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I'I'IIIIJIII'I."II"1iIIII"IIIII‘n’J mm m 111“ mil 1| 1 31 1 ll 1| mtg 1| 1 'K.’ \ (My, \..ah-.. i“ “BEAM/‘1 F deinganfhnhc ‘Jhuvennty ~1— This is to certify that the thesis entitled MODAL-COHERENT EQUIVALENTS DERIVED FROM AN RMS COHERENCY MEASURE presented by Jack Stewart Lawler has been accepted towards fulfillment of the requirements for Ph.D. degreein Systems Science ; Major professor Date 8/6/79 0-7639 OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: ______._._..————— Place in book return to remove charge from circulation records MODAL-COHERENT EQUIVALENTS DERIVED FROM AN RMS COHERENCY MEASURE BY Jack Stewart Lawler 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 MODAL-COHERENT EQUIVALENTS DERIVED FROM AN RMS COHERENCY MEASURE BY Jack Stewart Lawler The modal and coherency analysis techniques have domi- nated the substantial research efforts which have been de- voted to developing dynamic equivalents that may be used to reduce the complexity of power system transient stability studies. The equivalents derived from these two techniques have distinctly different structures and properties. Although both approaches have desirable features, each approach has been the subject of criticism and neither technique has gained complete acceptance. The objective of this research is to show that a dynamic equivalent which combines the best features of both modal and coherent equivalents can be de— rived based on a coherency analysis using the rms coherency measure. An algebraic formula is derived which relates the expect- ed value of the rms coherency measure, evaluated over an infinite observation interval, to the parameters of the power system state model and the statistics of the system disturbance. This expression is used to establish an important link between the modal and coherency analysis approaches to Jack Stewart Lawler power system dynamic equivalents by showing that the inertial- ly weighted synchronizing torque coefficients, which determine the system modes, are also the basis of coherency aggregation when a particular probabilistic disturbance, called the modal disturbance is used to identify coherent groups for coherency based aggregation. This result allows a coherent equivalent to be derived which closely approximates a general purpose modal equivalent based on the same coherency measure and disturbance. An example system is used to show that the ei- genvalues of the coherent equivalent derived from the modal disturbance closely approximate the system eigenvalues retain- ed by the corresponding modal equivalent and that both of these equivalents are excellent general purpose equivalents, suitable for studying many different system contingencies. The coherent equivalent based on the rms coherency mea- sure and the modal disturbance is called a modal-coherent equivalent. This equivalent has the theoretical soundness and general purpose applicability of a modal equivalent and the power system component structure of a coherent equivalent. An efficient computational algorithm, applicable to large scale systems is developed for constructing the modal-coherent equivalent. It is shown that the computational effort re- quired to construct the general purpose modal-coherent equiva- lent is competitive with the total effort required to construct the set of coherent equivalents which would be needed for a transient stability study of a relatively small number of dis- tinct system disturbances. For Pauline ii ACKNOWLEDGMENTS I would like to express my appreciation to the Chairman of the Department of Electrical Engineering and Systems Science, Dr. J. B. Kreer, for the financial support of the department during my first year of study. Drs. G. L. Park and R. A. Schlueter and the Division of Engineering Research are thanked for providing continuing financial support over the final two years of my doctoral program. Special acknowledgment is made of the contribution of Dr. R. O. Barr, who had confidence in me and encouraged me to pursue advanced education. The participation of Drs. P. McCleer and A. N. Andry on my guidance committee is also appreciated and acknowledged. I am grateful to Dr. G. L. Park for the opportunity to par- ticipate on his wind energy research projects which contributed greatly to my overall professional development. Dr. Park is al- so thanked for his significant service on my guidance committee. I would like to express my sincere appreciation to my the— sis advisor, Dr. R. A. Schlueter, for his guidance throughout the course of my doctoral program. Dr. Schlueter served not only as advisor but also as a friend and his friendship will never be forgotten. Most of all, I would like to thank my wife, Julie, for her love, patience and gentle encouragement. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . CHAPTER 1 INTRODUCTION. . . . . . . . . . . . . . . . . . 2 POWER SYSTEM MODEL, GENERALIZED DISTURBANCE MODEL, AND THE RMS COHERENCY MEASURE. . . . . . 2.1 Linearized System Model . . . . . . . . 2.2 Generalized Disturbance Model . . . . . 2.3 RMS Coherency Measure . . . . . . . . . 2.4 Coherency-Based Aggregation Technique . 3 THE RMS COHERENCY MEASURE AND DYNAMIC SYSTEM STRUCTURE O O O O O O O I O O O O O O O O O O O 3.1 Algebraic Relationship Between S (w), System Structure and Disturbance Statistics. . . . . . . . . . . . . . . 3.2 Disturbance-Independent Coherent Equivalents Derived from the RMS Coherency Measure . . . . . . . . . . . 4 COMPARISON OF MODAL AND COHERENT EQUIVALENTS DERIVED FROM THE RMS COHERENCY MEASURE. . . . . 5 A MODAL-COHERENT EQUIVALENT . . . . . . . . . . 6 COMPUTATIONAL ALGORITHM FOR CONSTRUCTING THE MODAL-COHERENT EQUIVALENT . . . . . . . . . . . 6.1 Evaluation of the RMS Coherency Measure and Identification of Coherent Groups . 6.2 Comparison of the Algorithms for Con- structing the Modal-Coherent Equivalent and Coherent Equivalents Based on the Max-Min Coherency Measure . . . . . . . 6.3 Computational Examples. . . . . . . . . 7 CONCLUSIONS AND FUTURE INVESTIGATIONS . . . . . 7.1 Thesis Review . . . . . . . . . . . . . 7.2 Future Research . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . iv PAGE vi 17 17 24 29 33 39 40 47 53 69 73 75 93 102 125 125 130 133 LIST OF TABLES Table Page 4-1 Coherency Measure CkR for the Unreduced System and the Modal and Coherent Equivalents Obtained with a ZMIID Disturbance in Mechanical Input Power. . . . . . . . . . . . . . . . . . . . 56 4-2 System Matrices and Eigenvalues of the Coherent Equivalent Derived from the ZMIID Disturbance. . . 57 4-3 Coherency Measure C for the Unreduced System and the Modal and Coherent Equivalents Obtained with a UD Disturbance, g = (l,1,l,l,1,l,1), in Mechanical Input Power . . 60 4—4 System Matrices and Eigenvalues of the Coherent Equivalent Derived from the UD Disturbance . . . . 61 4-5 Comparison of the Coherency Measure Ckl of the Unreduced System and the Modal and Coherent Equivalents Obtained with the UD and ZMIID Disturbances for Two Particular Determi- nistic Disturbances in the Mechanical Input Powers of Generators 1 and 2 . . . . . . . . . . . 64 6-1 Suggested Ranking Table Structure. . . . . . . . . 89 6-2 Ranking Table for the Modal Disturbance, MECS" Example System . . . . . . . . . . . . . . . . . . 113 6-3 Coherent Group Identification Based on the Ranking Table and the Commutative Rule, MECS Example System-Modal Disturbance . . . . . . . . . 115 6-4 Coherent Group Identification Based on the Ranking Table and the Transitive Rule, MECS Example System-Modal Disturbance . . . . . . . . . 116 6-5 Modes of Intermachine Oscillation for Modal- Coherent Equivalents Derived from the Commutative Rule . . . . . . . . . . . . . . . . . 121 LIST OF FIGURES Figure Page 6-1 Three Generator Example System . . . . . . . . . . 