due to load shedding (iii) changes in load injections due to line switching The modeling of these three disturbances requires determination of El and possible modification of the net- work before determination of matrices A and B. The procedure used is a modification of one used in [6] when the coherency measure was used for developing an equivalent for the external system. generator droppigg - the transient reactance of the generator dropped is omitted from the network and the deviation in the generator output PM. of the generator drOpped is set equal to the loss of generation. load shedding - the load deviation PL for all buses k where load is shed should be set equal to the change in load caused by the load shedding operation. line switching - the network is modified to repre— sent the system after the line switching operation is performed. The load deviations, PL and PL , at buses to which this line is connected, are set equal to the changes at that bus which occur due to the particular line switching operation. Note that in each case above all variables in El are zero unless otherwise specified. The operating point 24 used to obtain matrices A and B is that obtained from the load flow after network changes are made rather than the base case load flow as in the case of using the co- herency measure for producing external system dynamic equivalents because in this case the coherency measure will be used to investigate effects of a contingency in the internal system where the load flow conditions are important. The vector function / _0_ t > T1 32(t) =( 32 o _<_ t 1 T1 (2.10) 0 t < O k— represents the affects of electrical faults where Tl represents the fault clearing time and APM 9. represents the step change in generation output equivalent to the accelerating powers due to a particular fault. This Change of mechanical powers ABM, which equal the accelerat~ iJlg powers of generators due to a particular fault cal- CHLlated by an ACCEL program [5], has been shown to adequately meiel the effects of that fault when a linearized model (2.2, 2~3) in the case of using the coherency measure to produce dynamic equivalents of the external system. The same 25 electrical fault model is used because it is reasonable in many cases for studying the effects of faults on gener- ators buses in the internal system and because it makes the model simpler for developing the coherency measure. The above model can be generalized to model the uncertainty of any particular disturbance and yet handle. specific deterministic disturbance as a special case. If the size and location of an electrical fault is not known and if the clearing time T1 for this fault is not known, then a probabilistic description of this electrical fault is l 1'12 E{u2} = = m2 _ O _ 5% O (2.11) T - - 1 where ml and 52 describe the uncertainty in acceler- 2 ating power on all generators due to this electrical fault. This mean and variance should be determined based on ob- served historical records or hypothesized based on the present network and present internal and external condi— tions. If 32 = Q, and m: = 92M for a specific fault, this generalized model then reverts to the deterministic model of a specific electrical fault. The uncertainty due to a generator dropping, line switching, and load shedding disturbance could be modeled by 26 ”-m —11 “31} ‘ = 39-1 m 12 (2.12) _ - T = _ .E{[p_l gillgl ml] } - 51 9.. .1322 where (l) m and R can describe the uncertainty In generatIdn changes due to generator dropping when the particular station, the generator in the station, and the power produced on the generator are unknown (2) m and describe the uncertainty in the location and2 magnitude of the load being dropped by any manual or automatic load shedding opera- tion (3) and R describe the uncertainty in the lOcation and2 the change in injections on buses due to any line switching operation It should be noted that APM and APL are assumed uncorrelated because this model is to represent only one specific type of contingency at a time. For the same reason 31 and 32 are assumed uncorrelated with initial conditions and E{§(0) 31} = g T (2.13) E{x(0) 32} = Q The uncertain model of 31 can handle the case of a specific deterministic disturbance by setting £1 = g and ml = 31 for the particular disturbance. 27 Care must be taken with this probabilistic model for generator dropping and line switching contingencies in order to make sure the network changes associated with the set of such contingencies being modeld probabilistically are properly performed. CHAPTER 3 A GENERALIZED MEAN SQUARE COHERENCY MEASURE The objective of this chapter is to develop a generalized measure of coherency which best reflects the overall system dynamics and which is broad enough to en- compass the deterministic and probabilistic contingency set already discussed in Chapter 2. This develOpment is the mean square coherency measure based on a similar development [6] on the rms co- herency measure. The mean square measure of coherency between gen- erator internal buses k and 2 based on the uncertain description of disturbances is _ l_ T _ 2 Ck£ - Tn E£f0 [A6k(t) A6£(t)] dt = i— E{fT[(A6 (t) - A6 (t)) - (A5 (t) - As (t))]2dt} Tn 0 k s Q s = —— E{fT XT(t)Q X(t)dt} (3.1) Tn O — —k£— where ka is a 2N-dimensional square matrix 2k, 2 Q = (3.2) —k2 2 2 and le is a N-dimensional square matrix where the ijth 28 29 elements are {9k£}ij = <-1 i = k, j = I and i = 2, j = k (3,3) L.O elsewhere Taking expectation inside the integral and recognizing that the coherency measure between internal generator buses k and l is an element of a matrix C, the co- herency measure becomes - _ ;_ T {9}k2 ' Cki " Tn f0 Tr{9k2 £3«(twat (3.4) = Trfgkzlifi-ffi Ex(t) dt]} T where Tr[-} is the trace operator (sum of the diagonal elements of the argument matrix) and Px(t) is E{§(t) §T(t)} = Bx‘t) = yx(t) + Ex(t) 33(t) Since the objective is to compute the N x N matrix and since {9}k2 = Ckz Ckz = Tr{g_k2 §X(T)} = {§x(T)}kk + {S§(T)}££ - {§x(T)}£k - {S§(T)}k2 (3.5) where 30 It thus can be seen that the coherency matrix Q is easily determined knowing §X(T). The matrix §X(T) is easily computed by substituting (At t Av €— §(O) + f0 5— dv §(El + 22) t < Tl x(t) =< eétx(0) + f3 évdv E El (3.6) A(t-T ) T - l 1 Av k + 5 f0 5— dv B 32 t > T1 8 (T) = l— IT E{x(t) xT(t)}dt ' (3-7) —x Tn O — — and taking expectation term by term using (2.7, 2.11, 2.12, 2.13) to obtain T _l_ T a At §X(T) - n f0 5 VX(O)€ dr T + l_ [Tl[fT évdv B][R + R + m mT + m mT + m mT + m mT] Tn o o E — —1 —2 -1—1 —2—2 #1—2 —2-1 x [f5 gévdv §1Tdr + 1 1T (UT eéVav B][m mT + R 1[/' eévdv BIT)dT <3 8) Tn T1 0 — -l-l —l 0 - A(T-T ) T 1 T — l 1 Av T T l A(T-T ) T x [e l fol Eévdv EiT)dT A(T-T ) T + l— IT ([5 l f l sévdv Bllm mTllfT Eévdv BlTldT n T o - —2—1 0 — T 1 A(r-T ) T + ifi'fg ([fg eévdv Ellfllflglle 1 f01 Eévdv ElT)dT T 1 31 The integer n is chosen to be one if a load shedding, line switching, or generator dropping contin- gency occurs and zero if an electrical fault occurs. This integer is chosen as one or zero so the integral will be finite and non-zero for an infinite observation interval. The matrix §x(T) becomes _ 1 §X(T) - T fg[fg Eévdv EJEIEEIfSE AVdv §]T)dr (3.9) if a specific load shedding, loss of generation or line switching disturbance occurs since in this case 31:32:0'E=0’El=u —l’ and VX(O) = Q and n = l. The matrix §X(T) has the form T _ 1 T Av T r Av T §X(T) - f0 [f0 5 dv glgzgzlfo E dv g] dr A(T-T ) T A(T-T ) T + I; ([e l fol eévdv §]gzgz[e l fol eévdv §]T)ar 1 if the specific deterministic disturbance is an electrical fault Since 51 = 52 = O, VX(O) = g, ml = Q and m2 = 32 and n = 0. This generalized mean square coherency measure can handle both deterministic as well as probabilistic des- criptions of power system disturbances. CHAPTER 4 A TAYLOR SERIES EXPANSION OF THE MEAN-SQUARE COHERENCY MEASURE The purpose of this chapter is to derive a Taylor series expression for the mean square coherency measure as a function of the observation interval T over which the measure is evaluated. The Taylor series expression for matrix 95(T) will be derived by first deriving a Taylor series expression.for §x(T), and analyzing the coefficient matrices in this series. The transient co- herency measure will be defined and analyzed in the next chapter based on the Taylor series expansion of the co- herency measure developed in this chapter. The mean square coherency measure can be computed using the matrix §x(T), which is defined in (3.7). If the observation interval T is less than the fault clearing time T vector §(t) is 1! _x(t) = eét'ym + I; eZ—de _1_3_(gl + u2) (4.1) and upon substitution the matrix §X(T) becomes _ t AT A r -x Tn f0 6— Zx(0)€— dr (4.2) + L ft [Igeévdwggu (O)B_T[fge§vdv]Tdr 32 33 where yx(0) = E{x(0)_ T(0)} (4.3) p (0) = E{u(0)uT(O)} = R + R + (m + m )(m + m )T(4 4) —u - - —l -2 —-l —2 -‘l -2 ° since 3(0) = 21 + 32 for t < Tl' Substituting m Aka 81-” ‘ 5 :RT‘ k=0 ' (4.5) k k+1 co A T IT eévdv = Z I 0 k=0 +1 ' into (4.2) and integrating,tflmamatrix §x(T) becomes w m . m j+m+1 l j T T .5; (T) = — I Z [5. Y (0):} 1 1 'T) x Tn j=0 m=0 x (j+m+le.m. (4.6) 00 00 m j+m+3 1 T T T +.—— Z 20[ [A B P (0)5 A 1 . . Tn j‘ _0 m-O — — —u —'— (j+m+3)(j+1)!(m+1)! Letting K = j+m, the §x(T) matrix becomes w k K+1 ..L_ j AT K- j T §x(T) ‘ Tn KEG jéotA Vx (°)— 1 (K+1)jl(K-j)! (4.7) 00 k . . K+3 l j T T K-j T + "E 1 .2 [é E 3u(0)§ 5 ] (K+3)Tfi+1)1(K-j+1)1 Multiplying numerator and denominator of each term in the expression by appropriate factors, matrix §x(T) can be expressed as TK+l K+3 _ m T §X(T) _ .E'Kio 2K (K+1)1 + ER (K¥3)1 (4'8) PSI-4 34 where coefficients K . . GK = Z (3) A3v (0)AT K 3 (4.9) _ . 3 __.X .— J=0 L K K+2 j T T K-j —K ) ('+1) A B P (MB A (4.10) 1%] “‘w ‘“ The matrices L and G 4K K can ea31ly be shown to satisfy the following recursive formulas: follows §K+1 = A gK + (A gK) K = 1,2,3,... (4.11) G0 - yx(0) L L - K_2. K o T , .IiK+]_ _ §(_I:'.K + A 2 )+ [A(LK + A 5-)] (4.12) _ T £0 28 gu(0)§ The derivation of the recursive formula (4.11) the derivation found in [9]. Since K v ' T K-' GK = X (9)21J v (0)A 3 (4.13) _ . J _ —X _— 3-0 T K K j T K+1-j GKA = Z (.)A v (O)A (4.14) _. _ ._ J _ —X _ j—O Replacing j by j-l in (4.13), K+1 . . G - Z (.k.)A3‘lv (0)AT K+1‘3 (4.15) -K i=1 3‘1 — “X _ and multiplying by A gives 35 K+1 . . _ k j T K+1-j A gK - .) (j_l)A yx(0)A (4.16) J=l The sum of (4.14) and (4.16) is G AT + A G - I (K)AjV (0)AT K+l'j —K— — —K _ .50 j — —x — 3 (4.17) K+1 . . ._ 3-1 — —X — j-l . K K K+1 _ Since (j) + (j-l) — ( j ) and ZX(O) - g0, T T K+1 K K+1 j T K+1-j K+1 G A + A G — G A + E ( . )A G A + A G —K— — —K —O— j=l j — —0 — —o (4.18) The terms on the right hand side of (4.18) can be expressed as a single summation = G (4.19) and therefore _ T §K+l " Exist + E. _G.K (4‘20) Using the symmetric property of g the expression K! (4.20) becomes G = A G T —K+1 —K I (5 9K) (4'21) Similarly the second recursive formula (4.12) is derived as follows: K . K+2 T T K-j _K = (4.22) )AjB P (0) j=o ““1 36 K . L . T _ K+2 ] :9 T K+l‘j AKA - .Z (j+l)A 2 A (4.23) :— Replacing j by j-l in (4.22) K+1 . . L = l (KI2)A3’1L AT K+l‘3 (4.24) —K 2 ._ j — —O— j—l K+1 . . A L = i F (KI2)A3L AT K+1 3 (4.25) _ _K 2.3. J _ _0_ 3—1 The sum of (4.23) and (4.25) is T 1 K K+2 j T K+1-j .121 = ~2- 2 (me. P. A 3-0 K+1 . 1=1 3 . K+2 K+2 = K+3 U51ng (j+l) + ( j ) (j+l)' (4.26) becomes T = A K+2 j T K+1-j A K+3 j T K+1-j —K5 2‘ 1 )5 £05 + 2 j§1(j+1)5 905 (4.27) A K+2 j T K+1-j + 2 (K+l)§'EOA = l(K+2)L AT K+1 + I § (K+3)AjL AT K+1 3 2 —O— 2 j=l j+l — -0— (4.28) + i (K+2)AK+1L 2 — —0 Now £K+l can be expressed as the summation K+1 . . _ A K+3 j T K+1-j EK+1 ’ 2 jgo (j+1)5 E 5 (4‘29) K _ A K+3 j T K+1-j A K+3 j T K+1-j ‘ 2( 1 )5 E0— + 2 Z (j+1)é E A 3—1 A K+3 j T K+1-j + 2 (K+2)§ gog (4.30) l T K+1 l K K+3 j T K+1-j = 2(K+3)§oé + 2 .E ('+1)§ LoA j-l + l c5 (T) CN+ (T) (5-9) k1 k2 T=T (5) k1 k2 T=TN (6) k2 k2 48 . s N+l . . . and Since Ck£(T) - Ckl (T) is a monotone 1ncrea51ng function of T, it follows that TE? (8) g, TEN“ (5.10) and the theorem is proved. The transient coherency measure is now defined (1 ck2(T) 0 §.T £.T:£(€) Chm =( (5.11) N N- Ck2(T) Tklle) 5_T :_T:£(E) N=2fiL... This transient coherency measure is an approximation of the mean square coherency measure over a sufficiently small interval 0£T£T* with a maximum error e. The number of terms (N) used in the approximation increase as T increases. This property of the transient coherency measure is necessary to analyze the changes in the power system dynamics after a disturbance occurs. This analysis of the changes in power system dynamic structure will be carried out for both a deterministic generator dropping disturbance at a single bus 31 = _1M1' 9.2 = 9' 31 = 9.! 32 = .9 (5'12) where T Se-1 (1,0, . . ., 0) such that the matrix 2 (4.37) is 2 =" diag (1,0, . . ., 0) (5.13) 49 and a probabilistic disturbance such that B = l (5.14) This probabilistic disturbance represents a zero mean 110 step change in initial shaft accelerations on all generators and could be produced by the generator dropping, line switching, load shedding, and electrical fault disturbance models. The explicit values of matrices A0, A2, A4, A6, evaluated from (4.42), which are required for this analy- sis, of the transient coherency measure for the deter- ministic disturbance (5.13) are 1:. = 2 [’6 m=n=i {L2} - {6P} = —mn LO otherwise ( o m#1.n¢i (23-15) N Ti. -30 , z: = Bil-Hf. mrii _ T (£4}m — {15(§_§_+ E 5)}mn = { Tni +15 M— m=i, naii n Tmi k +15 Er— m#i, n=i 50 T 2 . __ P+702 PXT+28PX2} . T . m1 51 +70 I O N T.. N T.. 70 {If .fi. if _u. 3’ ‘i 3‘ Mi ”9' 4 T.. N T + 56(_.Z —11 ..Ll z 11 2 , 3:]- ((4.154 + (jzl “.2 ) nm=1 #1 3 l 1' T.. T NgT . T ~70 .29 —-l‘ “l *- 28 .2 —3—“ -J—1 3=l Mi Mn 3:}. M1 M jxi ‘ 3 Ja‘n Ng Tn n Tn, Mg Ti. m=i, .. (21 ...l ).....L... - _:_ ( E 1.1..) , 3-1 M M M 3-1 .4. n¢1 n n n J. N T N T T '1 g ' _70 ml(.zg 1_) 28,-; m1 :1 M 3=1 112 L3=1 Mn :4. iii ‘ 3 3m Ng T T . T Mg T _ - (z 21.1,.n11..-m1(: .31) i=1 M :4 ‘4 3:1 M " m m m 51 . 2 The tran51ent coherency measure for TeE0,Tkz(e)] and the deterministic disturbance (5.13) is now analyzed. This transient coherency measure evaluated over the first interval OETET:Q(€) when n=l is 4'1“ >43... (5.16) / O kaéi, 9,1 0 kzir =... ) -4 N - - f 1,! - ... - 4 6 5' -3O(j=l *1 + :.'-‘) 1,6 k-l, ff]. 1'] — 1 £ ' i 14 N m , F— -3C)( 2 ‘44 TK' \ . b f ’51 j=l-;i‘+ 1' I—- < 1 ”=1 ‘ M. M 71 1. k The coherency measure between generators k and R, where neither k nor 1 is the disturbed generator (i), indicates that all these generators kfi swing coherently as a single generator over the first inter- val. The coherency measure ka(T) = C*ki(T) is non zero over this interval and indicates all generators kfi swing together as an infinite bus against generator i that is accelerated by the disturbance. Generator i is a stage 1 bus since it is the only generator accele- rated in the first interval. The part of the system dynamics not directly connected to bus i can be neglected over this interval because it does not affect the coherency measure (5.16)for any pair of buses in that system. The transient coherency measure over the second interval Til(e)5TgT§£(g) for n=l is ‘ 4 6 8 ~T * T . T . T ~ T . C* = - . —+ :—+ -—-—+ 32(T) :k1(EO 31 E2 5: E4 7: £6 9.) 9kg T 2 T *- T T 8 f 1 o 9 2 xx 21‘ T k¢i n0(_.__-+ "'"""“"' ‘ ’ _T' ’ ) ’ 2 .,2 M M Mk “1 k 1 y) 0 k=2=1 A* T = 3x£( ) 1 ”8 1 .: 5’ 1 _ ‘ 1 _ Iv u— - , C ”(T) + [116111. + +56 99 {£61 b6}11]9 k l 8 1.. rr‘) +rf.