103 vi CHAPTER 1 INTRODUCTION The extensive research efforts to develop dynamic equiva- lents suitable for use in simulations of power system response to disturbances have been dominated by the modal and coher- ency analysis approaches [l,2]. These two techniques have developed independently and appear to be distinctly different concepts, although both approaches have the common goal of reducing the complexity of power system analysis. Each ap- proach has been the subject of criticism and neither tech— nique has gained complete acceptance. This thesis explores the relationship between these two approaches and proposes a new approach to power system dynamic equivalents which combines the best features of modal and coherency analysis. In order to define some relevant terminology, the tran- sient stability problem for large scale systems and the need for dynamic equivalents are briefly reviewed. An overview of modal and coherency analysis is also given in order to provide a perspective for the contribution of this research. The specific objectives of the thesis are then discussed. The Need for Dynamic Equivalents in Transient Stability Studies Consider a large geographical area where the electric power needs are served by many interconnected utilities. Let one particular utility be designated as the internal system an: study area) and let the remaining utilities in the interconnected network be designated as the external system. The transient stabilityyproblem is to determine how well-behaved the generators in the internal system will be when a disturbance occurs in the internal system. The types of disturbances considered in transient stability studies fall into four basic categories; generator dropping, load shedding, line switching or electrical faults. Recognizing that the internal-external system model may contain several hundred interconnected generating units, each of which is characterized by nonlinear differential equations, it is apparent that the determination of the transient behavior of the internal system generators for any particular disturbance will entail the solution of hundreds of nonlinear, coupled, differential equations. Thus, the transient stability problem for modern power systems is characterized by a severe dimensionality problem. Histori- cally, transient stability analysis was performed using the classical, second order, synchronous machine equations, which model only the rotor dynamics of the synchronous machine, to represent each generator in the system. This model was used in order to minimize the number of differential equations to be solved. More recently, it has become apparent that there is a need to increase the level of generator modelling to in- clude the exciter dynamics and the effects of the turbine- governor system, in order to improve the accuracy of transient stability results. An increase in the level of generator modelling naturally aggravates the dimensionality problem. Since the number of generators in the internal system model is generally a relatively small percentage of the total number of generators in the composite internal-external system, it is clear that the study area of interest contrib- utes only fractionally to the over-all complexity of the transient stability problem. Put another way, the bulk of the computational effort expended by a particular utility in performing a transient stability study is consumed in computing the impact that the large external system has on the behavior of the much smaller internal system. A crude approach to ease the dimensionality problem would be to simply neglect the external system and perform the transient stability study on the isolated internal system. In general, the coupling which exists between the internal and external systems, due to tie lines, is not sufficiently weak to permit such an approach. In fact, one of the reasons for installing the tie lines which couple neighboring utilities is to improve system performance under disturbed conditions and to neglect these lines in contingency analysis would ig- nore their important function. A more reasonable solution to the dimensionality problem is to find some method to reduce the size of the external sys- tem model. Ideally, the reduction would be done in such a way that the impact that the external system has on the behavior of the internal system, would be preserved with respect to disturbances that occur inside the study area. A reduced order model of the external system which meets this objec- tive is called a dynamic equivalent (or simply an equivalent). Two approaches that have been developed for constructing dynamic equivalents for power systems are the modal and coher- ency analysis approaches. The characteristics of these two techniques are now discussed. The Modal Analysis Approach to Dynamic Equivalents The modal analysis approach [1] assumes that a reason- able simplification of the transient stability problem would be to use a linearized model of the external system. The linearization can be justified, since the response of the external system to remote internal system disturbances would be at most weakly nonlinear. The order of the linearized model is then reduced by performing a modal reduction based on controllability and observability considerations. The re- duced order linear model of the external system is called the modal equivalent. The procedure for constructing the modal equivalent is 1. Establish the linearized model of the unre- duced external system. 2. Compute the eigenvalues and eigenvectors of the linear model. 3. Put disturbances at the boundary between the internal and external systems and use rules of mode elimination [1] based on the eigenvectors and eigenvalues to eliminate any mode of the external system model which i) is relatively unexcited by any boundary disturbance (uncon- trollable modes). ii) is relatively undetectable at the boundary (unobservable modes). With respect to the coherency analysis approach which will be discussed later, the modal approach has two signifi- cant advantages. These are 1. The approach is theoretically sound. 2. The modal equivalent is general purpose and may be used to study any disturbance in the internal system. The disadvantages of the modal approach (with respect to co- herency analysis) are l. The eigenvalues and eigenvectors required to perform mode reduction are expensive to compute for large systems and are virtually impossible to compute for system models larger than one hundredth order. 2. The mode reduction process destroys power system structure, that is, the reduced order linear model obtained with the modal procedure is no longer recognized as a linearized power system model. Because the modal equivalent is not represented in terms of power system components, present tran- sient stability programs would have to be modified to accept the linear model representation. To overcome the difficulty in computing the eigenvalues of a very large system model, it has been proposed that the external system be divided into several computationally 6 manageable "multiple external system" models [3]. Such an approach is undesirable unless weakly coupled regions in the external system can be easily identified, otherwise there would be no guarantee that the eigenvalues of the multiple external system models would be the same as the eigenvalues of the undivided system model. A general pro- cedure for producing modal equivalents which preserve a meaningful physical structure is presently being pursued [4]. At this time, it appears that any approach to power system dynamic equivalents which requires eigenvalue calculations is not practical. The Coherency Analysis Approach to Dynamic Equivalents The coherency analysis approach to power system dynamic equivalents is based on the coherency phenomena associated with the behavior of synchronous machines. Coherency is a disturbance-dependent phenomena and for a particular system disturbance, two generators are said to be coherent if the coherency measure between them, evaluated for that distur- bance, is less than some prescribed threshold. There are various ways of defining the coherency measure between two generators all of which are a function of the difference be- tween the voltage angles (swing curves) at the internal gen- erator buses of the two generators. Two or more mutually coherent generators define a coherent group and the genera- tors in a coherent group are said to "swing together" in response to the system disturbance for which the coherency measure was evaluated. More than one coherent group may be observed for a single disturbance. It has been shown [5] that the coherent groups cor- responding to a particular system disturbance may be accu- rately identified from a coherency analysis of the response to that disturbance of a simplified, linearized model of the internal-external system. Once coherent groups have been identified, a coherent equivalent is constructed using the coherency-based aggregation technique [2] to form one equiva- lent generator per coherent group observed in the external system. Although coherent groups are identified with a simplified, linear model, the aggregation technique is applied to the full nonlinear system model and the coherent equivalent obtained is a reduced order, nonlinear model. The transient stability study for the disturbance used to identify coher- ent groups is then performed using the coherent equivalent identified for that disturbance. There is no theoretical justification for power system model aggregation based on a coherency measure dependent only on voltage angle. The foundation of the coherency analy- sis approach rests on the intuitive appeal of the concept that generators which swing together function as essentially a single generator and may therefore be aggregated. Although the coherency analysis technique lacks the sound theoretical basis of modal analysis, it offers significant advantages. 