‘ _,;} ~11).)3— m, ( cki(‘) + “E6’kk ‘-6’1; “‘5 ki —6 1k 91 where £6 mn are taken from (5,15), This transient coherency measure between genera- tors k and 2, where neither k nor 2 is the disturbed generator, is dependent on the relative stiffness of equivalent lines Tki Tgi I Mi Mi connecting generators k and i and 2 and i. If there are no equivalent lines connecting generator 1 to gene- rator k and to generator 1, Tki=T9i=0 * then the transient coherency measure Ck£(T) continues to be zero over the second interval just as it was over the first interval. However if there is an equiva- lent line connecting generator k to generator 1, (5.17) 53 T O ki" then c;£(T) is non zero over the second interval for all generators 2. Thus, the set of all generators k connected to i by equivalent lines T 0 kif will leave the group acting as an infinite bus swinging against generator i during the first interval and be accelerated by the disturbance over this second inter- val. This set of generators are called stage 2 generators. It should be noted that all generators (k) not connected to generator 1 by an equivalent line Tki=0 will not have been affected by the disturbance in the first and second interval 0§T§T§2(€) and act as an infinite bus. That portion of the system not in stage 1 or 2 or connected to stage 2 but not in stage 1 (stage 3) does not affect the coherency measure during the second interval. If the above analysis were carried out for the Nth interval for each N it could be shown that (1) stage 1 generators are those that feel the affect of the disturbance immediately at t = 0; (2) stage N+1 generators are connected to stage 1 generators in the shortest path by N branches or through (N-l) generators as shown (3) the N+1St stage is accelerated beginning at Tig-l(e) since C§E(T)N are nonzero for any generator K in stage N+l but lower This (1) (2) 54 order approximation Cfij(T) for K = l,2,...,2N are zero when j belongs to stage N+l,N+2, {ij(T)}1:_l are all non zero for anyN k in stage N+l since every term {ij(T )}j_ -1 de- pend on {L4N+2}kk which is not zero because 2 {L4N+2}kk N (XkL) - N N (5.19) {X }k°= (.2 _2 °-- 2 Xi. X. . X. . X. ) 1 32 J3 jN 32 3233 3334 ij and because {éy}ki is nonzero for any k in N+l. {XN}k2 indicates both the generators and lines affected by the disturbance and the stiff- ness of the interconnection between the dis- turbed generator in stage 1 and any generator k in stages 2 to N+l in the 2Nth interval. Note that {AN }ki is the sum of all N branch connections between k and i and thus relative stiffness of the interconnection between k and i depends on an increasing number of paths and a larger portion of the system as T and thus N increases. information is important because it disects the transient response into discrete intervals Tig 1(e) < T < Tififl(€) when the N+lSt stage is accelerated indicates the portion of the system affected by the disturbance and provides a measure of the relative stiffness of that interconnection 55 {éN}ki between the disturbed generators and the portion of the system affected by the disturbance during each of these intervals which begin when even number terms E2N+2 are added to the coherency measure approx- mation. (3) permits the cause of significant oscillation at any location in any time interval, which could lead to loss of stability, to be easily determined. This could be accomplished with- out the guesswork since contributions to the transient response at any time and loca- tion can be easily determined analytically without repeated simulation. This type of analysis would be helpful in trans- mission or generation expansion planning to isolate the effects on transient stability of adding lines or generation. It would also be helpful in contingency analysis for operation planning. This completes the analysis of the power system transient response for a deterministic disturbance at a single bus. The above results, which are confirmed using computational results in the next section, also indicate that changes in dynamic system structure, weakness in that structure, and coherent groups could be expected due to the changes in the stiffness of 56 the interconnection which occur because the disturbance has spread. This can be clearly seen for the determi- nistic disturbance at a single generator but the results obtained then depend on the location and magnitude of that disturbance. A more complete understanding of the changes in the coherency measure, coherent groups, and weakness in dynamic structure as a function of time is possible if the transient coherency measure is analyzed for a probabilistic disturbance, which is chosen so that the transient coherency measure only IT and does not depend on the distur- depends on 5: -g- bance. The disturbance model that makes this possible is from (4.37) 2 = 1 -(5.20) T '. T_r- T1 _00. §2u(0)§ ' §L§1+52 + (ml+fl2)(ml+flz) QE ‘ 9 E. This disturbance produces a zero mean independent, identically distributed (IID) step change in shaft acceleration at all generators in the system and is thus an excellent disturbance to investigate the dynamic structure and structural weaknesses during the transient interval. This disturbance (5.20) could be produced by a generator dropping, load shedding, line dropping or electrical fault disturbance model. The generator dropping disturbance required to obtain g = I is 57 m —11 _ _ . 2 2 2 = 511 O 0 IO R1 32 =2 (5-22)-“111=9 (5.24) The generator dropping disturbance that produces zero mean IID step change in shaft acceleration on all generators requires the mean of the step deviation in mechanical power on each generator to be zero, the correlation between the step deviation on any two gene— rators to be zero, and the variance of the step change in mechanical power deviation on any generator to be proportional to its inertia squared. The transient coherency measure over the first interval for this probabilistic disturbance at every generator is now given as: AT 4 * C(T)=e[6IT+1r(+xT5‘ l ' (5.25) 12 4 N9 N9 6 T T . T = ———---30 .3 k + ,2 2° .l. J; T 51 (3=1—-l 3:1 —3-M + Tkr (Mk + M93)? jfk J#2 9 This transient coherency measure reflects the stress on a power system during an initial interval where the disturbance at each generator has not yet propagated to neighboring generators. This coherency measure is quite independent of the kind of disturbance 58 or the magnitude or correlation between disturbances so that the true system structure and weakness in that structure are indicated for this first interval. The structural coherency of generators k and over the initial interval, that is independent of the disturbance and only dependent on system dynamic struc- ture, not only requires a relatively stiff interconnection of k and 2 measured by k (— + —) (5.26) but also that these generators be stiffly connected to the system N N 9T 9T 2' z j .27 3.51 Mk 3.31 M2 (5 ) 2 A group of generators will only be mutually cohe- rent if all the generators k are stiffly tied to most of the generators in the group because then both (5.26) and (S.27)will be large for all members of that group. However, a generator could be coherent with one member of a group and not other members of the group if it is very stiffly tied to that one member. The form of the transient coherency measure CE£(T) over the first interval also shows how the first swing stability of generators k and 1 also depends on the relative stiffness of the equivalent line connecting k and £(5~26)andthe relative stiffness of all lines 59 connected to k and to 2. This is extremely important information since it is not disturbance dependent and indicates regions where there is relative structural weakness and susceptibility to transient stability problems over that initial interval where first swing stability is determined. These expressions (5.26) and (5.27) can be considered measures of the relative stiffness of any equivalent line in the system and the stiffness of the intercon- nection to any bus k or 2 respectively. These could be used to determine weakness of any equivalent line or the interconnection to any generator bus. The transient coherency measure over the 2Nth interval TEN-1(a):T§TiN (e) is c =-——* __ T 2 T T T (5.28) N . T. 2N+4 [.2 a .XN-Jx J]EL___. é 3:0 NJ-' - 2N+5! -k£ This transient coherency measure for the 2Nth interval has terms that have form §N XNT which indi- cates the disturbance on each generator has propagated to N+l stage generators for each disturbed generator and thus the transient coherency measure depends on generator inertias in stage 1 to N+l for each disturbed 60 generator and the synchronizing torque coefficients of lines that connect them. The last term added to the coherency measure N . j . PFJ T (5.29) = _ X £2N+2 jgl am 5 should indicate the relative stiffness of the inter- h interval as did {2(_N}ki for connection during the 2Nt the deterministic disturbance. Thus, the transient coherency measure for the probabilistic disturbance indicates structural weakness, the relative stiffness of the interconnection between generators and the coherent groups, which are independent of disturbance th interval for each N. location at the N The dynamic coherency measure defined as c121; ._. Cir”) Ta. (5.30) measures the power system structure in steady state where the disturbance propagation and reverberations have subsided. This dynamic coherency measure can not be evaluated for the synchronous frame power system model (4,6) since the g matrix is singular. The dynamic coherency measure will now be developed and compared with the transient coherency measure. The dynamic coherency measure is evaluated for a uniform machine angle frame model 61 _3 y.— :3. l 51322 -a;_ “T .. 6 — (61,52, &=(&,w, — l 2 61 = 51 - 6N wl=mi-mN r-- |-1-1 l-l -1 2N-1| l-l L— | _J F‘- -— 5N4 66667"'6 L.— O O -l -l -_ £1 4 512:. I N-l) 62 where g, T and E are defined in (2). The dynamic coherency measure [39] for this model and the step disturbance model V = _ym) 9_ _fl - ‘ O P |0 I i P (O) = R +m mT u .— has the form XX Ski - 9kg s (m) in where f l j=k -l j=2 k#N 2%N O j#k,£ l j=k é . = =L {_kfl}J ({0 ji‘k mm 2 v 1 3:2 K4) . T S (m) = 11m - fE{y_(t)y_ (t)}dt Tea T o T l T A = E!” e: vdv]G P (O)GT f e5 Vdv (31' o --u - T = F 1c; P (meTF'l — —w _— T -l -l -l -l [M 23.1: 321 3m 212221 0 O O :2 63 The dynamic coherency measure is proportional 1211 for the transient to [g'lglg gzj'l rather than [5‘ coherency measure. The generators will be coherent in the transient coherency measure if the equivalent line which connects them is stiff since this measure depends on g-II. However a pair of generators will only be coherent for the dynamic coherency measure if (1) the two generators are extremely stiffly connected that the first generator will be coherent with all generators in the group to which the generator is stiffly connected to even though this first generator is not stiffly connected to the other generators in that group or (2) both generators are stiffly connected to most if not all of the generators in the coherent group to which they both belong. In both cases all generators in a coherent group for the dynamic coherency measure should be mutually coherent since the dynamic coherency measure depends on [g’lglg §23-l and not on 5'13. This transient and dynamic coherency measure can be used to indicate power system structure and weaknesses in the dynamic structure for some T. It should be noted that the transient and dynamic coherency measures and thus the system structure depends on the unit committ- ment network configuration, load flow conditions, and interval T given. With the disturbance, unit committment, 64 network configuration, load flow, and interval specified, the dynamic or transient coherency measure can be used to (l) (2) determine the groups of generators that are coherent and thus are relatively stiffly tied together. These coherent groups and of course incoherency of the generators in different groups defines the structure of the system; determine the relative weakness in the struc- ture by examining the coherency measure Ck£(T) on equivalent lines that connect internal generator buses k and 1 that lie in two different coherent groups. If their coherency measure Ck2(T) is very large, the apparent weakness in the equivalent lines, measured by T l + 1 k2(—- f) (5.31) Mk 2 for the transient and dynamic time frames respectively, connecting the internal gene- rator buses could be diagnosed as due to the (i) overload on the equivalent lines (ii) relatively small capacity of the lines for the generator groups they are expec- ted to connect; 65 The stiffness of the interconnection to generator k for the transient and dynamic coherency measures are [M'1T1 and 1 1 Ir 2, 2 221 The former measures the size of the contin- gency that a bus could withstand without loss of synchronism for the transient interval. Weakness in the interconnection to a generator in the transient or dynamic time frame could be due to loss of lines due to a contingency, maintainance, or due to an inadaquate design. This transient coherency measure (5.25) over the first interval can be shown to be an excellent transient security measure because (1) (2) (3) it is defined for the initial interval where first swing stability is determined it measures weaknesses in system structure for this interval which are independent of magnitude location, correlation, or kind of disturbance. It could also be evaluated for a probabilistic or deterministic decsrip- tion of a specific contingency if that were desired. each term Tkj can be shown to be proportional 66 to the energy capacity of that line to withstand a fault. This will be done later in Chap- ter 7 using an equal area criterion argument because all generators connected to the distur- bed generator act as an infinite bus for the disturbance during the first interval. (4) it depends on the present unit committment, network, and load flow condition in the system. This transient security measure (5.25) can be used as a performance index for optimal power dispatch or optimal load shedding problems to force a more secure transient stability related operating condition out of the optimal power or load dispatch in the alert and emergency operating states in a manner similar to that used in (28, 29). The use of the transient cohe- rency measure as a transient security performance measure and for developing transient security constraints for the optimal dispatch and load shedding problems will be discussed in a subsequent publication. This transient and dynamic rms coherency measure could also be evaluated off-line for system planning purposes (1) to assess the security and structural weak- nesses for the present power system with a particular network configuration, unit committment and load flow, which are greatly (3) 67 affected by forced outages and maintainance schedules. assess dynamic and transient system security and structural weakness for particular trans- mission and generation expansion alternatives. assess particular relaying strategies and their effect on system security and structural weaknesses. 68 COMPUTATIONAL RESULTS 5.2 The purpose of this section is to use a computer program, which can calculate the Taylor series approxi- mation of any order for any observation interval, to (I) analyze and justify the basis for using the transient co- herency measure to analyze the transient response of power systems and (2) confirm the difference in the structural properties of a power system measured by the transient and dynamic coherency measures. The justification of the basis for using a transi- ent coherency measure to analyze propagation of distur- bances will be accomplished using the simple system shown in Figure 0 with system matrices F—l66.66 66.66 -0 -0 -0-__( 66.66 -85.00 1.66 16.66 -0 §?_M- 2? -0 1.25 -76.25 25.00 50.00 -0 12.50 25.00 -87.50 50.00 1__ -0 -0 400.00 400.00 -800.00 .1 -1 _ g_ = d1ag{3.334,3.334,2.SOOO,2.5000,20.0016} and the deterministic step change in shaft accele- ration at generator 1. 659 Ix F! H N I H] 3| H N I! O O :: ._. fl —--—-" 0 U .— r! p. M II C N C II o U ’2 'L 4 e 9. 9‘5 2’ 9 35 it 20 01. :2 ”‘ . n E: U‘ T fill o 0 1 (T12 ‘ T23 * 724) T23 T24 M2 M2 M2 T (T + T T T34 23 _ 23 34 35 6““ M3 M3 3 :31 321 - (724 ‘ 34 + T45) M4 M4 M‘ o L35 L42 M5 M5 fl 6 1 --- C) M 2 "' l P = (3 C) c J Figure 0 O '30 u U‘ | I *3 70 For this disturbance and power system model, the analysis based on the transient coherency measure indi- cates (1) generator 1 is a stage 1 generator accelerated over the first interval and this acceleration . , l is observed in Ckf (T), (2) generator 2 is a stage 2 generator accelerated beginning in the second interval and this 2 acceleration is first observed in Cki (T); (3) generator 3 and 4 are stage 3 generators and accelerated beginning in the fourth interval and this acceleration is first observed in 4 Ckz (T); (4) generator 5 is a stage 4 generator and accele- rated beginning in the 6th interval and this 6 acceleration is first observed in Ck2 (T). because stage N+l was shown to be first accelerated over the 2Nth 2N k2 (T). interval and this acceleration first appears in C The Taylor series approximation for the mean square coherency measure of orders N=l,2,...,9 for T=0.025, 0.050, 0.075, and 0.100 is shown in Table l. The results shown in this table show that ‘)\ (1) stage N+l is accelerated in C“I (T) but is k2 K not accelerated in {Cki (T)} :EII for every T- (2) (3) 71 The order of the approximation at which acceleration of a stage begins to appear is found by noting the order of the approximation when the coherency measure between buses in ’this stage and every other stage are non zero for stages 1, 2, and 3. that all stages are accelerated at any value of T no matter how small if the order of the approximation is high enough; that the effective movement between buses in stage N+l is significantly less than in stage N for all N as measured by Oil (T) when T is .very small and that the motion in each stage N begins to become significant (> E) as T increases. This is the basis upon which the definition of the transient coherency measure is based. This definition adds terms to the coherency measure approximation as their effect becomes significant with increasing T. Neglect- ing the very small motion and acceleration in stages far from the disturbance location until their effect becomes significant permits (1) disecting the transient response into the dis- crete sequence of events which are the acceler- ation of successive stages, (2) the determin- ation of lines and generator in each stage, (3) (4) (5) 72 the determination of stiffness of the inter— . N connection {é }ki generator and any generator k in stages 2 between the disturbed to N+l that is affected by the disturbance in the 2Nth interval and (4) the effect of any particular lines synchronizing torque coeffi- cient or generator inertias on the transient response at any time and location. the determination of the exact values of TEE (e) is difficult for any k, but can easily be approximated even from Table 1. It is clear stage 2 does not become significant (€=30x10-l3) until Ti£(e)=0.25 seconds and that stage 3 does not become significant for the same 8 until T§£(€)=.10 seconds. Ti£(s), which is the time that stage 2 (generator 2) begins to accelerate, is found by noting when 2 2 which is the time stage 3 (generators 3 and 4) C R (T) is greater thane:for 2=3,4,5. Tfi£(€), begin to accelerate is found by noting when C§£(T) is greater than 6 for i=5. the approximation for the 2N model with zero damping, which has two zero eigenvalues,breaks down for T=0.15 as would be expected as can be seen from the convergence difficulties for C§4(T). The value T* for which a Taylor series 73 approximation would be satisfactory should only be T*§.lO seconds if N is small. TABLE 18 T-.025 k-: Ci: :: Cii 3:; Ci; Ci: CZ: 'Z: 1-2 53900-08 53901-08 53901-08 53901-08 53901-08 53901-08 53901-08 539:.-:5 1-3 54115-08 54115-08 54115-08 54115-08 54115-08 54115-08 54115-08 54:15-09 1-4 54115-08 54115-08 54115-08 54115-08 54115-08 54115-08 54115-38 54.-s-05 1-5 54115-08 54115-08 54115-08 54115-08 54115-08 54115-08 54115-08 54*;5-23 2-3 0 31142-13 31029-13 31030-13 31030-13 1030-13 31030-13 31:3‘-13 2-4 0 31142-13 31018-13 31019-13 31019-13 31019-13 31019-13 31:-9--3 2-5 0 31142-13 31031-13 31031-13 31031-13 31031-13 31031-13 31:3--:3 3-4 0 0 0 11305-20 11:60-20 11:60-20 11260-20 :1:e:-;0 3-5 0 0 0 13950-22 12827-2: 12858-22 12855-2: 1:535-22 4-5 0 0 0 13957-00 13793-20 13795-20 13795-20 13‘9s-z0 TABLE lb T-.05 C k" C1141 C126 c1301 c1401 C1541 CE: CZ: C; 1-2 16859-06 16867-06 16867-06 16867-06 16867-06 1686“-06 16867-06 16==‘-06 1-3 17134-06 17137-06 17137-06 17137-06 17137-06 17137-06 17137-06 1‘-“-06 1-4 17134-06 17137-06 17136—06 17136-06 17136-06 17136-06 17136—06 .'-35-06 1-5 17134-06 17137-06 17137-06 17137-06 17137-06 17137-06 17137-06 1‘:3'-06 2-3 0 15945-10 15674-10 15677-10 15677-10 15677-10 15677-10 15607-10 2-4 0 15945-10 15651-10 15654-10 15654-10 15654-10 15654-10 15:54-10 2-5 0 15945-10 15677-10 15679-10 15679-10 15679-10 15679-10 155'9-10 3-4 0 0 0 92609-17 90948-17 90963-17 90962-17 90952-17 3-5 0 0 o 11433-18 77097-19 81219-19 81044-19 8.045-19 4-5 0 o o 11433-16 10876-16 10893-16 10893-16 1:593-16 74 TABLE 1c T-.O75 k" c1111 cii C131; Ci: C21 C: i c1711 Ci, 1-2 12297-05 12332-05 12331-05 12331-05 12331-05 12331-05 12331-05 1233.- 5 1-3 12768-05 12779-05 12779-05 12779-05 12779-05 12779-05 12779-05 12779-25 1-4 12768-05 12778-05 12778-05 12778-05 12778-05 12778-05 12778-05 2776-25 1-5 12768-05 12779-05 17‘79-05 12779-05 12779-05 12779-05 12779-05 12779-35 2-3 0 61297-09 58843-09 58896-09 55896-09 58896-09 58896-39 5869--‘9 2-4 0 61297-89 58645-09 58707-09 58706-09 58706-09 58706-09 587C--‘9 2-5 0 61297-09 58865-09 58910-09 58910-39 53910-09 58910-09 589l--'9 3-4 0 0 0 18024-14 17273-14 17286-14 17286-14 ‘728é--4 3-5 0 0 0 22251-16 59140-17 99969-17 96037-17 962':-.' 4-5 0 0 0 22251-14 19780-14 19942-14 19942-14 19942--4 TABLE 18 T-O.10 1-. cfu a; c): c; c; c; c; c: 1-2 48811-05 49296-05 49272-05 49273-05 49273-05 49273-05 49273-05 49273-05 1-3 52338-05 . 52491-05 52485-05 52486-05 52486-05 52486-05 52486-05 52456-35 1-4 52338-05 52483-05 52478-05 52478-05 52478-05 52478-05 52478-05 52475-05 1-5 52338-05 52492-05 52486-05 52486-05 52486-05 52486-05 52486-05 52466-05 2-3 0 81637-08 75690-08 75926-08 75920-08 75920-08 75920-08 75923-08 2-4 0 81637-08 75222-08 75495-08 75487-08 75487-08 75487-08 75487-03 2-5 0 81637-08 75742-08 75945-08 75943-08 75942-08 75942-08 75942-08 3-4 0 0 0 75866-13' 70130-13 70345-13 70340-13 70343-13 3-5 0 0 0 93661-15 28713-15 25746-15 16391-15 17386-15 4-5 0 0 0 93661-13 75051-13 77337-13 77124-13 77143-13 Table 1. Taylor series approximations of the mean square coherency measure evaluated for obser- vation intervals T=0.025, 0.050, 0.075 and 0.100 seconds. 75 i-j Dynamic Coherency Measure Transient Coherency Measure T=.25 sec. 1-2 201549-01 63503-04 - 169508-01 53364-04 1-4 233710-01 68186-04 - 199762-01 58412-04 - 198108-01 57644-04 - 169895-01 48244-04 -3 103027-01 47835-04 - 164842-01 59355-04 - 101774-01 49620-04 2-6 102282-01 48980-04 2-7 861094-02 37962-04 3-4 784689-02 45758-04 3-5 305593-02 37054-04 3-6 288078-02 35828-04 3-7 169590-02 29936-04 4-5 637864-02 48205-04 4-6 636804-02 47248-04 4-7 910123-02 42924-04 5-6 188861-03 30501-04 5-7 330258-02 28038-04 6-7 318956-02 28047-04 Table 2. Transient and Dynamic Coherency Measures for the MECS example system. 76 A second example is used to show the difference in the structural properties measured by the transient and dynamic coherency measures. These coherency measures are evaluated for the seven station model of the Michigan Electric Coordinated Systems 010 that was used to derive modal [4] and coherency'CBT based equivalents based on the rms coherency measure evaluated for an infinite observation interval. The system matrices for this model are — -38.51 5.33 8.41 3.12 22.45 -89.70 9.88 4.92 b .55 .97 12.11 .80 .36 34.25 31.12 8.68 -145.90 23.92 24.40 27.14 30.98 §--§'13- 8.79 3.29 18.02 -80.87 1 .14 17.72 16.88 12.86 5.89 18.62 16.15 -137.50 45.57 37.97 14.03 6.27 28.72 17.73 45.37 -140.80 36.47 L14.23 22.99 23.68 16.91 38.02 36.21 -172.00_j & a1 \0 -1 . 5 I dxaqtl.2375,5.4228,4.7612,3.6319,3.6340,3,6340,3.6340) and the disturbance used is the zero mean IID step change in shaft accelerations (5.20). From analysis of the Matrix g, one would conclude (3,5,6,7) would be mutually coherent and generator 2 would be coherent with generator 7 but not with other generators in that group (3,5,6) the transient interval. This is confirmed by analyzing the Taylor series approximations of the mean square coherency measure for various values of T. The analysis of the previous section would indicate that the equivalent line connecting generator 2 and 7 77 would have to be extremely stiff for generators 2 and 7 to be coherent in the dynamic coherency measure because generator 2 is not very stiffly connected to generators 3,5, and 6. This analysis would also suggest that if 2 and 7 were coherent in the dynamic coherency measure, generator 2 would be coherent with 3,5 and 6. The results from Table 2 shows that generator 2 is not coherent with 3,5 and 6 in the dynamic coherency measure. This would suggest that generator 2 and 7 are not extremely stiffly connected which is confirmed by observing the (2,7) element of the system matrix g. CONCLUSIONS This chapter discusses (l) a definition of the transient coherency measure which is a Taylor series approximation of the mean square where the order of the approximation increases with the observation interval in order to keep the approximation error within some bound (2) the use of the transient coherency measure for a deterministic disturbance at a single bus to (a) disect the transient response into discrete events which are the acceleration of successive stages further from the disturbance location; (b) determine the lines and generator in each 78 stage; (c) determine the stiffness of the interconnection xN ki between the dis- turbed generator and any generator k in stage 2 to N+l in the 2Nth interval; and (d) the effect of any line synchronizing torque coefficient or generator inertia on the transient response in any location and any interval. (3) the use of the transient coherency measure for analysis of changes in dynamic structure weak- ness in that structure and coherent groups as the observation interval increases for the transient coherency measure. Comparison of these structural properties for the transient and coherency measures is also made (4) the use of the transient and dynamic coherency measures as transient and dynamic security measures for security assessment in both system planning and system operation application. The use of the transient security measure as a per- formance index and as transient security con- straints for the optimal power dispatch- optimal load shedding problem. A very important application of the transient co- herency measure, which has not been mentioned but is extremely important, is the development of coherency based equivalents for transient stability application. 79 The differences between the transient and dynamic coher- ency measures determined in this chapter could be used to better understand the differences and the appropriate applications for modal [4] and coherent.[3T equivalents derived based on the dynamic coherency measure and the transient coherent equivalents derived based on the co- herency based aggregation procedure and either the max-min [5] or transient coherency measures. CHAPTER 6 AN OPTIMAL COHERENCY BASED SECURE DISPATCH FORMULATION OBJECTIVES: The principal objective of this part of the re- search is to show how an improvement in the formulation of the optimal secure dispatch problem can be achieved by including the transient coherency measure into the formulation of the optimal dispatch problem. METHODOLOGY: The methodolOgy of this chapter shall consist of meeting the above objective by (i) (ii) (iii) identifying the deficiencies in the current formulation of the optimal secure dispatch prob— lem through documentation of the literature. selecting a formulation approach (nonlinear vs. linear programming) based on both the trends in the current literature and the requirements for real time implementation. The limitations of the chosen approach are also mentioned. proposing a formulation of the optimal secure dis- patch problem which includes a coherency based performance index that should help improve the security, reliability and stability of the power system. 80 81 This develOpment is now pursued in chronological order in the following subsections of this chapter. 6.1 THE NEED FOR AN IMPROVED DISPATCH: The literature [12, 13] substantiates the fact that there exists the need for an optimization criterion for optimal dispatch which incorporates comprehensively the properties of security, reliability and stability. This need for a better index of performance is supported by Hajdu and Podmore [13] who indicate that for enhancing the security of a power system, it is desirable to express the results of on-line transient-stability studies in a more concise form such as a transient-—security index, for reasons which are directly quoted from [13]: . The evaluation of transient stability from visual inspection of swing curves is a time- consuming task and requires a certain de- gree of judgement, particularly for systems with many generators. . In case of transient insecurity, the swing curves do not give the operator an apprecia- tion of the nature or the severity of a potential stability problem. This is un- like the case of steady-state insecurity where the magnitudes of the potential over- loads caused by a certain contingency give the operator an immediate appreciation of the severity of the problem and its location. Byerly and Sherman [28] have proposed a stability index 8 = max max 8.. (t), (6.1) k t i,j ijk with 0.. (t) g angular displacement between generator 13k i and j at time t due to fault k 82 5k = max value depending upon system which is a maximum angular swing between a pair of gen- erators and is similar to the max-min coherency measure [ 5]. However, this index is beset by the difficulty that the critical value which separates the stable and unstable cases is dependent upon the fault and therefore upon the operating conditions as a result of that fault. This dependence on the fault is undesirable because a stability index should be able to handle not only a specific deter- ministic disturbance but also a probabilistic description of any disturbance. In addition, this max criterion does not yield any significant meaning in the case of instability. El-Abiad, et a1 [32] and Saito, et a1 [33] have also proposed transient security indices using the method of pattern recognition. The pattern recognition methods, however, are heuristically based and further advances in the accuracy and efficiency of Liapunov methods are neces- sary for making these tools useful for on-line studies. The deficiencies in the current formulation are further accentuated in [13] by the lack of a definition of proper constraints for transient system security. This "much more difficult, and to date unresolved problem" is according to current practice handled by formulating transient system security constraints on line phase-angle differences or power flow on lines. In addition, Dyliacco, et al [14] point out that using a phase angle constraint is insufficient to prevent 83 thermal overloading and that determining the bound on the angular difference lei - ejl which ensures a stable transient in case of a subsequent fault or combination thereof remains "a difficult and largely unsolved problem." Consequently the overall approach will be one in which an effort is made to address some of the aforemen- tioned difficulties by justifying the transient coherency measure (5229 as a transient security measure and modifying the current formulation of the optimal secure dispatch prob- lem to include a transient coherency measure which is shown to enhance the transient security of the system. Accordingly, the transient coherency measure (5.25) over [0, Ti£(€)] will be shown to be a useful security index since it improves system security by (1) dispatching to decrease angular differences 5ij and thereby move the system further below the static stability limit (2) increase the system stiffness, and (3) move the system away from thermal overload by limiting the power flow, P... 13 Security enhancement of the system is thus shown to be achieved through preventive and corrective controls reSpectively which are described in Chapter 8. 