1. Eigenvalues and eigenvectors are not required to construct the coherent equivalent, and the overall computational procedure has been shown to be efficient and applicable to very large systems [5]. 2. The form of the equivalent obtained is a reduced set of equivalent generators and lines and may therefore be used with existing transient stability programs. The coherency based aggregation technique presently uses a max-min coherency_measure to identify coherent groups. The technique has been criticized because 1. The max-min coherency measure used to identify coherent groups has not been shown to be proportional to system parameters and thus the coherency mea- sure must be determined by simulation [6]. 2. The equivalent produced using the max-min measure is dependent on the particular disturbance used to iden- tify coherent groups for aggregation such that a unique equivalent must generally be constructed for each disturbance to be studied [7]. Satisfactory responses to these criticisms have not yet been given. One additional criticism is directed at both modal and coherency analysis. Quite often there are alternative solu- tion approaches for solving a given engineering problem, but when this is the case, the various solution techniques are related and the solutions which are obtained are essential- ly identical. Dynamic equivalents derived from the present modal and coherency analysis techniques [1,2] have such dif- ferent structures and characteristics that it is reasonable to question whether or not these approaches are mutually consistent. The rules of mode elimination, based on controllability and observability considerations, used in modal analysis [1] are designed to preserve the states of the internal system model. Mode reduction based on preserving states, and not a coherency measure, does not guarantee that intermachine behavior is preserved and may also be sensitive to the choice of reference generator used to establish the state model. Thus, if present modal equivalents are also coherent equiva- lents, they are coherent equivalents by accident and not by design. Similarly, the present coherency approach [2] makes no effort to insure that system modes are preserved. The fact that coherent equivalents must be recomputed each time the location of the system disturbance is significantly changed is sufficient evidence to indicate that coherent equivalents do not preserve system modes. Thus, the present modal and coherency analysis approaches are not mutually consistent. Neither modal nor coherency analysis has gained complete acceptance by the power industry. The criticisms of these approaches shed considerable light on the properties of a dynamic equivalent which the industry would find highly desirable. The characteristics of an "ideal" dynamic equiva- lent are now discussed. 10 The Ideal Dynamic Equivalent An ideal equivalent should be a general purpose equiva- lent, that is, a single ideal dynamic equivalent would be suitable to study any disturbance which might occur within the study area for which the equivalent is derived. Therefore, the ideal equivalent should be based solely on system structure and not on the type or location of any particular system disturbance. In addition to the general purpose property, an ideal equivalent should be both a modal and a coherent equivalent. Coherency is an important power system phenomena which is strongly related to transmission line power flows and pecu- liar to the behavior of synchronous machines. Any approach to power system model aggregation which produces an equiva- lent which significantly alters the coherency measure between generators would upset the power transfer characteristics of the system and neglect the essential nature of synchronous machine interaction. The characteristic modes of oscillation are similarly important. If an equivalent does not closely preserve system modes, then there is little chance that the equivalent can accurately predict the time response of the full system modeliknra large class of disturbances. Thus, an ideal dynamic equivalent should preserve system modes and the coherent behavior of generators. Two final characteristics that an ideal equivalent should have are dictated by practical considerations. First, the equivalent shouldkxarepresentedixiterms of normal power system components so that it can be used with existing 11 transient stability computer programs. And second, the approach for deriving the equivalent must be computation- ally efficient and applicable to large scale systems. The discussion of the characteristics of an ideal power system dynamic equivalent indicates that such an equivalent should be a general purpose equivalent which is both a modal and a coherent equivalent, which can be derived from an efficient procedure that preserves power system component structure. Neither modal analysis nor coherency analysis leads to an equivalent which has all of these properties. However, the characteristics of an ideal equivalent are a composite of the most desirable features of present modal and coherent equivalents. This suggests that a unified ap- proach to dynamic equivalents which is consistent with the objectives of both modal and coherency analysis may result in a dynamic equivalent with ideal or nearly ideal properties. An early effort to unify modal and coherency analysis is now briefly reviewed. A Coherent-Modal Approach to Dynamic Equivalents Recognizing the need to insure that modal equivalents preserve the coherent behavior between generators, a recent paper [8] established rules of mode elimination for construct- ing modal equivalents based on a modal analysis of an rms co- herency measure evaluated over an infinite interval. The rules of mode elimination are designed to eliminate modes which do not significantly change the rms coherency measure between any two generators by more than some arbitrary amount. 12 Modal equivalents which are derived to preserve a coherency measure may also be considered coherent equivalents and the approach suggested by [8] may be termed a coherent-modal approach. The coherent-modal approach would not lead direct- ly to an ideal equivalent, since eigenvalues are required and since the equivalent does not retain power system component structure. However, the approach does provide some indica- tion of how an ideal equivalent might be obtained. The rules of mode elimination based on the rms coherency measure [8] were derived for both deterministic and probabil- istic system disturbances. In a companion paper [9], those rules were applied to an example system, and modal equivalents were constructed for various step disturbances in mechanical input power. It was observed that when the step disturbance in the mechanical input powers on the system generators were zero mean, independent and identically distributed (ZMIID), that the process of mode reduction closely resembled a coherency-based aggregation since, the only modes eliminated by the rules of mode elimination were associated with the intermachine oscillations within a coherent group. It was further observed that the rules of mode elimination applied to deterministic disturbances did not resemble a coherency aggregation. Two hypotheses were proposed based on these observations. The first was that dynamic system structure can be identified through an rms coherency measure if proba- bilistic disturbances are used to identify coherent groups. The second hypothesis was that the eigenvalues of the coherent equivalent derived using the coherency-based aggregation 13 technique to aggregate the coherent groups identified by the rms coherency measure and the ZMIID disturbance should closely approximate the system eigenvalues retained by a modal equivalent based on the same coherency measure and disturbance. The above hypotheses, based on empirical observations of the behavior of coherent-modal equivalents, provides a major direction for this research. The specific objective of this thesis is now described. Thesis Objective The objective of this thesis is to develop the justifi- cation and the means for constructing a modal-coherent equiv- alent whose properties closely approach those of an ideal dynamic equivalent. It is emphasized that the term "ideal" is meant to be taken in the context of the preceding discussion. The modal-coherent equivalent proposed in this research is constructed using the coherency-based aggregation tech- nique [2] to aggregate the coherent groups identified by the infinite interval rms coherency measure and a particular probabilistic step disturbance in mechanical input powers called the "modal disturbance". Two properties of an ideal equivalent are realized by the modal-coherent approach as a natural consequence of the use of a coherency analysis and coherency-based aggregation to identify and construct the equivalent. These are 14 l. The coherent behavior of generators is preserved by the equivalent. 