6.2 SELECTING A GENERIC FORMULATION: In formulating the optimal dispatch problem for power system security control applications, the choice of method lies essentially between two generic approaches [15]: (a) Nonlinear Programming and (b) Linear Programming (and its extensions to Quadratic and Convex programming). 84 The techniques used in category (a) comprise essentially of nonlinear system models handled by nonlinear solution tech- niques such as the elegant nonlinear programming formula- tions which shall be briefly surveyed later. This approach is, however, beset by three major impediments to on-line implementation which concern, as quoted from [13]: (l) the computational and memory requirements of the ac-power flow program (2) the relatively slow convergence properties of the various nonlinear optimization tech— niques, and (3) the large real—time data base requirements at a central location. The techniques used in category (b) comprise essentially of linearized system models handled by linear solution techniques such as the various linear programming formulations which shall also be surveyed later. The com- mon feature of these LP formulations is that they enjoy [16] (i) complete computational reliability (ii) very high speed (iii) ability to track deviations accurately making it suitable for both real-time or study mode pur- poses. Since the "fundamental requirement", as quoted from [17], for on-line application of an optimal dispatch strategy, is for the technique to be "computationally very fast", the choice of an LP formulation for on-line imple- mentation becomes a very practical proposition. In selecting linear prOgramming as the generic approach for formulating the optimal dispatch problem, it 85 is important to point out that this choice, reflecting the more favorable convergence characteristics and guaranteed solutions to feasible configurations, is subject to important modeling simplifications. These simplifications are essentially the lineariza- tion of both the performance index and constraints. This linearization places the following limitations [22] (i) since a linearized model is used, the accuracy of the model is sacrificed (ii) no attempt should be made to model large disturbances such as losing a large block of generation (iii) only small dimensional load-shedding prob- lems should be attempted. In addition to these model limitations, one recommendation regarding reactive power optimization is that only real power optimization should be attempted as the extra computational effort involved in including reactive optimization buys little additional information [23]. 6.3. THE FORMULATION Having selected the generic approach for reformula- tion, an improved LP formulation of the optimal secure dispatch problem is proposed. This formulation is referred to as a quadratic programming (QP) formulation when an addition to the LP objective function of a quadratic transient security index is demanded by the security assessment functions. 86 The proposed formulation reflects the work of many eminent researchers beginning with the early work in this area by SRI in the application of Operations Research tech- niques to planning and reliability studies of the EPA grid. The many contributions and refinements including those of Stott which should permit effective real-time application of the LP approach, are given in a most generalized version in a recent publication by him [16]. In this formulation, which is based on the inclusion of the transient coherency measure in the performance index, only the more important contributions in Stott, et al [161,[17] which can affect an improvement of system security, stability and reliability, are included. In addition, the formulation being developed applies only to thermal and nuclear generation and excludes hydrostations for obvious reasons. The development of the formulation is divided into three parts which consist of: (l) The general framework (2) The linearized performance index (3) The linearized constraints 87 6.3.1 The General Framework In order to proceed with the development of the aforementioned parts, it is necessary to provide a gon- ceptual framework around which the formulation of the optimum secure dispatch is completed. To this end, a brief statement identifying the various technical and mathematical components of the optimum dispatch problem is provided. 6.3.1.1 The Optimum Dispatch Problem Minimize the cost of real power generation NG Min {.2 Fi(PGi) = PGi'QGi'ViI‘Si 1"]. NG 2 .1 ai + biPGi + ciPGi} (6.2) i=1 subject to equality constraints due to power system energy balance requirements that the real and reactive power de- manded be supplied by the generations N PG. - PL. = pN = v. 2 Y..V. cos(0. - 6. - 6..) (6.3) i i i 1 j=l ij j i j 13 N N QGi ' QLi = Qi = Vi jgl Yijvj Sln(6i - éj - aij) (6.4) and the inequality constraints which arise due to machine rating and service quality requirements PG? : PGi 1 PG? real generations (6.5) 111 GM . . QGi : QGi i Q i reactive generations (6.6) 88 PL? 3 PL : PL? real loads (5.7) m M . QLi : QLi : QLi reactive loads (6.8) V? : Vi : V? voltage magnitudes (6.9) tm. < t.- < tM. tap changers (6.10) ij — 13 - ij ¢m < ¢.. < ¢M. phase shifters (6-11) ij — ij — 1] s9. < s.. < 8”. thermal (short lines) (6.12) 1];— ij —' ij lai - 6j| : wfj stability (long lines) (6.13) where m,M are the lower and upper limits on variables PG, QG are the real and reactive generations PL, QL are the real and reactive loads V, 5 are the voltage magnitude and its angle Y, a are the admittance magnitude and its angle t, ¢ are the values for tap changing and phase shifting transformers S, W are the line flows and phase angle differences NG, N are the number of generating stations and number of nodes Superscript N denotes net power injection. 6.3.1.2 Modus Operandi of Optimum Dispatch Having stated the optimum dispatch problem (6.2 - 6.13) in its most general form, it is desired to complete the diagnosis of this problem before the details of the proposed in parts on-line tracking secure formulation are developed 2 and 3. 89 This objective is realized along the lines of the following diagnostic steps: (a) Statement of the optimum dispatch problem in com- pact mathematical programming (MP) notation (b) introduction to the tools (constraints) for classify- ing the security level of operation of the power system (c) discussion of current dispatching practices in the utility industry in the context of system security (d) discussion of Operator limitations when system security is completely under the control of the operator (e) identification of the features of an on-line track- ing implementation of this optimal secure dispatch problem (f) off-line to on-line conversion characteristics and an overview of parts 2-3. 6.3.1.2(a) MP Statement of the Optimum Dispatch Problem This formulation framework is now stated in mathe- matical programming format using compact state space nota- tion. min F(x,u) (6.14) u ll 0 s.t. g(x,u,p) (6.15) (6.16) /\ O h(x,u,p) 90 where x is a vector of dependent controlled variables u is a vector of independent control variables p is a vector of perturbations or disturbances. The optimum dispatching problem stated in this form re- presents a constrained optimization problem having a non- linear objective function subject to nonlinear equality and inequality constraints. This compact mathematical programming format will be adhered to hereafter as it greatly facilitates in the formulation of an on-line secure dispatch algorithm. It will also be used ex- tensively in Chapter 8 for formulating the sub- control problems in terms of power system operating states. 6.3.1.2(b) Tools for Security Level Classification It is appropriate at this juncrure, therefore, to precisely define the constraints (6.15) and (6.16) which have hitherto been dealt with only in general terms. In addition, it is necessary to introduce the concept of security constraints, which form the backbone of the off- line security assessment functions, as well as the on- line tracking problem under formulation. With this motiva- tion, the constraints (6.15, 6.16) are now defined and described in terms of the role they play in maintaining the integrity of the power system. The load constraints The load constraints are represented mathematically by the equality constraint (6.15) and are the power system 91 network flow equations for a particular network con- figuration. In terms of an energy balance, these repre- sent the conservation of energy in the interconnected power system and impose the requirement that load demands be met by supplied generations. The load constraints are, perhaps, the most important constaints as it is a dis- crepancy between supply and demand which generally leads to power imbalances and possible cascading outages. The operating constraints The operating constraints are represented mathe- matically by the inequality constraint (6.16) and are the machine rating and service quality requirements demanded by the power system for reliability. In physical terms, these constraints impose upper or lower limits on the range of operation on variables associated with the component parts of the system. Accordingly, these con- straints are mathematically expressed by inequalities on such quantities as equipment loading, bus voltage, gen- erator real and reactive powers and phase angle differences. The operating constraints are further classified into two categories which are . soft constraints . hard constraints The soft constraints are associated with those variables whose inexcessive violations may be generally allowed on a short time scale. Therefore, these include bus voltage limits (6.9) and thermal line loading limits (6.12). 92 The hard constraints are associated with those variables whose violations are undesirable on any time scale as they could cause equipment damage or lead to eventual system break-up. These, therefore, include tap changer limits (6.10) which cannot be exceeded due to mechanical limitations; power transmission limits (6.13), which could lead to loss of steady state stability or transient stability; and power generation limits (6.5). The security constraints The security constraints are the additional con- straints on the present operating condition which are needed to include security into the formulation of the on-line secure dispatch problem. These are, therefore, none other than the additional load and operating con- straints considered vulnerable based on engineering ex- perience with the particular power system or off line security assessment programs. These constraints are imposed in order to ensure that there will be no viola- tion of the current operating conditions if a contingency were to occur. In the security context, the soft security con- straints (6.12) represent steady state Operating limits on power transfer across short transmission lines to account for thermal overloading and voltage constraints that prevent over or under voltages on system components. Since a system is steady state stable only if it is transient stable, the hard security constraints represent transient stability limits on line phase angle differences to ensure there is no loss of synchronism or oscillations 93 increasing in amplitude, leading to cascading and eventual system splitting. The current industry practice in this regard is to incorporate these security constraints by imposing empirical steady state limits on line phase angle differences across selected transmission lines. This approach, although partially adequate for off-line security studies of small systems does not include a broad class of possible contingencies and is insufficient to meet the growing level of security desired of modern inter- connected systems. The lack of appropriate transient se- curity constraints that ensure system stability, security and reliability has been documented previously. Security constraints are generaged by outage simulation tests or postualted next contingencey tests, forming an integral part of the security assessment functions,desired for providing information to the operator as to the security level of the power system. In this context, therefore, the security constraints are the set of additional constraints declared vulnerable by the security assessment functions. 6.3.1.2(c) Pgesent Security Approach: The current utility approach to power system security is characterized by two essential features which are off-line economic dispatch . operator controlled security since the off-line secure dispatch is used in only a few systems. The objective of the off-line economic dispatch 94 is to minimize fuel cost as follows min F(x,u) (6.17) u s.t. g(x,u,p) = 0. The solution to this problem is easily obtained by using the classical Lagrange multiplier technique for convert- ing the constrained optimization problem (6.17) to an un- constrained one, and solving the necessary conditions thereof to obtain what is popularly referred to as the optimum incremental "lambda" dispatch. 6.3.1.2(d) Limitations of Off-Line Secure Dispatch: It is important to observe that the off-line economic dispatch formulation (6.17) does not include the inequalities (6.16). Security of operation is, therefore,left up to the system operator, who modifies the economic dispatch depending upon the violations of the inequality constraints indicated by the security monitor. In this off-line mode, security is clearly not the chief objective and security of the system is totally the responsibility of the operator whose control actions are attributed to pperator limitations which are char- acterized by . insufficient information as to the operating state of his system 95 . lack of comprehension of the changing control requirements and controls which are suitable to the changes in the operating state of the system as it moves continuously in response to the changing security level of the power system. 6.3.1.2(e) Identification of Desired On-Line Tracking Features for Security: These limitations of the operator point to the need for an on-line secure dispatch approach having the following features in that it should possess . complete system information from a static state estimator . ability to change the control objectives from economic to security based objectives in response to the changing level of security . ability to change the controls from generation dispatch to load shedding, etc. as the security level of the system diminishes The proposed on-line secure dispatch is aimed at addressing essentially the need for a better performance index while the need for improved constraints are dealt with indirectly as a result of the revised performance index. 96 6.3.1.2(f) Off-Line to On-Line Conversion Characteristics The off-line secure dispatch as previously formulated is an improvement over economic dispatch operator controlled security because security constraints can be handled automatically with minimum change in fuel cost. Operating constraints can not be handled in the off-line secure dispatch problem since the time required to solve the dispatch problem is large compared with the time frame required to perform corrective action. An on- line secure dispatch problem is not only required to make possible corrective action for operating constraint viola- tions but should also be characterized by: . converting off-line problem to an on-line tracking problem . decomposition of the real and reactive prob- lems . addition of a coherency based performance index . monitoring deviations from base case economic dispatch . eliminating need to re-run complete load flows at each iteration of on-line security dispatch 97 An overview of the need for some of these on-line tracking characteristics is now provided. In View of the recent power outage in New York, it is clear that current operating practices are in- sufficient to prevent future outages and that the security of power system operations will have an additional and more profound impact on the total economic dispatch prob- lem as a whole [15]. However, the inclusion of rigorous modelling techniques cannot be justified based solely on economic incentives as the results of recent studies [15] point out that no dollar savings are involved when cost minimization is the chief objective. Consequently the use of more advanced techniques is justified based on the need for improved system security for which more rigorous models are required in order to execute the different func- tions associated with operating security. As a result, the approach taken here is one of designingcn0formulating a strategy in which the security of operations receives a high priority when the system is declared vulnerable by existing violations or by the security assessment func- tions and one which yields the economic dispatch solution as a by product of the optimum secure dispatch When no load security or operating constraints would be violated. The rest of this chapter will deal primarily with a reformulation of the above off-line problem into an on- line tracking problem. In order to achieve this goal it becomes necessary to modify the performance index and to 98 incorporate the operating and security constraints into the optimum secure dispatch formulation. A brief over- view of the performance index for the on-line secure dis- patch problem is now provided before getting into its details. The performance index of the economic dispatch problem is a quadratic cost function and is assumed dependent only upon the real power generation. This assump- tion does not result in any loss of generality because the reactive generations do not have any measurable or significant effect on fuel cost as a result of the de- coupling between real and reaetive powers. Fixed costs are excluded from this economic objective. The sole objective of this criterion is to minimize the total fuel cost while simultaneously satisfying the demands. The augmentation of the off line economic objec- tive function for the on line secure dispatch problem is executed when the system is declared vulnerable by the security assessment functions. Depending upon the level of security the power system is operating at, preventive or corrective rescheduling is introduced in the form of penalty functions which minimize the deviation from the economic generation schedules. This rescheduling tells the system operator where to introduce generation shift and load curtailment controls. It is to be noted that load curtailment or load shedding is designed in this 99 formulation framework as a last resort action by intro- ducing large coefficients into the corresponding penalty functions. While these preventive and corrective controls are useful in optimal rescheduling for improved security, they still lack a comprehensive capability of enhancing the security of the system as has been documented earlier. Therefore, a coherency based performance index is intro- duced into the objective function. This addition, as will be theoretically shown in Chapter 7 I has a profound impact on enhancing the transient security of the power system by adaptively improving the security, reliability and stability of the system, a comprehensive feature here- tofore not found in previous work. The net effect of the introduction of the coherency based performance index following a postulated transient next contingency or an actual emergency condition, is to-increase the security margin by selecting the generation and load corrections that result in stiffening the ties within the system or between connecting arease thus raising the level of se- curity the power system is operating at. In the reformulation it is assumed that a base case is available for the off line problem. This off- line problem in then converted to an on-line tracking dis- patch with the aid of an on-line estimator which eliminates the need for solving load flow equations (6.3, 6.4) to solve the secure dispatch problem as long as (1) an 100 equation requiring total generation satisfies total load is included and (2) the inequality constraints are ex- pressed in terms of deviations in bus injections using the Jacobian matrix generated by the static state estimator. The power flow is decoupled and the reactive power flow and associated voltage and reactive power constraints are eliminated because the reactive power and voltage have little effect on results of the secure dispatch problem and the voltage and reactive power dispatch is handled automatically in present system implementation. Having converted the off-line problem to an on- line problem, it is possible to enhance the accuracy of the model by successive linearizations about the operating point wherein the triangular factorization of the Jacobian available from the base case is obtained using sparsity techniques [33]. Sensitivity could also be incorporated intotflmamodel, making it possible to update the state of the network continuously without the use of repeated load flows. Control Variables and Priorities Since real power control is used in this formula- tion, the set of controllable activities considered within the domain of real power control consists of those control actions which can be represented analytically as bus real power injection changes or equivalent real power generation changes. All such controls which are considered here are 101 listed in order of priority as: thermal units on control (2) thermal units off control (3) emergency start-up (4) load shedding 6.3.2 Development of the Linearized Performance Index: In this step of the formulation, the given quadratic objective function (6.2 of the off-line economic dispatch problem is augmented by additional performance indices which are needed in order to convert it to an on-line tracking problem and to improve the level of security of the power system. These additions comprise of a penalty function for generation shifts . a penalty function for load curtailment a transient security index. Designing a proper objective function is philo- s0phically the most important aspect of any optimization strategy. The objective being designed here will involve a careful tradeoff between two factors (a) Economy (b) Security A generalized objective incorporating this tradeoff is now formulated as follows: NG N k MIN J = .2 Fi(PGi) + .2 piIAPGiI + .2 oilAPLiI i=1 i=1 i=1 N N + E {amok}, (6.18) where C 'lZEfi-3O[1§E]5—l+1§fii+T (-—+-1---)]T6 —k£ 51 j=l Mk j=l M2 k2 Mk M2 7. 32k 324 V.V. Ti]. = X—:—j-l <:6s(6i - 63.) - line stiffness (6.18a) connecting internal generator buses i and j APGi = a vector of changes in controllable power generations or equivalent bus generation shifts APLi = a vector of changes in interruptible loads (Si - (Sj = a vector of differences in bus voltage angles pi'oi’aij = positive weighting cost coefficients Pending a discussion of the control objectives of each term in (6.18) in terms of the role it plays in enhancing the security of operation of the power system, the linearization of the performance index is conducted. Linearization of the performance index Clearly the performance index (6.18) is nonlinear owing to the quadratic nature of the economic objective, the discontinuous nature of the two penalty functions and the inherently nonlinear characteristic of the coherency term. Therefore, in order to use (6.18) in an LP algorithm, it is necessary to linearize it. Since the incremental changes in the power system are small owing to 103 the fact that changes in load occur in small steps, linearization is a viable assumption and the expression (6.18) is now linearized about the operating point. From (6.2) the economic dispatch objective in its original quadratic form is given by the cost of generation per hour as Q II II MZ N F.(PG.) = X a. + b.PG. + c.PG7 . (6.19) i l l l i: l l l l l Linearizing (6.19) and using the relationship 3J1 AJl = .é—P—é: APGi (6.20) the equation (6.20) can be written as AJl = "642 (bi + 2ciPGi)APGi (6.21) 1 i Evaluating the coefficients at the operating point and representing the resulting coefficient by Ki’ the equa- tion (6.21) becomes AJl = "642 KiAPGi (6.22) l i which is the first part of (6.18). From (6.18), the second part of the objective is given by J = Z piIAPGi . (6.23) 23.1 Penalty function (6.23) represents a discontinuous func- tion and must be linearized before it can be used in the 104 LP algorithm. Since APGi represents a free variable, it must first be converted into a nonnegative variable to comply with the nonnegativity requirements of a linear program. Accordingly, APGi should be converted into a difference of two nonnegative variables as follows APG. = 0987 - APGT (6.24) l l where APGI > 0 l— APGi > 0. Equation (6.24) is then rewritten as N + _ 32 = iglpilAPGi - APGiI (6.25) Applying the triangle inequality IA - 8| 3 |A| + [8| (6.26) to equation (6.25), the result is N .1. - 02 - i£16i(apei + apci) (6.27) It is to be noted that the mathematical formalism of (6.24-6.27) is equivalent to recognizing that (6.23) can be intuitively represented by the sum of two simple . . + - linear functions in two variables APGi and APGi, 105 N — +_ - J2 - i£l(piapci piAPGi) (6.27a) where APG. Z 0, APGi i 0. + - 1 Equation (6.27a) can directly be converted to meet the nonnegative requirement resulting in (6.27). Similarly, linearization of the third term in (6.18) results in g ll "64w 6.(APLT + 3917) (6.28) . l l l i l where APL: 3 0, APLT > 0 l— From (6.18), the last part of the objective which is the transient security index is given by N {owe k=l i=1 k“ k“ N J=1 4 (6.29) From (5.25), this weighted coherency term can be written in terms of the synchronizing torque coefficients, T.., 13 as N N 4 NT. NT. T k 2 J = Z X 6 {12-—- 30[ Z ——l + X __l 4 k=l 1:1 kg 51 j=le j=l ”2 36k 375 (6.30) 6 1 1 T +T (—+—))—} k1 Mk M2 7: Expanding (6.30) the result is N 1%] T4 I; Tk. 31 T2‘ J = {a 12—- - a 30[ -—1-+ -—1 4 k=l 2:1 kg 51 k2 j=1 Mk j=1 M2 jfk j#1 6 (6.31) 1 l T + T (—— + ——)]——} k2 Mk M2 71 Recognizing that constants do not affect the optimization process, the effective term needed from (6.31) is given by N § § Tk. § T1. 1 1 T6 J = - a 30[ ‘ -—1-+ ——l + T (__ + __)]__ 4 k=l 2:1 kz j=1 Mk j=1 M2 .k£ Mk M2 7! 3¢k jgg (6.32) where the synchronizing torque coefficients Tkl’ sz, T represent line stiffness as defined in (6.18a) which is kl V.V. = _i_l - Tij xij cos(5i dj) (6.33) Approximating the cosine in (6.33) with the first two terms of the power series, each synchronizing torque co- efficient above may be approximated Viv' (6i - sj)2 Tij = —§:% {1 - 2! } (6.34) which makes (6.32) a quadratic. The function J4 is the quadratic transient security index which when combined with the linearized J1, J2 and J3, gives the following complete form of the security oriented objective function 107 The function J4 is the quadratic transient security index which when combined with the linearized J l I J2 and J3. gives the following complete form of the security oriented objective function . N + _ N + min J = lclAPG.-APG. + - - + _ 1:1 1 1 l) .£101(APGi + APGi) APG ,APG l APL+,APL' k 6. - 6. + - 1 3 + Z o.(APL. + APL.) i=1 1 l l N N N T . N T _ kg 2' 1 1 6 Z X a 30[ i + X ——l + T (—— + —_ E_ k=l 2:1 k2 j=1 Mk j=1 “2 k2 Mk ”()17’ (6'35) J¢k j#2 Control Objectives: The first term in the objective function (6.18) reflects the desire to minimize fuel costs subject to equipment limitations and predetermined schedules. In this context this portion of the objective function re- presents the usual economic dispatch objective. Since each terms depends only upon one independent variable, it is often referred to as a separable economic objective function, the cost coefficients a. bi’ 1, Ci being deter- mined empirically [24]. 108 The second term in the objective function (6.18) is a penalty function for the generation shifts which reflects the desire to bring a vulnerable system into a secure operating condition in the least costly manner. The cost coefficient pi is chosen depending upon the power system operating state and the criteria for this choice shall be discussed in Chapter 8. Similarly the third term in (6.18) is a weighted penalty function for the load shifts and is used to pro- vide the least costly load correction based on a priority Oi, the choice of which will also be discussed in Chapter 8. Earlier in this chapter, a diagnosis of the optimum dispatch problem (6.1-6.13) revealed that the performance index was lacking in its ability to respond effectively to transient insecurities and that there existed a need for a comprehensive security, reliability and stability measure. This improvement in the objective function is realized by the addition to the performance index (6.18) of a measure which is based on stiffening the transmission network and thereby adaptively raising the security level of the power system. The specific controls for altering the stiffness of the power system are discussed in Chapter 8 within the context of the overall power system security problem. 