2. The equivalent is represented in terms of normal power system components. It will be shown, that two additional properties of an ideal equivalent are achieved by the modal-coherent equivalent be- cause the rms coherency measure and the modal disturbance can identify coherent groups which reflect the dynamic structure of the system. These properties are 3. System modes are closely preserved by the equivalent. 4. The equivalent is general purpose and may be used to study a large class of disturbances. It will also be shown, that the final property of an ideal equivalent is realized by the modal coherent approach because 5. The infinite interval rms coherency measure can be evaluated for the prob- abilistic modal disturbance using an efficient technique which is applicable to large scale systems. Thus, this thesis will develop a modal-coherent equivalent which is a general purpose coherent equivalent and an approx- imate modal equivalent that may be derived from a computa- tionally efficient procedure which preserves power system component structure. The justification for the modal-coherent equivalent is developed in Chapters 3 and 4. In Chapter 3, an algebraic relationship is derived relating the expected value of the 15 rms coherency measure, evaluated over an infinite interval, to the parameters of the state model and the statistics of the system disturbance. This relationship is used to show that the rms coherency measure and a particular probabilis— tic system disturbance can reflect dynamic system structure. Thus, the first hypothesis proposed in [9]ij;verified theoretically. Using the same example system as [9], it is shown in Chapter 4, that the eigenvaluescflfthe coherent equivalent constructed using the coherency-based aggregation technique to aggregate the coherent groups identified by the rms coherency measure and a particular probabilistic system disturbance closely approximate the system eigen- values retained by the modal equivalent based on the same coherency measure and disturbance. It is further shown that both the modal and the coherent equivalent derived from this probabilistic disturbance are suitable for studying the ef- fects of any disturbance which might occur outside the areas of the system aggregated to form these equivalents. Thus it is shown that a general purpose coherent equivalent which is an approximate modal equivalent can be derived from the rms coherency measure when an appropriate probabilistic disturbance is used to identify coherent groups. The appro- priate probabilistic disturbance is shown to be the modal disturbance. The example system used is shown to be a special case where the above results apply to the ZMIID disturbance as well as the modal disturbance and thus the second hypothesis proposed in [9] is empirically verified. 16 Chapter 5 discusses the need for an approach to dynamic equivalents which is consistent with the objectives of both modal and coherency analysis and retains the structure of a coherent equivalent. The modal-coherent equivalent is proposed to meet this need. The means for constructingtflmamodal—coherent equivalent is developed in Chapter 6. An efficient method for evaluat— ing the infinite interval rms coherency measure for the probabilistic modal disturbance is developed which is appli- cable to large scale systems. It is shown that the computa- tional effort required to construct the general purpose modal- coherent equivalent is competitive with the total effort re- quired to construct the set of coherent equivalents which would be needed to perform a transient stability study for a very modest number of distinct system disturbances. The final chapter summarizes the contribution of this thesis and proposes topics for future investigation based on this research. CHAPTER 2 POWER SYSTEM MODEL, GENERALIZED DISTURBANCE MODEL, AND THE RMS COHERENCY MEASURE The objective of this chapter is to present the mathe- matical models used in the development of the modal-coherent equivalent. The linearized power system model, generalized disturbance model and rms coherency measure used for coher- ency analysis are defined, and the mechanism by which co- herent groups are aggregated to form a coherent equivalent is briefly discussed. 2.1 Linearized System Model Recent work on coherency-based dynamic equivalents at System Control Incorporated (SCI) has shown that a simpli- fied model for coherency analysis can be derived with the following assumptions 1. 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. 2. 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. l7 18 3. 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 radi- cally affect the more basic characteristics such as the natural frequencies and mode shapes. The third assumption recognizes that the generator accelerating powers are approx- imately constant during faults with typical clearing times. The above assumptions and their justifications are quoted from [11]. A linear model can be obtained by starting with the classical synchronous machine representation for each generator Mi 5% wi(t) = PMi(t) - PGi(t) - Diwi(t) (2-la) i=l,2,...,N EL 6.(t) = w.(t) i=1 2 ... N (2-1b) dt 1 1 ' ' ' ' where, i a subscript for generator i N the number of generators in the system Mi inertia constant (p.u.) Di damping constant (p.u.) mi generator speed (rad/sec) Si generator rotor angle (rad) l9 PMi mechanical input power (p.u.) PGi electrical output power (p.u.) mo nominal synchronous generator speed (rad/sec) Equations (2-1) are nonlinear due to the nonlinear relation- ship between PGi and the bus angles in the interconnected network. For a lossless transmission network, the system network equations can be written as § PG. = l j=1 j#i 7! + ‘lLIIMW where, Iv.l.lvjl IVklprQI Ivillv.l -—————l— sin(6.-6.) X.. 1 j 1] i=l'2'OOO'N Ivillv X 2| . . $1n(6i-6£) 12 (2-2) IVKIIV-I ———l— sin(6 -s.) Xk. k j J k=l,2,...,K Ivkllv I x Q sin(6 —e ) k2 k 2 magnitude of the complex voltages at generator buses i and j magnitude of the complex voltages at load buses k and 2 voltage angle at load buses k and 1 is the number of load buses is the impedance of the line directly connecting any two specified buses The synchronous machine equations may be linearized using a first order Taylor series expansion, by introducing the deviations Adi, Ami, APMi, APGi about the nominal load 0 flow conditions 5:, w , PMi' PGE. The resulting linear model 0 has the form 6 _ . - _ _ Mi 5? Awi(t) — APAi(t) APGi(t) DiAwi(t) (2 3a) i=l,2,...,N d _ -_ - 3E AS;(t) - Awi(t) , 1-l,2,...,N (2 3b) The network equations may also be linearized with real and reactive powers decoupled and written in polar form as APG BEG/SQ BEG/36 AR ——— = (2-4) APL BEL/86 BEL/36 99 where, PG = (PG PG PG )T —— l’ 2""' N T BE = (PLl,PL2,...,PLK) _ T g — (61,62,...,6N) T g = (81,62,...,6K) K is the number of load buses APLj deviation in power injection at load bus j Aej deviation in voltage angle at load bus j The decoupling of real and reactive powers is justified for transmission systems which exhibit high X/R ratios (i.e. low loss networks). Line losses should have little impact on co- herent behavior between generators and for the purpose of co— herency analysis a lossless network will be assumed. The power-angle Jacobian matrix in the network equations (2-4) is a sparse, symmetric and singular matrix. The network equations are not of full rank since the entries