109 The augmentation of the dispatch strategy is executed only when the security assessment functions in- dicate or insecure condition with respect to a possible next contingency or an actual violation of the transient security operating constraints. The objective (6.35) subject to linear constraints poses itself as a quadratic programming (QP) problem which can be solved by various QP algorithms, typically the Wolfe's method which converts the QP to a linear pro- gram using Kuhn Tucker conditions. The use of a QP approach for the economic dispatch problem has also been proposed by Nicholson and Sterling [31] but on-line implementation was impractical due to the memory require- ments of the computer prOgram. However, recent advances i11computational.techniques by Dayal, et a1 [31] have made the use of Quadratic Programming a viable proposition for on-line implementation although not as fast as the con- ventional LP packages. 6.3.3 The Linearized Constraints Having formulated the linearized performance index, the next step in the formulationis to write the linearized constraints which comprise of the linearized load flow constraints . linearized operating constraints linearized security constraints 110 In order to proceed with this formulation, it is necessary to recognize first that the constraints (6.3 - 6.13) which were formulated for the general optimum dis- patch problem, represent in effect a full set of constraints comprising of two generic categories which are: . Real Power"- Phase Angle Constraints . Reactive Power - Voltage Constraints Of this full set, the load flow contraints which are re- presented by (6.3, 6.4), amount to 2N in number for an N bus system. For any optimization algorithm, the task of selecting an optimum generation schedule and simul- taneously satisfying the 2N contraints translates into a rather formidable computational requirement, especially for larger networks. However, this difficulty is alleviated by making the observation that the full set of constraints are necessary only in an environment in which the dispatch objectives are to minimize fuel costs and to minimize losses while assuring that all equipment is or will, for a set of possible contingencies, operate within present voltage and thermal limits, generation, load and line capacity and stiffness constraints. As suggested by Carpentier [18], for small perturba- tions, the interaction between real power and bus voltage, and between reactive power and phase angle is weak, there- by allowing a decoupling of the real and reactive problems into two separate problems which deal independently with lll . Real Power Optimization . Reactive Power Optimization ReCOgnizing that the reactive component does not affect in any major way the fuel costs or security, the de— coupled reactive power flow can be omitted from the on- line tracking problem formulation. Accordingly, the modeling process is freed of the need for monitoring con- straints associated with the reactive power balance (6.4) and associated voltage profile constraints (6.8, 6.9, 6.10). The constraint set (6.3 - 6.13) is thereby reduced to (6.3, 6.5, 6.7, 6.12, 6.13). Note that the phase shifter constraint (6.11) has been excluded here since only generation shifting and load shedding is being con- sidered for controlling the security level of the power system. The voltage levels used for the on-line secure dispatch problem are thus obtained from the static state estimator and actions required to adjust voltage profile and real power flows are done automatically by existing voltage control equipment and methods. Having stated and discussed the conditions under which the full optimization problem (6.2-6.13), consisting of the real and reactive dispatch, can be reduced to an on-line secure dispatch, it is important to recognize that the solution to the decoupled real power flow optimiza- tion represents a trade-off between computational require- ments, accuracy, and the information content of the solu- tion when compared to the rigorous full ac-network solution. 112 With the aforementioned decoupling process in perspective, tfluaon-line real power dispatch may be stated as N + _ g + ‘min _J = Z k.(APG. - APG.) + p.(APG. + Ape?) 196*,Aps i=1 1 l 1 i=1 1 1 l APL+,APL- 6.-6. k + - 1 3 + Z o.(APL. + APL.) ._ 1 1 1 1—1 N N N T . N T . 6 _ Z Z ak£30[ 2 _£1 + ; Mil + Tk£(%— + §—)]§T (6.36) k=l i=1 3:1 Mk j=1 2 ‘k 1 ° 3%k j#1 Where 2 Viv' (<5l - 6 ) Tij - x.. {l ‘ 21 } 1] s.t. N N - P . = . = . .. . . - . - PGl Ll Pl vl j£1 Ylej cos(6l 63 aij) (6.37) m M PGi i PGi i PGi (6.38) m M PLi _<_ PLi : PLi (6.39) m M Sij : ij i Sij (6.40) M '51 ‘ 6j' i wij (6.41) The next requirement points to the need for eliminating the need for a recomputation of the load-flow at each iteration of the on-line tracking dispatch when- ever a preventive or corrective control action is desired 113 in response to the need for improving the security level of the power system. The objective is then achieved by . converting the real power load flow constraint (6.3) into a linearized real power balance con- straint for an area . relating the real power injection changes in the network to the changes in the line phase angle differences representing real power flow along transmission lines by obtaining a linearized network model . formulating the linearized operating and security constraints for the oneline problem using this linearized network model. 114 6.3.3.1 The Linearized Load Flow Constraints The decoupled on-line secure dispatch (6.36-6.41) resulted in a reduction of the power flow equations from 2N to the N constraints on real power flow represented by (6.37). Since the on-line algorithm obtains its load flow information from a static state estimator, it will now be shown that it is not necessary to resolve (6.37) each time a preventive or corrective control action to re- schedule the power system is desired for improved security. However, any corrective action in terms of gen- eration shifts, APGi, or load corrections, APLi, must be such that it maintains a balance between supply and demand in the interconnected system. In this regard it is imperative to restate that it is precisely the discrepancy between supply and demand which leads to a drop in system frequency and consequential deterioration of the security level of the power system. Obviously, even though the state estimator is providing the necessary state informa- tion, neglecting the power balance constraint would dis- patch in a manner to compound the security problems of the network. Accordingly, a preservation of that balance for the on-line tracking problem must necessarily be reflected in an incremental relationship between the corrective actions, APGi and APLi. With the supply and demand power balance relation- ship in focus, the constraints (6.37) can be shown to 115 satisfy this balance relationship by expressing (6.37) as N PG. = PL. + v. Z Y..V. cos(6. - 6. - 6..) (6.42) l l 1 j=1 ij j l j ij Summing the generation and demand for the entire network, (6.42) can be expressed as N k E PG. = _E pLi + PLosses (6.43) Clearly this relationship states that the sum of the real generation equals the sum of the real demand plus the real losses. Neglecting the losses as they amount only to a few percent of the total demand, (6.43) becomes PL. = 0 (6.44) 1 IIMZ "U C) I "MW Constraint (6.44) represents only one constraint compared to the N constraints (6.37). In addition, in this formulation in which the control actions are composed of generation shifts, APG, and load corrections, APL, this energy balance requirement translates to a conservation of power in the interconnected network by representing (6.44) in incremental form which is the desired power balance equation for the on-line dispatch problem N k f APG. - Z APL. = 0 (6.45) . l . .1. i=1 i=1 116 In order to use the constraint (6.45) in an LP algorithm, the free variables are converted into nonnegative vari- ables and the resulting constraint is N + N _ k + k _ ( X APG. - Z APG.) - ( ) APL. - X APL.) = o (6.46) O l I l I - l I 1 i=1 i=1 1-1 1:1 Rearranging (6.46) the resulting linearized real power balance constraint corresponding to (6.37) of the decoupled model is N + k + N _ k _ Z APG. - Z APL. - X APG. + X APL. = o (6.47) . l . l . l . 1 i=1 i=1 i=1 i=1 where PGT, PGT, PLT, PLT > o J. l l l — 6.3.3.2 The Linearized Operating Constraints The objective of this section is to write in- cremental constraints corresponding to . real generation (6.38) . real load (6.39) . line loading (6.40, 6.41) The incremental constraints on generation and load are simple constraints and do not require any information be- sides their maximum and minimum limits on system operation. These constraints are of the general form: APGMIN : APG < APGMAX APLMIN 3 APL < APL (6.48a) MAX (6.48b) 117 The incremental constraints on line loading, however, are not as simple and require establishing a relationship be- tween the line phase angle differencesand.injection changes. The general form of these constraints is MIN MAX M 1.9— :J (6'49) with .1 =AA_I (6.50a) where AW = a vector of r line phase angle differences A = r x n network matrix AI = a vector of n bus injections In order to find the matrix A, a linearized network model is developed. This is done by (a) obtaining the voltage angle differences in terms of the bus injection changes using a decoupled dc-model which gives AI (6.50b) where _H = Jacobian matrix (b) relating the voltage angles to the line phase angle differences by a bus incidence matrix giving 91 =_§T_§_ (6.51) where _§ = n x r bus incidence matrix The remainder of this section is concerned with (l) formulating d.c. load flow (6.50b) 118 (2) form linearized network model (6.50a) (3) derive the limits APGm, APGM, Ame, APLM, 44m, ATM for the incremental constraints (6.48a, 6.48b, 6.49). 6.3.3.2(a) This component of the modelling process is equivalent to obtaining a decoupled model linearized about the operating point (V°,6°) which would conform with two requirements the LP requirement of obtaining a linear net- work model . the on-line tracking dispatch requirement of obtaining a deviational, incremental type of a model. Writing the ac power flow equations (6.3, 6.4) in func- tional form Pg = f (v,e) 1 1 (6.52) 0114: f2(v,e) where fl,f2 = nonlinear functions of the network para- meters Relating small changes in nodal power injections to small changes in complex node voltages [17] and obtaining the operating point from either . a static state estimator for real time studies, or an ac-load flow for study mode purposes, 119 the linearized system of equations in vector form are: r- “ r- s r- H O 4.2” $3 3% .49. = (6.53) N 312 12 A—Q 36 av A—Y L .J B ._J L. .4 Recognizing the Jacobian matrix, (6.53) may be written as r- -\ r- fir- a 41°.“ 41 = o AQN g (6.54) —— AV \— d \— _JQ—J where g° = Jacobian matrix evaluated at operating point (V°,6°) Recalling from section (6.3.3) that a decoupled model can be obtained for small perturbations, the decoupled model is: (6.55) where ICE IZ ICC [O 13.1: 9.2 Since the decision to eliminate the reactive model has al- ready been made earlier in this chapter, the real part of (6.55) may be written in terms of the notation of (6.50) as 120 A1 = g 6 (6.56) The system (6.56) is now rewritten without the slack bus as A A_ (6.57) (> H II |m> where AI = asparse (n-1) vector of all bus injection changes except slack whose nonzero elements are the controlled bus generation shifts or load shed (or phase shifts if desired) g a (n-l) x (n-l) Jacobian submatrix of H Solving (6.57) for Ag, the resulting equations are fi'l A1 (6.58) _§_=_. _ This result gives the changes in voltage angles, Ag, in reSponse to a shift in bus injections, AI. 6.3.3.2(b) To relate the bus injections to the line phase angle differences as in (6.51), it is important to recognize that the assumption of a lossless network results in the branch power flows being directly proportional to the branch phase angle differences. This requirement is ex- pressed mathematically by A AW = 6 T _E, (6.59) where £3 = a vector of r line phase angle differences a vector of (n-l) bus voltage angle changes D [00) a» (n-l) X r bus incidence matrix 121 Here the elements corresponding to the slack bus have been omitted by eliminating the nth row resulting in (n-l) linear independent equations. I Substituting (6.58) into (6.59), the model be- come 8 A)? = ET fi’1 43‘: (6.60) [r X l] = [r X (n-l)][(n-l) x (n-l)][(n-l) x l] Expressing (6.60) in compact form A A? = A: (6.61) |>> where A_= §?§-l, an r X (n-l) matrix Comparing (6.61) with (6.51), the desired network matrix A is given by 'o 50 4= -- 434:1: s ...... :0 :0 Representing this matrix in the general form of the de- sired model (6.51), the resulting model is 0 AI (6.63) m o 0.0 122 It is important to point out that the triangular factorization of A, in the definition of A is obtained by sparsity techniques developed by Tinney et a1. [33]. However, since the optimal power flow is being operated on-line wherein state estimation is being used to provide the data base, the Jacobian is automatically available and a single factorization of g is sufficient for the con- tingency tests to be discussed later. In addition, the network matrix A is dependent only upon existing net- work conditions which makes it possible to utilize (6.63) for writing the linearized operating and security con— straints. The Operating Constraints: The operating constraints, which are the constraints on physical limitations on equipment, before the occurrence of an actual contingency consist of two parts: . constraints on correction limits . constraints on line loading Correction limit constraints All such constraints are expressible in general form by the vector inequalities APGMIN : APG < APG APLMIN 3 APL < APLMAX (6.64) Let the system operating point for the real power generation be given by PG° and let APG be the 123 corresponding generation shift or correction needed to improve the security level of the power system. Repre- senting this correction in terms of upper and lower physical limitations on generation, the appropriate con— straint is MIN MAX PG 3 PG° + APG 3 PG (6.65) Subtracting PG° from (6.65) PGMIN - PG0 3 APG 3 PGMAX - PG° (6.66) Expressing (6.66) in terms of deviations APGMIN 3 APG 3 APGMAX (6.67) Substituting into (6.67) the relationship APG = APG+ - APG‘ the resulting constraint is APGMIN 3 APG+ - APG- 3 APGMAX (6.68) Expressing (6.68) as two separate constraints which agree with physical intuition 0 3 APG+ 3 APGMAX (6.69) APGMIN < APG- < 0 Substituting into (6.69) the relationships 124 MAX MAX APG PG - PG° MIN _ PC_MIN _ PG° APG the resulting constraint set is + MAX 0 Observing from (6.75) and (6.76) that M,MIN = kyMIN _ 4° (6.81) M,MAX = KyMAX _ 3° constraint (6.80) can be expressed as {MIN - 3° 3 5(33_ - £33) 3 WMAX - 4° (6.81a) Substituting APG = APG+ — APG" and 3° = 5 £6 WMIN - AI° 3 AAPG+ - AAPG' - AAPL+ + AAPL- 3 (MAX - AI° (6.82) Expressing (6.82) as simple inequalities 127 MAX + - + - AAPG - AAPG - AAPL + AAPL W - AI° | A (6.83) -AAPG+ + AAPG- + AAPL+ - AAPL- < -PMIN + AI° Since there are r branches in the network there will be 2r constraints of the type (6.83). 6.3.3.3 The Security Constraints The incremental security constraints are composed of . line outage constraints . generator outage constraints The security constraints are, therefore, the incremental line loading operating constraints which have been declared vulnerable by the security assessment function based typically on a violation of 90% of the bound. Specifically these would include the con- straints given by (6.83) and would be activated based on information provided by the assessment functions. Line outage constraints Let the system be subjected to "L" line outages. The matrix A is obviously affected by such outages by a change in the network configuration and the constraint for each such outage is given generally by , AWMIN < A.AI < AWRAX j = l,...,L (6.84) 128 Using a development similar to (6.83) the "L" line outage constraints are _ + - ‘PMIN - AjI° 3 AjAPG+ - AjAPG - AjAPL + AjA PL < PMAX - A.I° (6.85) — J Expressing (6.85) as simple inequalities + - + - AjAPG - AjAPG - AjAPL + AjAPL < WMAX - AjI° (6.85a) _WMIN + AjI° IA + - + - There would be up to 2L constraints (6.85a) for "L" line outages. Generator outage constraints Let the system be subjected to NG generator outages. Obviously the A matrix is unaffected but the injections undergo changes and the appropriate constraint is of the form MAX A? In this case it is important to assume that the frequency and voltage dependence of the loads is neglected other- wise the injection relationships under discussion would be nonlinear. In a manner similar to (6.79) the corrected in- jection vector is defined in terms of the scheduled normal injection, 1°, the correction, APG, APL and the vector g] of loss in generation due to the jth generator outage I = 1° - G3 + APG - APL (6.87) The constraints are obtained using a procedure similar to (6.85) and they are ¥MIN - A(I° - G3) < AAPG+ - AAPG- - AAPL+ + AAPL- <8W“-Au°-<§) (68m Expressing (6.88) as simple inequalities MAX AAPG+ - AAPG- - AAPL+ + AAPL- < w - A(I° — G3) (6.88a) + _ _ . -AAPG + AAPG + AAPL+ - AAPL < -PMIN + A(I° - G3) Similarly (6.88a) would constitute 2NG constraints. Summarizing, the on-line secure dispatch formula- tion with a quadratic transient security index is given in mathematical programming format by: 130 N N +MIN _J = .) Kl (1PG3 - APG ) + E P.(6PGT + APGT) APG ,APG 1:1 1 i=1 1 l l APL+,APL-~ k (6.89) 6. - 5 + ‘ 1 + Z 0.(APL. + APL.) 3 i=1 1 1 1 N N N T . N T 6 v k 2 1 l T - A X a 30[ X -—l + E ——l + T (—— + ——)}—— k=l i=1 k2 3:1 Mk j=1 M2 k2 Mk M2 7! 3%k j#1 where 2 ViV. (6l - 6 ) T13 — —§7%-{1 - 21 } 1] s.T. N k + N _ k _ Z APG - Z APL. - Z APG. + X APL. = 0 (6.90) . . l l . l 1=l 1:1 1:1 1:1 APG+ 3 PGMAX - PG°\ -APG‘ _ PG° - PGMIN (6.91) APL+ 3 PLMAX - PL° F -APL' 3 PL° - PLMIN/ AAPG+ - AAPG' - AAPL+ + AAPL- 3 TMAX - AI° - _ (6.92) -AAPG+ + AAPG + AAPL+ - AAPL 3 -PMIN + AI° _ _ O A.APG+ - A.APG - A.APL+ + A.APL < PMAX - A.I 3 J J J — 3 (6.93) -A.APG+ + A.APG- + A.APL+ - A.APL' < -WMIN + A.I° 3 J J J - J AAPG+ - AAPG- - AAPL+ + AAPL- 3 WMAX - A(I°-G3) (6.94) _ + .. 1 -AAPG+ + AAPG + AAPL - AAPL 3 -8NIN + A(I°-G3) + - + APG , APG , APL , APL- < 0 131 Constraint (6.90) is the incremental load flow (n: power balance constraint and is a single constraint. Constraints (6.91) are the operating constraints associated withtflue limitations on the range of correc- tions and are (2N + 2k) in number. Constraints (6.92) are the operating constraints associated with line loading and are 2r in number. Constraints (6.93) are the security constraints .associated with line outages and amount to 2L in number. Constraints (6.94) are the security constraints associated with generator outages and amount to 2NG in number. The maximum number of possible constraints is therefore (1 + 2N + 2k + 2r + 2L + 2NG) and there exist several efficient computer packages for handling this large number efficiently. CHAPTER 7 PERFORMANCE INDEX JUSTIFICATION FOR SECURITY DISPATCH OBJECTIVES: The principal objective of this part of the re- search is to show that the transient coherency measure is an improved measure for power system security and thereby raises the security margin of the electric energy system. METHODOLOGY: The methodology of this chapter shall consist of meeting the above objectives by (1) developing of the equivalent-line concept and appropriate relationships for use in the analytical development (ii) analytically justifying the addition of the new performance index and showing it to be a measure of (a) security and reliability (b) stability 7.1 The Equivalent Line The concept of an "equivalent-line" connecting a specific generator to an infinite bus, which is appropriate 1 k2 for the analytical development, leading to the proposed 132 over the interval 0 3 T 3 T (e), is used in this chapter 133 justification of the performance index as a measure of security, reliability and stability. 3 The use of this concept is, therefore, explained and the appropriate expressions are develOped within the context of this development. il 12 *3 i3 [N INFINITE Figure 1 BBS From elementary electric energy systems theory, the synchronizing torque coefficient between buses i and j, representing the stiffness of the transmission line, is given as in (6.18a) by the expression Tij - xij cos(6i - 0j) . (7.1) However, if more than one transmission line connects the rest of the system to the disturbed generator 1, the equivalent synchronizing torque coefficient, connecting bus 1 to the rest of the system, is given by the sum of the synchronizing torques for each connection k, weighted 134 by the inertia of the disturbed bus i as N V.V %_ Z l k cos(6. - 6 ) (7.2) i k 1 1k i k 1 Xik l k "N42 _ 1 T. M Representing the rest of the system by an infinite bus (or equivalent bus), as in Figure 1, it is desired to represent buses k by an infinite bus as l N l Viv' MT E Tik = NT ‘27; cos(6i - 6j) (7.3) 1 k—l 1 13 where j is the infinite bus. Note that if the voltage magnitudes and angles in equation (7.2) were all equal, the equivalent synchronizing torque would reduce to the simple problem of obtaining a parallel equivalent as in classical network theory. How- ever, this is generally not the case. In developing the equivalent—line concept, the problem, therefore, reduces to one in which it is desired to find the appropriate expressions for Vj, xij’ and 6j in (7.3). The solution to this is obtained by equating (7.2) and (7.3) and noting N Vk V. ——— cos(6. - 6 ) = —l— cos(6. - 6.) . (7.4) Xik 1 k Xij 1 3 "54 k 1 Invoking the trigonometric identities 135 cos(6i - 6k) = cos 6k cos 0i - Sln 6k s1n 6i (7.5) cos(6. — 6.) = cos 6. cos 6. - sin 6. sin 6. l J J l J l and substituting into (7.4) the result is ”2‘ 1 ‘E ‘9 [———-cos 6 ]cos6. - [——— sin 6 ]sin6. k=l Xik k l k=l Xik k l (7.6) 11; ‘_’j_ = [x.. cos 6j]cos 6i - [x.. Sin cj]51n 6i 13 13 Comparing coefficients in (7.6) the result is Ii] Vk V. -——-cos 6 = —1— cos 6. (7.7) k=l Xik k xij 3 and 1%] Vk V. ——— sin 6 = —1- sin 6. (7.8) k=1 Xik k Xij 3 Taking ratios of (7.7) and (7.8) the result is N V k sin 6. 2 ET— COS 51k _ k-l 1k tan 6. — = (7.9) 3 cos 6. N Vk j 1 ——— sin 6. X. 1k k=l 1k V. and solving (7.7) or (7.8) for X1— the result is ij N V N V V 2 X£— cos 6k Z XK- sin 6k j = k=l ik = k=l ik (7 10) Xij cos 6j sin 6j ' Equations (7.9) and (7.10) give the desired relationships .defining the equivalent line which are written in final 136 form as (N E ZE— cos 6. 6 = arctang k=l Xik 1k j N v 1 k . §—— Sln 6ik (7.11) N Vk Z -——— cos 6 v. = k=l xik k Xij cos 6j 7.2a From the analysis of the transient coherency measure as a function of T, it is clear that generators which are not directly connected to the disturbed generator act as infinite busses in the interval right after the disturbance. Since the generators k act as infinite bus j for any generator 1 over this initial interval, the "equal area criterion" can be applied. This criterion which deals with the principle by which stability under transient conditions is determined does not require the need to use the digital computer for solving the swing equation. Although not applicable to the study of multi- machine systems, it does provide tremendous insight and understanding of the stability and dynamics involved in a system where one machine is swinging with respect to an infinite bus. Having justified the use of the equal-area criterion during the initial interval, the analysis then proceeds as follows using Figure 2. 137 Assume Gij and Pij are the present angle at bus i and the power over the equivalent line connecting generator 1 and j,tfim3power being given by the expression Pij = 3%:; sin(8i - ej) (7.12) l] where Vi = voltage at generator 1 Vj = voltage at generator j Xij = reactance of equivalent line ij. Furthermore, assume that the electrical fault very near bus i is cleared when the power on the line connecting ij is ng and angle on generator i is 8;. 63. = e! - e. K P.. l] l J P 13 en = en _ 6 ij i j m A 8: 2 m ". O KPpij ’7 B : K P'.. 5 P 13 3 8!. 8?. w - 0?. N 0.. 13 13 ANGLE 13 13 Figure 2 138 The equal area criterion would state that A2 must be greater than or equal to Al in Figure 2 in order not to lose synchronism. In order to meet the objective (iii)it is necessary to show that mean square coherency is a measure of the capacity of the system to withstand a disturbance. From elementary mechanics and Figure 2 ENERGY = f Ida; r = P/ZNN (7.13) where I is the net torque on the machine 6 is the angular displacement N is the speed constant P is the net power on the machine. Letting KP = ZWN the above energy equation is rewritten as ENERGY = KP f Pde . (7.14) This is the fundamental equation which will be used in the development of the relationships for A2 and Al in Figure 2. Since the analysis is being conducted in terms of energy it is instructive to introduce the concepts de- veloped by Podmore et a1 [25] and Magnusson [26] to assist in the justification of the transient coherency measure as a measure of security and reliability. 139 The concept of transient energy has unparalleled intuitive appeal in that it provides a very useful physical interpretation of the mathematical equations used in the study of power system stability. This transient energy consists of two components - a kinetic energy term dependent upon rotor speeds and a potential energy term dependent on rotor angles. Con- sistent with equation(7.l3), the integral of a torque with respect to the angle through which it acts is the work done, or energy transferred, in the process. Thus the area Al represents the potential energy inherent in the initial displacement of 6 from eij to egj at the beginning of the transient. This potential energy is transformed into kinetic energy when 0 equals egj, and is returned to the potential form as 9 increases toward n - egj. Note, however, that w - egj is simply a theoretical upper limit whereas the real upper limit would be at the point where the machine once again reaches synchronous speed. Angle 6 then will decrease back to eij, during which time the energy goes into the kinetic form again, and then back to potential. With these preliminary concepts in perspective it becomes quite simple to understand the physical meaning of the analysis in which it is desired to show that the ability of a system to withstand a disturbance is governed by the upper limit of its available energy capacity. 140 Expressions are now written for areas Al and A in the post-fault state. Following this a relation- 2 ship between A and A2 will be developed into an 1 equation for A2. This resulting equation for A2 in the post fault state will then be compared to the A2 equation for the pre-fault state and the results analyzed to lead to the objective (ii). The area A is given using Figure 2 l r‘;-ej 1 V3 ez-ej EiE. = KPPij(6ij - eij) - K [ ' -§T% Sln eijdeij (7.16) 8.-0. 1] l J EiE' 8£j = KPPij(0ij - eij) + KP _ET; cos eij ' (7.17) 13 8.. 1) EiE' ._ KPPij(0ij - eij) + KP —§;% {coseij - cos eij} (7.18) and area A2 is given by w-(BE-ej) A = . . . - I}. _ l.l_ . _ ...-I. . 2 KPJe"-8 P136813 KPPUH (al 8]) (81 83)} l 3 (7.19) 2KPEiE. = Xij cos eij - KPPij(n - Zeij) (7.20) In order to develop the relationship between A1 arms A it is desired to express A2 in terms of Al 2! as follows 141 1° E.E. = 2 _l'__l e 'f ' _. H _ n A2 KP Xij cos 1] KPPij(n zeij) (7.21) EiE. - 2 K P9. 9. - 3. + " - ' 1 { P 13(013 813) KP —§;% (cos eij cos eij)‘+ 2A Eliminating terms, A2 is rewritten as E.E. = _i_l ' .. " _ I . A2 ZKP Xij COS eij KPPij(W 29ij) + 2Al . (7.22) Since the pre-fault synchronizing torque coefficient is given by EiE' T3. = ———l cos 8'.. ij Xij 1] the expression (7.22) may be rewritten in terms of Tij as = I ... n _ ‘ I A2 2KPTij KPPij(N 29ij) + 2A1. (7.23) Consistent with the energy concepts thus far introduced, A represents the energy capacity of the line to with- 2 stand a fault, Tij is proportional to the energy capacity of the line before a loss of synchronism can occur, KPP;j(n - Zeij) represents the energy of the fault which does not get transferred, and A represents the fault 1 energy transferred onto the line as a result of the con- tingency. The above expression for A2 after a fault with energy Al has occurred can be written as 142 = I. _ I I _ '. _ I II _ I _ F. _ {RP(Pij Pij)(fi Zdij) 2A1} (7.24) where ' - ' - " represents the prefault energy the line can absorb due to a fault without loss of stability and KP(Pij - Pij)(w - zeij) - 2Al . (7.26) represents the reduction in the fault energy absorption capacity of the line due to the fault. From the pre-fault equation (7.25) it is clear that Pij(w - Zeij) is relatively small in practice com- pared to ZTij due to the fact that the pre-fault operat- ing angle Sij is chosen as small as possible in practice. This approximation translates to the following important conclusion that "the pre-fault energy capacity is essentially proportional to the pre-fault synchronizing torque coefficient, Tij' representing the stiffness of the equivalent line ij." Furthermore, since the post fault equation (7.24) reduces to the pre-fault (7.25) in the absence of a fault and since the reduction in the energy capacity due to any fault will be uncertain before any fault occurs, the energy capacity post-fault is propor- tional to the energy capacity pre-fault which has already been arguedtx>be proportional to pre-fault line stiffness. 143 In conclusion,therefore,it is stated that the pre-fault line stiffness Tij the line to withstand a fault. The significance of this represents the energy capacity of conclusion is demonstrated by the fact that any design criteria for line stiffness would be specified before the fact and not after i.e. prior to the occurrence of a fault. From the above analysis, it is clear that the transient coherency measure is an excellent measure of security because (1) the transient coherency measure decreases proportional to the increase in stiffness of the equivalent line connecting internal generator buses i and j, and this pre- fault stiffness coefficient was shown to be proportional to the energy capacity of that line to withstand a contingency without loss of synchronism in the initial interval 0 3 T 3 T:£(€) after a contingency. This shows that the transient coherency measure is proportional to the energy capacity to withstand a fault and is thus a measure of security and reliability. (2) the transient coherency measure decreases proportional to the increase in stiffness of the equivalent line connecting generators k and 2 and this stiffness increase is proportional to the decrease in the dif- 144 ference in phase angles at the two internal generator buses. Thus minimizing the co- herency measure will increase stiffness of the equivalent line and reduce the pre- fault angle difference further below the static stability limit. 7.2b In the second part of the analysis it will be shown how the transient coherency measure is a stability measure as well. By showing that (7.27) P~1 0 II N + 71 M >2 5 i j and thus decreasing Ci amounts to increasing the com- 3' plex part of the eigenvalues of the system indicating that reduction of this coherency measure increases the stiffness of the electrical network and thus the stability of the system. The forementioned result is now derived. First a relationship between the eigenvalues of -1 A and -M T is established. '1 (7.28) 3’ u Ixslcn ...