in any row, or column, sum to zero (the diagonal elements are the nega- tive of the sum of the off-diagonal elements in any row or column). Thus a unique solution for AR and £9! given APG and 21 APL, cannot be obtained. This minor dilemma is solved by an angle referencing scheme which is discussed shortly. Equations (2-3) and (2-4) are said to be a synchronous frame model since the deviations in bus angles and generator speeds are measured with respect to an external reference rotating at the nominal system speed, mo. If a step input in mechanical input powers, which is not balanced by an equiva- lent change in load, is applied to the synchronous frame model, the speed of the generators in the model will change. The deviations in generator angles in response to such a dis- turbance will appear as ramp functions when measured with respect to a reference that always rotates at the nominal synchronous speed, mo. Thus, the synchronous frame model has an eigenvalue at the origin (step input, ramp output). The analysis presented in this thesis requires a model which has all nonzero eigenvalues. Such a model may be obtained by referencing all of the bus angles in the system to the angle of an arbitrarily chosen reference generator. Referencing angles is a common practice in power system analysis and re- sults in no loss of generality. Selecting generator N as the system reference, equations (2-3b) and (2-4) may be rewritten in the machine N reference frame as d_ A d d . - a-E AOi(t) - a—t— A6N(t) (2'5) D 0') p ('1' l Awi(t) - AwN(t) , l=1,2,.1.,N-l and, 22 AFC BEG/3g BEE/32 A_ = A A A (2"6) APL BEL/36 BEL/86 _g where, 6i = 6i - 6N , 1=l,2,...,N-l ej = ej - 6N , j=l,2,...,K In addition to eliminating the eigenvalue at the origin, the angle referencing scheme allows the network equations, in the generator N frame, to be uniquely solved for £2 and £2 given the values of APE and 92L. When each of the generators in the system is character- ized by the same damping to inertia ratio, that is Mi . 57 = o (a constant) , 1=l,2,...,N (2-7) 1 then the differential equations describing the generators may be written as d A _ -l -1 5E Awi(t) — Mi (APMi(t) - APGi(t)) - MN (APMN(t) - APGN(t)) - OAwi(t) (2-8a) i=l'2'ooo'N-l d A _ “ ._ _ 3E A6i(t) — Awi(t) , 1—1,2,...,N l (2 8b) where, (Di = (Di - (UN ' 1=l,2,...,N-l Equations (2-8) are referred to as a uniform machine N frame model. The uniform damping assumption, (2-7), does re- sult in some loss of generality. However, the assumption is partially justified by recognizing that damping to inertia ratios are typically small and in general the value of the 23 damping constant is not accurately known. A thorough dis- cussion of the synchronous frame, machine N frame and uni- form machine N frame models may be found in [10]. A state variable model may be derived by writing the 2N-2 equations in (2-8) in vector form and using the network equations (2-6) to eliminate APE from the expression. The resulting model has the form int) = A 3<_(t) + g yt) (2-9) where, [9: _A___PM x = A 3: (2-10) E_g. APL 9 _ 9 9 A = r E = “E I ’01 E E E and, V'- 1 l— _l _ M1 1 —MN (2-11) -1 -1 M2 : ’MN M = . ' “ l l l -1 ‘ -1 MN-l: -M'N L. ..a g; = egg/3g; - (agg/ag) [3111/3ng (egg/as) g = 483/39) [ail/82H”L The synchronizing torque coefficient matrix, T, and the re- flection matrix, L, are found in the process of solving the network equations for APG in terms of Ag and APL with the result APG i g 6 + p APL (2-12) 24 A linearized power system model has been developed and attention is now focused on a generalized disturbance model. The disturbance model will be shown to model deterministic as well as probabilistic system disturbances. The model was developed by Schlueter, in [8], and the presentation here follows that development. 2.2 Generalized Disturbance Model The initial conditions of the linear model (2-9) are assumed random with E{§(O)} 9 (2-13) E{§(O) _T<0)} = yxm) since the expected deviations from any operating state is zero but the variance of such deviations is nonzero. The rms coherency measure will be shown to depend on the covari- ance matrix yx(0). The initial conditions are included not to reflect any specific type of disturbance but rather the effects on the state from some hypothetical disturbance whose statistics (2-13) may be inferred from internal and ex- ternal operating conditions. The input, composed of the deviations in the mechanical input power, AEM, and the deviations in load power, AAA, can be used to model i) loss of generation due to generator dropping 11) loss of load due to load shedding iii) line switching iv) electrical faults 25 These contingencies can be modelled In? an input 3(t) that has the following form 3(t) = 31(t> + 32(t) <2-14) The vector function 21 t 31m = (2-15) 9 t<0 IV 0 represents 1) 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 modelling of these three disturbances requires deter- mination of E and possible modification of the network before 1 determination of matrices A and A. The procedure used[ll] for each disturbance type is discussed below: generator dropping - the transient reactance of the generator dropped is omitted from the net- work and the deviation in the generator output PMi of the generator dropped is set equal to the loss of generation. load shedding - the load deviation PLk for all buses k where load is shed should be set equal to the change in load caused by the load shed- ding operation. line switching - the network is modified to re- present the system after the line switching operation is performed. The load deviations, PLk and PLm, 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. 26 Note that in each case above all variables in 31 are zero unless otherwise specified and the operating point used to obtain matrices A and A is that obtained from the base case load flow even if network changes are made. The re- sults obtained without finding the post disturbance load flow conditions is apparently satisfactory because the ef- fects due to changes in the load flow are assumed to be con- fined to the study system and thus should not affect the coherency of the external system being equivalenced. The vector function A t>Tl 32(t) 32 OStSTl (2-16) A ‘t< 0 represents the effects of electrical faults where T1 repre- sents the fault clearing time and ABM 32 — 0 (2-17) represents the step change in generation output equivalent to the accelerating powers due to a particular fault. The change of mechanical powers, ABM, which corresponds to the accelerating powers on generators due to a particular fault is calculated by an ACCEL program [11], and has been shown to adequately model the effects of that fault when a linearized model based on pre-fault load flow conditions is used. Again the results obtained neglecting faulted and post fault load flow conditions is apparently satisfactory due to the fact that the effects due to changes in load flow conditions are 27 assumed to be confined to the study system and should not effect the coherency of the external system being equivalenced. The above model can be generalized to model the uncer- tainty 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 known, then a probabilis- tic description of this electrical fault is m E{u } — "21 - m (2-18) -2 —2 0 R O T —21 MHz-1221 [22-1221 }- O 0 ‘52 where AZIand BZldescribe the uncertainty in accelerating power on all generators due to this electrical fault. This mean and variance should be determined based on observed histori- cal records or hypothesized based on the present network and present internal and external conditions. If 32 = g, and m21==A§M 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 switch- ing, and load shedding disturbance can be modelled by m E{gl} = ’11 = $1 (2-19) I312 R o T _ —11 — _ E{[El 51] [31 ' 91] } ‘ 0 R ’ B-1 28 where all and all can describe the uncertainty in generation changes due to generator dropping when the particular station, the generator in the station, and the power produced on the generator are unknown. 512 and 322 describe the uncertainty in the location and magnitude of the load being dropped by any manual or automatic load shedding operation. r1‘12 and 322 the location and the change in injections describe the uncertainty in on buses due to any line switching operation. It should be noted that APM and APL are assumed uncorre- lated 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{§(O) HI} = 9 (2-20) E{§(0) 3?} = g The uncertain model of 31 can handle the case of a spe- cific deterministic disturbance by setting 51 = A and £1 = 31 for the particular disturbance. The probabilistic descriptions of generator dropping and line switching are made assuming the network changes associ- ated with the deterministic disturbances of these types can be omitted. This assumption seems valid since the effects of retaining these elements in the network should be confined to 29 the study system and should not seriously effect coherency of the external system being equivalenced. 2.3 RMS Coherency Measure Early work by SCI [2] was aimed at identifying the par- ticular coherency measure of voltage angle differences that would produce the best dynamic equivalent when the generator buses classified as coherent by each measure were equivalenced. Two particular measures compared at that time were the max—min and rms coherency measures CkR = max{5k(t) - 6£(t)} - min{6k(t) - 6£(t)} (2-2l) t€[O,T] t€[0.Tl 2 1 2 Ck2 =J/T 1? [A6k(t) — A6£(t)] dt where A6k(t) = 6k(t) - 6k(0) The max-min coherency measure was shown to produce bet- ter dynamic equivalents than those constructed based on an rms measure of coherency when both measures are compared based on the same short observation interval. The result is not unexpected because the max-min measure should insure there are no large deviations between the equivalent and the unreduced model whereas the rms measure should only insure that average energy in the deviations is small over this short observation interval. Another way of viewing this difference in the two measures is that a max-min measure of coherency tends to measure the sum of the amplitudes of dominant modes of the system dynamics where rms coherency tends to measure 30 the sum of the energy in dominant modes of angle differences 6k(t) - 6 (t) k, 2 = l,2,...,N. The max-min measure of 2 coherency is thus clearly a better measure over a short ob- servation interval when a dynamic equivalent for one partic- ular contingency occurring in one location is desired. However, if an equivalent is desired that best reflects the overall dynamics of the external system, the rms measure will be shown to be superior. The rms measure of coherency between generator internal buses k and 2 based on the uncertain description of distur- bances is defined by _ 1. t _ 2 _ Ck2(t) —J/;5 1) E{[A6k(r) A5£(T)] }dT (2 22) The integer n is chosen to be one if a load shedding, line switching, or generator dropping contingency occurs and zero if an electrical fault occurs. This integer is chosen as zero or one so that the above integral will be finite and non-zero for an infinite observation interval.» The computation of the rms measure is facilitated by constructing the intermediate quantity, §X(t), which is a 2N-2 dimensional symmetric matrix which is defined in terms of the state vector of the linear model as s (t) = 3; ft E{X(T)XT(T)}dT (2-23) —x tn () - — The coherency measure between any pair of generators k and 2, defined by (2-22), depends only on the generator angles. Therefore, the value of any Ck2(t) will be determined by the 31 upper left (N-l))<(N-1) submatrix of §X(t). Defining the upper left (N-l))<(N-l) submatrix of §x(t) to be §X(t), the coherency measure Ck2(t) is related to the entries in §x(t) by /{§X(t)}kk - {§-x(t)}k2 - {§X(t)}gk + {§x(t)}u A k,2#N Ck2(t) = J1§x(t)}kk k#N. £=N (2-24) \fisxunM 2%N, k=N The matrix §X(t) can be computed for the disturbance 3(t), given by (2-14), by substituting the solution to the state equation EAT §(O) + [T eév dv A (Al + AZ) T=9. T1=9v El=9 <3-6> 42 into equation (2-26), in which case §x(t) becomes T '1' 5V T T 5" T s (t) = f 1 [I e dv B] [R +m m ] [f e dv B] dT —x 0 0 — -2 —2—2 0 — A(T-T ) AV t — l T — T + I [e f l e dv A] [A2+m2m2 ] T 0 l A(T-T ) Av [€_ 1 [T1 6 dv A]T dT (3-7) 0 Since the coefficient matrix A of the linear system model is nonsingular, the interior integrals may be evaluated as T AV -1 QT [ e dv = A (e -A) (3-8) 0 Defining _ T T _ E2 ’ §[52+Ezflz ]§ (3 9) and substituting (3-8) and (3-9) into (3-7) / W T AT ATT AT Air {)1 [5 A28 -e AZ-er +A2]dr T t AT A T A(T-Tl) A T + f [5 W28 -6 wze l T " — -1T sx — A < 1 >3: AT AT(T-Tl) - e— W E— —2 T §(T-Tl) é (T-Tl) \ + 8 A25 ]dT (3-10) I Combining the first term in the first integral of (3-10) with the first term of the second integral, evaluating the remain- ing terms of the first integral, and making the change of variable s='r-Tl in the remaining terms of the second integral (3-10) becomes t AT ATI -1 ATl W I e w e dT-A ( -I)W —2 — — —2 0 AT -1 — T - E2[A (E -l)] HflzTl It-T as 5T5 §TT1 T _ -l - l e E 8 ds 8 -l _ §X(t) — A < 0 2 >5 (3 11) an m As .198 - e f l e fizz ds 0 t-T 5 5T5 + f l e E25 ds L 0 J Defining, t a a% y = lim f e EZE dv tmw () T (3-12) t-T 5V 5 V = lim I l E E26 dv t+00 (3 which satisfies the Lyapunov equation _ T I ‘82 - A X + Z a (3-13) and substituting (3-12) and (3-13) into (3-11), §X(m) becomes \ 3’ P3 __1 2+EH€11H52+2§> AT —1 + (A y + y A?)[A 1(6— 1-;)1T 1T gxm = A < T >A (3-14) T 5 T1 -@y+y§>q-ye AT - €_ 1 V + V K — _ x 44 Using the series expansion 5 = Z _i_ 5 (3-15) equation (3-14) may be written as 3 ’ ooT \ v + [ Z —% An'l] [A v + v AT] _. n.— —— —— n=1 T m T1n n-l T 1 + [A Y + Y a 1 I 21 jfr'é; 1 1T _ - n= - _X<)-A< >5 -a m T n n T m T n n ' Y[ Z Tfi’ A 1 ‘ [ Z 75— 5 12 + Y n=0 ' n=0 ° \ / (3-16) which, after cancellation of terms, may be written as n 0° T T (w) = Z —%— (An-2V + v A“’2 ) (3-17) —X n=2 1" _’ — - ’ If the fault clearing time, T1’ is sufficiently short, only the first term in the series will be required, and under this assumption (w) = 3 T12 (3-18) Since 2 is the solution to the Lyapunov equation (3-13), 2 will depend on system structure due to the dependence of the solution on A and A and on the statistics of the pulse dis- turbance (32 and m2) and the same conclusion will hold for §x(m) and the rms coherency measure. 45 Step Input Disturbances For step inputs, an explicit formula will be derived re- lating the rms coherency measure to system structure and the statistics of the step input. Substituting :Q’R =0 (3’19) n=1:\_7x(0)=_grlfl _2 _ 2 into (2-26) the desired expression for Ax(t) is AV 1 T — T g (t) =.EOIT1 [IOT e— AVdv B] ][Rl+mlmlT ] [£)€ dv A] dT AV AV T + i It [fTe— dv B] [R +m m T] [ITE— dv B] dT t — —1 —1—1 — T 0 0 1 (3-20) Defining _ T T _ El §[Bl+rfllr_“_l “.3. (3 21) and evaluating the interior integrals of (3-20) using (3—8) 1 -l t AT ATT AT ATT Sx(t) = E A fl) [a Ale -6 W1- file +fl1]dTA -1T (3-22) As t approaches infinity, the first three terms in the time averaged integral (3-22) vanish, since the system model is asymptotically stable, leaving T s (00) = A [TA-l = [5'1 g] [R +m [g'lng (3-23) 1 ml T] For the form of A given by (2-10), A.1 becomes -o(g T)'1 -(g T)'1 A = <3-24) ; o and using the form for A also given in (2-10) 46 1 w 9%. 494. TF1»: a g’ g = (3-25) 0 o For simplicity, step disturbances in mechanical input power and load bus power injection will be considered separately. For step disturbances in mechanical input power, A and A as 1 1 defined by (2-19) become m R 0 T1 = —11 I 31 = -11 — (3-26) 9 9 9 where 911 is the mean value of the step disturbance in mechan- ical input power and All is the covariance of the disturbance. Substituting (3-26) and (3-25) into (3-23), Ax(w) for step disturbances in mechanical input power is given by -l T -l T SM): HEP wigfigflnllml)fll 9 ’X 9 9. (3-27) For step disturbances in load bus power injection, m1 and Al become 9 9 9 m = , R = (3‘28) 12 9 512 Substituting (3—28) and (3-25) into (3-23), §x(m) for step disturbances in load bus power injection becomes -1 [(M T)-1M L] [R12+m12m12T] [(M T) M L]T o S (m) = ’ — ’ ’ ’ ‘ - — “ — - — ‘X 9 9 (3-29) Thus, for step input disturbances, an explicit formula has been derived which relates the rms coherency measure, through its dependence on Ax, to the parameters of the linear system 47 model and the statistics of the disturbance, eliminating the need for simulation to determine the coherency measure as is required to determine the max-min coherency measure. It has been shown that the matrix Ax(w), which defines the infinite interval rms coherency measure, can be related algebraically to system structure and disturbance statistics. For random initial conditions and the pulse type disturbance, the determination of Ax(w) was shown to entail the solution of Lyapunov equations. For step inputs, an explicit formula for Ax(m) was derived. In the next section, the significance of the relationship between the rms coherency measure and system structure for step disturbances in mechanical input power is discussed. 