-4.....- I [0: |H Ix n I 3 F-l The eigenvalues of A are obtained from the characteristic equation derived from equating the determinant of the matrix [1; - A] to zero, and solving for the values Xi that solve the equation, i.e. 145 det 1; - A = 0 (7.29) )1 -1 det — = det g = 0 (7.30) -2<_ 3.1. 2N where {Ai}i=l are the eigenvalues of A. Since the matrix M is written in the above N x N block matrix form, the 2N eigenvalues of A can be computed using IMI = I) I - é) = 0 (7.31) A Similarly the N eigenvalues )i of X are computed by solving the following equation INI = Ii; - 5| = 0 . (7.32) Observing that (i) i = ‘14-]? (ii) )2 )2 12 are the solutions of [M] = O 1' 2"°°' 2N (iii) XI’XZ""'XN are the solutions of [NI = 0 equation (7.31) can be rewritten as 1421-§I=0=|i1-2£l so that 1 = X (7.33) Thus the square of the eigenvalues of A equals the eigen- values of A or -M-lT. Secondly a relationship between the synchronizing torque coefficients and the system eigenvalues is established. 146 Invoking the well known result from matrix analysis relating the trace of a matrix W to its eigenvalues N Tr{w} = Z A. (7.34) — . l 1=l and applying it to (7.31) N A Tr{§} = X Ak (7.35) k=l and using the relationship (7.33), equation (7.35) can be rewritten as 2N 2 Tr{§} = 2 1k (7.36) k=l Since A = -A—l$, Tr{§} becomes -1 Tr{§} = -Tr{M T} N N Tk. = - 1 Z ._l. (7.37) k=1 j=1 Mk 375k Substituting (7.37) into (7.36) the following expression is obtained N N Tk' 2N 2 Z 2 M J = - X )k (7.38) k=l j=1 ’k k=l j#k From (5.25), the transient coherency measure for a dis- turbance which produces a zero mean, independent identically distributed (IID) step change in shaft accelerations at all generators is with A = A given by 147 4 N T . N T . 6 T k 2 1 1 T g = 12 —— - 30[ X __l + Z __l + T (—— + ——)]—— (7.39) kg 51 j=1 Mk j=1 M2 k2 Mk Mg 71 j7‘k #1 This transient coherency measure may now be expressed in terms of the eigenvalues of the system matrix A by sum- ming over all 2 and k and by making the observation that the sum of the diagonal elements of the AC1 2 matrix equal the negative of the sum of the off diagonal elements, so that (7.39) becomes N N 4 N N T 6 X 2 0kg = 12(N2-N)§%-- 30(2N+2){ Z Z —££ $7 (7.40) kel i=1 ' k=1 2=1Mk ° k#£ £#k Substituting (7.38) into (7.40) this becomes ,N N 4 6 2N _ 2 . : ~ 2 E Ckz — 12(N -N)§T + 30(2N + ‘)%T X A: (7.41) k=l 22—1 - k=l k#£ The result shows the transient coherency measure summed over all pairs of internal generator buses is there- fore proportional to the sum of the square of the eigen- values of the power system with zero damping. Since these eigenvalues are imaginary and the imaginary part of these eigenvalues measure system stiffness much like the eigen- values 41'A2 =:j\/§ for a harmonic oscillator Mx + KX = 0 (7.42) the sum of square of the eigenvalues and thus the coherency measure is a measure of the overall stiffness of the system. 148 Since the loss of synchronism and loss of stability at any point in the system due to any possible contingency is dependent on the stiffness of the overall system, the probability of a loss of stability in a system for any location and any shaft accelerating contingency is thus measured by summing the coherency measure over all possible pair of generator buses in the system. CHAPTER 8 THE SECURITY CONTROL PROBLEM OBJECTIVES: The principal objective of this chpater is to pro- vide an integration of the on-line tracking secure dis— patch formulation within the scope of the overall power system security problem. METHODOLOGY: The methodology of this chapter shall consist of meeting the above objectives by (l) (2) (3) (4) (5) identification of the power system security control problem in terms of the generic operating states of a power system representation of the overall control problem in terms of sub-control problem formulations associated with each of the operating states formulating an on-line tracking algorithmic structure for solving the optimal secure dispatch problem identification of the time-frame for controls involved in the integrated power system security control problem reformulating the sub-control problems in (2) using the augmented transient coherency measure and dis- cussing the choice of weighting coefficients. 149 150 8.1 IDENTIFICATION OF THE SECURITY PROBLEM: Power System Opgrating States: The security identification problem is presented by representing the power system operating conditions in terms of power system operating states. The literature [12, 13] makes reference explicitly to three operating states - normal, emergency and restorative, and includes a normal insecure or alert state implicitly as an insecure subset of the normal state. However, a better understand- ing of the security problem has led to the representation of the power system operating conditions in terms of five operating states [34] which are: . normal operating state . alert operating state . emergency operating state . extremesis operating state . restorative operating state Operating State Description Tools: The power system operation is governed by three sets of generic equations [12] which are: (i) differential equations (ii) algebraic equalities (iii) algebraic inequalities The set of differential equations (1) are those which describe the dynamics of the system components in terms of well known physical laws. 151 The algebraic equalities (ii) describe the energy balance and thus refer to the system's total load and gen- eration. The algebraic inequalities (iii) state the upper and lower limits on the system's components and thus refer to limitations on system variables such as voltages and currents. However, only the algebraic equations are useful in describing the five operating states and therefore these are translated respectively into . load constraints g(x,u,p) = 0 . operating constraints h(x,u,p) 3 0 . security constraints s(x,u,p) 3 0 These constraints have been discussed in Chapter 6 and are stated here as they form a very important part of the operating state identification process. Having shifted the scenario to the algebraic equa- tions dealing with load, operating and security constraints, it is now possible to describe the five states in terms of those three constraints with the help of mathematical pro- gramming nomenclature. Normal State: The power system is said to be in this state when the load and operating constraints are satisfied, i.e. g(x,u,p) = 0 (8.1a) A O h(x,u,p) (8.1b) Consequently the intersection of (8.1a) and (8.1b) defines 152 the set of all feasible normal operating states in which the power system may be operated. When the system moves from one normal state to another in response to the con- tinuously changing load profile, it is referred to as being in a quasi-steady state condition. Alert State: The power system is said to be in this state when the load and operating constraints are completely satisfied but the security constraints are violated, i.e. g(x,u,p) = 0 (8.2a) h(x,u,p) 3 0 (8.2b) s(x,u,p) 3 0 (8.2c) Emergency State: The power system is said to be in this state when some of the operating constraints are in actual violation, i.e. g(X.U.p) = 0 (8.3a) h(x,u,p) 3 0 (8.3b) s(x,u,p) 3 0 (8.3c) This type of violation takes place generally as a result of two types of contingencies: (a) steady state (b) transient Violation of soft operating constraints such as thermal overloading, which have already been discussed in detail in Chapter 6, lead to steady-state emergencies. Similarly, violation of hard operating constraints, generally lead to transient emergencies. 153 Extremesis State: The power system is said to be in this state when both load and operating constraints have been violated, i.e. g(x,u,p) # 0 (8.4a) h(x,u,p) 3 0 (8.4b) s(x,u,p) 3 0 (8.4c) This generally happens if emergency control measures in the emergency state, subject to time constraints, are un- able to respond fast enough and the system is in the midst of complete collapse. Restorative State: The power system is said to be in this state once action has been taken to halt the disintegration in the extremesis state. In this state, once again, the load and operating constraints are described by i.e. g(x,u,p) # 0 (8.5a) h(x,u,p) < 0 (8.5b) s(x,u,p) < 0 (8.5c) Having represented the power system conditions in terms of operating states, it is now possible to state the power system security control problem. Statement of the Security Control Problem: Given an insecure system, to find the best secure operating point satisfying the load, operating and security 154 (logical) constraints for the constrained Optimization problem formulated in general form as min f(x,u) objective function s.t. g(x,u,p) = 0 load constraints h(x,u,p) < 0 operating constrainé:.6) s(x,u,p) < 0 security constraints where x is a vector of dependent state variables u is a vector of independent control variables p is a vector of disturbances or perturbations Some examples of these variables x = A8, AV (angles, voltages) u = APG, AQG, At, A¢ (gen. shift, tap changers, phase shifters) p = APL, AQL (load shift) The overall objective of stating and identifying the security problem in this way is to enhance the security of the power system by utilizing the available generation and transmission capacity, Spinning reserves and tie-line interchange capacility in an optimal fashion. Figure 3 provides a pictorial view of various concepts and the operating states thus far described [34]. 8.2 THE SUB-CONTROL PROBLEMS: With the overall objective and the five operating states in perspective, it is now possible to decompose the 155 E,I NORMAL SECURE Cost Minimization Load Tracking System Coordination Restorative Preventive Postulated Control Control Next Contingency L, E'I Restorative 5'1 RESTORATIVE Control ‘ ALERT Load Restoration 3} INSECURE Resynchornization Preventive Controls Eggizggiis Corrective Actual Control Violation of Inequality Major 3L3>§onstzaint Load Loss/ ’ _ §,I System E,I Splitting (L EXTREMESIS ‘ EMERGENCY Major Gen- Load Shedding eration Loss/ Controlled Islanding Tie Loss Operator SYSTEM SYSTEM NOT INTACT INTACT Figure 3 STATE TRANSITION DIAGRAM 156 security problem into sub-control problems which deal in- dependently with distinct control objectives depending upon the level of security and power system is operating at. These sub-problems are illustrated in Figure 4 and the discussion of each problem formulation follows. Decomposition of the Security Control Problem: Subproblem I: Economic DiSpatch This subproblem deals with the security level in the normal state. Since the system is operating free of contingencies, the chief objective is to dispatch the generation based solely on economic considerations. Mathe- matically, this can be formulated as N min f(x,u) = _Z KiAui A“ 1‘1 (8.7) s.t. g(x,u,p) = 0 where Aui = a vector of changes in controllable generations Subproblem II: Economic Dispatch with Preventive Control This subproblem deals with the security level in the alert state. In this stage the security assessment functions have indicated a greater probability of an actual violation of the operating constraints via a series of postulated next contingency tests. The chief objective of the dispatch is to execute the appropriate preventive con- trol in anticipation of a potentially dangerous situation. But since the contingency has not actually taken place, 1157 MIN 2 ki Aui G(x,u,P) = 0 s.t. w* 1w MAX 2 01(3 i b i) s.t. G(x,u,p) = H(X.U.P) 3 MIN 2 oilAuil+ ZaiIAWiI Soto I o G(x,u.p) H(x,u.p) 3 S(x.U.p) 3 0 -IN )kiAui s.t. G(x,u,p) = 0 O 0 H(x.u,p) < < S(x.u.p) + ZOiIAui' [[11 Figure 4 MINZki Au. i+z°i [Aufl z oi [AWi s.t. G(x, u, p) H(x,u,p) IA!“ S(x,u,p) + SUB-CONTROL PROBLEMS 158 there is time for preventive action by implementing pre- ventive strategies which minimize the deviation from the most economic generation point. Mathematically, this en- tails superimposing on the economic dispatch formulation, the set of violated security constraints and augmenting the performance index with a penalty function for minimum deviation from economic generation as follows: N N min f(x,u) - 1:1 KiAui + 1:1 wilAuiI s.t. g(x,u,p) = 0 h(x,u,p) < O (8.8) s(x,u,p) < 0 where pi positive cost coefficients of the penalty function. Subproblem III: Economic Dispatch with Preventive- Corrective Control This subproblem deals with the security level in the emergency_state. At this point an actual violation of the operating constraints has taken place and the appropriate control strategy is to take corrective action as fast as possible. Since time is the essential ingredient in the emergency state, the dispatch is based essentially on re- moval of the insecure condition and therefore economics takes a low priority. Load shedding is thus a possible control action in addition to generation dispatch due to 159 the need for rapid control action and a lower priority on economics. Mathematically this may be stated as t) )1 I I If I I min f(x,u) = K.Au. + 0. Au. + O. Aw. Au,Aw i=1 1 1 i=1 1 1 i=1 1 1 s.t. g(x,u,p) = 0 h(x,u,p) < O (8.9) s(x,u,p) < 0 where |Awi| = a vector of changes in interruptible loads Oi = a positive cost coefficient This formulation provides the system with the capability of executing least costly preventive and corrective control actions in a manner similar to that dis- cussed for the alert state formulation. Subproblem IV: This problem deals with the security level in the extremis state. In this state the emergency control is directed at saving as many pieces of the system as possible from total collapse. This can be stated mathematically as N K .332. =2 .14.) +3 0.14.) s.t. g(x,u,p) = 0 h(x,u,p) < 0 (8.10) s(x,u,p) < 0 160 with load shedding taking a higher priority. This part of the control is mostly operator oriented. Subproblem V: This subproblem deals with the security level in the restorative state. In this state extremis has been avoided through saving some parts of the system and the objective is to resume normal operation by slowly re- scheduling the whole system. Since the objective is to maximize the supplied demand, this can be stated mathe- matically as K K * min Z 0.(Aw. - Aw.) or max 2 0.Aw. Aw i=1 1 l 1 i=1 1 1 s.t. g(x,u,p) = 0 (8.11) h(XIUIP) < 0 where * Awi is the connected demand. An on-line secure dispatch tracking algorithmic structure to solve these sub-control problems is pre- sented in the next section. 161 8.3 ON-LINE TRACKING ALGORITHM: In order to solve the optimal on-line secure dis- patch, an algorithmic structure is proposed as in Figure 5. The algorithmic steps are: STEP 1: Assume base case load flow (x°,u°) from static state esimator is available STEP 2: Obtain Jacobian matrix J from static state estimator STEP 3: Obtain J”l matrix by factorization techniques exploiting Sparsity of YBUS matrix STEP 4: Solve fast decoupled load flow Ax = J-lAu STEP 5: Apply Ax to optimization process which provides the control correction Au. STEP 6: Using the fast decoupled d.c. load flow from step 5 to determine constraints,solve the on- line secure dispatch problem. If cost > e reiterate thru step 4 STEP 7: Reschedule power system STEP 8: If any more insecurities detected by security assessment, go to step 2. Otherwise STOP. 162 flmDEUDZEm UHZZEHmOcAC UZHKU<¢B MZHAIZO Bmqom _. u xa Hun .tsumosoce . macaw . _ . . . . _ . (((((((((((((((((((((((((((((((( .quewa.)..-- -)..( yr yr zoatnosoa lemma“. ez