3.2 Disturbance-Independent Coherent Equivalents Derived from the RMS Coherency Measure The relationship between the infinite interval rms coher- ency measure and dynamic system structure is now discussed for the special case of step disturbances in mechanical in- put power. It will be shown that if the step input is a ZMIID disturbance then the rms measure is determined strictly by the synchronizing torque coefficients of the linear system model. When the disturbance in mechanical input power at each generator bus is zero mean, independent of the distur- bance at every other generator bus, and has a variance pro- portional to the square of the generator inertia, it is shown that the rms coherency measure is dependent solely on the inertially weighted synchronizing torque coefficients. These 48 results indicate that coherent equivalents which depend on system structure, and not on the location of any particular system disturbance, may be derived by using one of the above disturbances and the rms coherency measure to identify the coherent groups for aggregation. Ax(w) and Dynamic System Structure The significance of Ax(w) for step disturbances in me- chanical input power is now discussed. From equation (3—27), the upper left quadrant of Ax(m) which determines the coher- ency measure between any pair of generators is defined as Ax(w) and is given by A §X(w)==[(g T)’lg] [ 1 +R T ElllflllT ~11] [(M Z)- If] (3'30) This expression is valid for both probabilistic and determi— nistic disturbances. For probabilistic disturbances _ . — _ — T — Ell — E[APM], Rll — E[(APM mll)(APM Ell) ] (3 31) and for deterministic disturbances T11 —11 As noted previously, equation (3-30) eliminates the need for simulation to evaluate the coherency measure as is required in the max-min approach. The coherency aggregation procedure has been criticized because the equivalents derived using the max-min coherency measure are dependent on the disturbance used to identify co- herent groups for aggregation. Inspection of equation (3-30) shows that the rms coherency measure will be a function of only system structure for any disturbance which satisfies 49 T _ _. T111311 + 311 " LI- (3 33) This condition is clearly met when the disturbance in APM is ZMIID, that is T11 = 9’ 511 = I (3'34) Substituting (3-33) into (3-30) 1 T s (w) = [(g T)'lg] [(g T)’ g] (3-35) —x Since ”A 9'19. T. = .1. 0-36) equation (3-35) shows that when the disturbance in mechanical input power is ZMIID that the rms coherency measure is a gen- eralized inverse function of synchronizing torque coefficients, such that the coherent groups are determined by line stiffness. Another disturbance of interest is the disturbance which causes T T _ _ l31313111311 +511”? ’ l (3 37) in which case (w) = [(g T)’l] [ = [<8 :>’181 [ Z 499k 429k 1 [<9 1>'181T <6-5> k=l N A = Z §:(w) k=l where, “k -1 k kT -1 T §x(m) = [(M I) E] [93% ARE 1 [(M I) 51 (6'6) The significance of equations (6-5) and (6-6) is that the A §x(w) matrix for the modal disturbance can be constructed by summing a sequence of Ex matrices, {é:(w) : k=l,2,...,N}, which corresponds to the sequence of N deterministic step dis- turbances, {APMR : k=l,2,...,N}, where each generator, in turn, is subjected to a disturbance in mechanical input power proportional to its inertia. Alternatively, it can be said that the expected value of the rms coherency measure for the modal disturbance is equal to the square root of the sum of the squares of the rms measures computed for the N deterministic 78 disturbances, {APMk}. The sequence of disturbances defined by (6-4) is referred to as the "modal disturbance sequence". Steady State Generator Angles and the Infinite Interval RMS CoherencyiMeasure It will now be shown that for any deterministic step input disturbance uk(t) given by uk(t) g for t< O k (6-7) k APM u = k for t220 APL that the matrix §:(W) depends solely on the steady state gen- erator angles exhibited by the linear system model in response to the step input. This result will allow the rms coherency measure for the modal disturbance to be computed from the steady state angle response of the system generators to each of the disturbances in the modal disturbance sequence. For the deterministic step input disturbance 33(t) given by (6-7) the matrix, §:(w) is by definition k l T §X(w) = lim T [T xk(t) Ek (t) dt (6-8) T+w 0 where xk(t) is the system state vector and the solution to the state equation for the step disturbance uk(t). If the linear system model is asymptotically stable and the magni- tude of each of the step inputs in the input vector, Bk, is bounded, then the system states will be finite for all time and will eventually converge to some finite steady state values, xk(w). Recognizing that the entries in the matrix integrand of equation (6-8) are well-behaved, finite functions of time, each of which converges in time to some constant 79 - value governed by xk(W), §:(w) may be written as T §:(w> = 33(9) 3k (w) (6-9) ’By definition, g_ E = A (6-10) Ag therefore, 7 _ k A_ = g§k(m) Agk (9) (6-12) Thus, the rms coherency measure evaluated over an infinite interval for any deterministic step input disturbance is de- termined by the steady state angle response of the generators to that disturbance. It can easily be shown that equations (6-6) and (6-12) are consistent expressions for §k(m) for the disturbances in the modal disturbance sequence. Letting 932k equal zero in (6-7) the disturbance 23 becomes the kth disturbance in the modal disturbance sequence when Agggijsdefined by (6-4). Setting the derivative of the state vector equal to zero as t approaches infinity in the state equation (2-9), §k(w) becomes k k -1 APM 2:. (co) = —A E (6-13) 0 80 Substituting (3-25) for A-1 E into (6-13) and extracting 93kt») from §k(°°) A0k(m) = (M T)’1M APMk (6-14) Substituting (6-14) into (6-12) equation (6-6) is obtained confirming that §:(w) depends on é§k(w). Equation (6-12) will be shown to be a convenient form for computing the infinite interval rms coherency measure for the modal disturbance. Computation of Steady State Generator Angles for a Step Disturbance The steady state generator angles required to calculate §X(w) for any step input, uk(t), can be efficiently computed using a triangular factorization technique to solve the sys- tem network equations (2-6), at time equal to infinity for 48k (w). The key to the approach is to show that the steady state deviation in the real power generations due to the step input, ég§k(w), which are required to set up the steady state network equations for solution, can be deduced from the en- tries in the disturbance vector and knowledge of the genera- tor inertias and damping constants. The procedures for con- structing éEEk‘m) from 23 and solving the steady state network equations for A§k(w) are now discussed. Prior to the occurrence of the step input disturbance, it is assumed that the system is in a power balance. The sum of the real powers generated by the system generators exactly balances the sum of the powers demanded at the load buses (neglecting line losses). The total mechanical input power to the system is just sufficient to maintain the balance 81 between generated and demanded powers and to allow the gen- erators to operate at the synchronous speed, mo. After the step disturbance uk(t) is applied, the power balance between generation and demand may be temporarily upset. However, if the linear system model is stable then as t approaches infinity the balance between real power gen- eration and real power demanded at the load buses must be restored. Thus, if there are no losses in the network 3] APG}:(°°) = - If APL]? (6-15) i=1 j=l 3 where APG:(w) is the steady state deviation in the electrical output power produced by generator i in response to the dis- turbance uk(t), and APL? is the change in the power demanded at load bus j (a negative value implies an increase in load) which is specified by the disturbance uk(t). If there is an excess (deficiency) of mechanical input power in the step in- put disturbance uk which is not balanced by a change in load, then the system generators will accelerate (decelerate) to some new system speed, wo + Awk, but remain synchronous. Each generator in the system is modelled by the linear, classical synchronous machine representation Mi §% Aw§ 0.4 N + 9 or, N + 22.5 32.5 > For a system containing 250 generators, the construction of the modal-coherent equivalent would be competitive with the construction of 9 coherent equivalents. It may be possible to reduce the number of disturbances in the modal disturbance sequence while retaining the essential 102 character of the modal-coherent equivalent. One approach would be to limit the disturbance sequence to include only those disturbances which correspond to generators in the internal system and a few external system generators which are first or second neighbors to the internal system. Such an approach would attempt to preserve the modal-coherent structure only in the internal system and in a boundary re- gion surrounding the internal system. The underlying assumption is that the dynamics arising from the remote re- gions of the external system have little impact on the behavior of the internal system and therefore accurate aggre- gation of the remote external system is not necessary. Since this method would significantly reduce the computational effort required to construct a modal-coherent equivalent, further research is needed to determine the validity of the approach. 6.3 Computational Examples The algorithm for evaluating the infinite interval rms coherency measure and identifying structurally coherent groups is now illustrated for two example systems. The first example illustrates the procedure described in Section 6.1 for the case when the load buses have not been eliminated from the network equations. Example number two demonstrates the pro- cedure when the network reduction necessary to eliminate the load buses has been performed and the synchronizing torque coefficients between internal generator buses are known. The second example system was treated previously in Chapter 4 103 where the coherency measure for the modal disturbance was determined by constructing the linear model matrix (M g) and computing (M g)-1. Example System 1 A schematic diagram of a 3 generator, 5 bus system is shown in Figure 6-1. The system data necessary to compute the rms coherency measure for the modal disturbance is given in the figure. generator 1 generator 2 M1=1,Dl=.l M2=2,D2=.2 bus 4 l/0° 1/0° bus 1 r— * , bus 2 Yl4-6-4 y24-6.4 y25=8.0 bus 3 l/0° .1___._ bus 5 1/00 3 3 D1 D2 D3 generator 3 O=M_=M—=M—= .l l 2 3 Figure 6-1 Three Generator Example System 104 The step by step procedure given in Section 6.1 to evaluate the rms coherency measure will be followed. 1. The synchronous frame network equations may be written as r— - r- '1;- fl APL4 20.48 0.00 -6.40 ’6.40 -7.68 A04 APL5 0.00 10.00 0.00 '8.00 ’2.00 A05 APGl -6.40 0.00 6.40 0.00 0.00 A01 APG2 -6.40 '8.00 0.00 14.40 0.00 A02 APG3 -7.68 -2.00 0.00 0.00 9.68 A53 L. ..J _ JL. ..- Using generator 3 as the reference generator, the network equations in the generator 3 reference frame may be written as r- - r- —-- APL4 20.48 0.00 -6.40 -6.40 A84 - A031 APL5 0.00 10.00 0.00 -8.00 A85 - A63 APGl = -6.40 0.00 6.40 0.00 A61 - A63 APG2 -6.40 -8.00 0.00 14.40 A62 - A63 L. J L. J).— ...J which is of the form fi=ié The triangular factorization of i may be accomplished by using the equations [15] ’L . Vi1 - Jil// 3‘11 1315N+K-l % j_1 . . . = — < .- Vij (Jij kgl Vikvjk)/ij 1< j< 1..N4-K 1 i=1 2 v,,=(J,.- 2 v. )2 l<1sN+K-l 11 11 1k k=l (6-34) to obtain r“4.525483 0.000000 0.000000 0.000000"1 m 0.000000 3.612278 0.000000 0.000000 2 — -1.4l4214 0.000000 2.097617 0.000000 L_:-l.4l4214 -2.529822 -0.953463 2.256304J 105 2. The modal disturbance sequence for this system is APMl = (1,0,0)T, APM2 = <0,2,0)T, APM3 = (0,0,4)T 3. Using equations (6-22b), the APGk(w) vectors corre- sponding to the modal disturbance sequence may be written as 9391(w) = (0.857143,—0.285714,-0.571429)T 9392(w) = (-0.285714,1.428571,-1.142857)T Agg3(w) = (-0.571429,-1.142857,1.714286)T 4. Recognizing that éggk = Q for each disturbance in the modal disturbance sequence, the vectors é§k(w) needed to solve the steady state network equations may be written as 931(m) = (0.000000,0.000000,0.857143,-0.285714)T 932(w) gg3>|>>|c>>u>>|>> U'l (.01627,-.00034,-.00013,-.00333,-.00203,-.00192)T (-.00123,.00962,-.00073,-.00124,-.00080,-.00079)T (-.00117,-.00065,.00597,.00006,.00004,.00010)T (-.00222,-.00117,.00052,.01085,.00035,.00042)T (-.00264,-.00140,-.00039,-.00053,.00621,.00072)T 112 (-.00254,-.00132,-.00025,-.00040,.00078,.00608)T |I>>|t>> (-.00646,-.00474,-.00498,-.00542,-.00454,-.00460)T A The steady state generator angles are the Ak given above. 6. The matrix §X(w) can be computed as A An A §_X(oo) = _-.32766 .02369 .02448 -.03941 -.02204 -.01895_- .02369 .02072 .01301 .00332 .00427 .00518 10_3 .02448 .01301 .06143 .03462 .02121 .02268 -.03941 .00332 .03462 .16016 .03246 .03404 -.02204 .00427 .02121 .03246 .06469 .03471 _:.01895 .00518 .02268 .03404 .03471 .06309 A 7. The rms coherency measures can be found from §x using (2-24). The values obtained are given in the ranking table, Table 6-2. In the following discussion it will be found convenient to represent the m generator coherent group consisting of generators n1,n2,..., and nm as (nl,n ..,nm). The data ob- 2,. tained from the MECS model for the modal disturbance is now discussed. It was stated in Section 6.2 that the coherent groups identified using the generator clustering algorithm, might be sensitive to the arbitrary choice of indices assigned to the system generators. This can be demonstrated using the data obtained with the MECS model. Let the threshold of coherency, Ec' be .0795 and use the clustering algorithm as described in Section 6.2 to identify coherent groups based on the rms 113 Table 6-2 Ranking Table for the Modal Disturbance, MECS Example System rank r kr'jlr Ck ,2 ____ r r 1 5-6 .007639 2 3-7 .007838 3 6-7 .007943 4 - .008043 5 3-6 .008897 6 3-5 .009149 7 2—7 .010987 8 3-4 .012343 9 4-6 .012457 10 2-3 .012495 11 4-5 .012646 12 4-7 .012656 13 2-6 .013170 14 2-5 .013299 15 2-4 .016560 16 l-7 .018101 17 1-3 .018443 18 1-2 .020025 19 1-6 .020704 20 1-5 .020891 21 1-4 .023804 114 coherency measures given in Table 6-2. Let the generators be processed by the clustering algorithm according to the generator numbering system adopted in this thesis, that is in the order l,2,3,4,5,6,7. The resulting coherent equivalent would be a 6-generator equivalent model consisting of gen- erators l,2,3,4,7 and an equivalent generator representing the coherent group (5,6). If the clustering algorithm pro- cessed the generators in reverse order, that is 7,6,5,4,3,2,l corresponding to a simple reassignment of the generator indices, then the coherent equivalent defined by the same coherency threshold would be a 5-generator model consisting of generators 1,2,4,5 and an equivalent generator represent- ing the coherent group (3,6,7). Thus, the coherent groups identified by the generator clustering algorithm are depen- dent not only on the selected coherency threshold but also on the arbitrary assignment of indices to the system generators. Based on the results of the ranking table, Table 6-2, a modal-coherent equivalent for the 7-generator MECS model can be identified which retains any predetermined number of gen- erators between 1 and 7. This can be accomplished using the commutative or the transitive rules described in Section 6.1. Table 6-3 shows the coherent groups identified as the commu- tative rule progresses through the ranking table. At each rank, r, the table indicates the coherent groups identified through that rank, the corresponding number of generators eliminated through that rank, ne(r), and the number of gen- erators which remain in the modal-coherent equivalent if the 115 Table 6-3 Coherent Group Identification Based on the Ranking Table and the Commutative Rule, MECS Example System- Modal Disturbance number of generators retained number of in the generators equivalent coherent groups eliminated identified rank identified through rank r at rank r r kr’gr through rank r ne(r) 7-ne(r) 1 5-6 (5,6) l 6 2 3-7 (5,6),(3-7) 2 5 3 6-7 (5,6),(3-7) 2 5 4 5-7 (5,6),(3-7) 2 5 5 - (5,6),(3-7) 2 5 6 3-5 (3,5,6,7) 3 4 7 2-7 (3,5,6,7) 3 4 8 3-4 (3,5,6,7) 3 4 9 4-6 (3,5,6,7) 3 4 10 2-3 (3,5,6,7) 3 4 11 4-5 (3,5,6,7) 3 4 12 4-7 (3,4,5,6,7) 4 3 13 2-6 (3,4,5,6,7) 4 3 14 2-5 (3,4,5,6,7) 4 3 15 2-4 (2,3,4,5,6,7) 5 2 16 1-7 (2,3,4,5,6,7) 5 2 17 1-3 (2,3,4,5,6,7) 5 2 18 1-2 (2,3,4,5,6,7) 5 2 19 1-6 (2,3,4,5,6,7) 5 2 20 1-5 (2,3,4,5,6,7) 5 2 116 Table 6-4 Coherent Group Identification Based on the Ranking Table and the Transitive Rule MECS Example System- Modal Disturbance number of generators retained number of in the generators equivalent coherent groups eliminated identified rank identified through rank r at rank r r kr'Qr through rank r ne(r) 7-ne(r) 1 5-6 (5,6) 1 6 2 3-7 (5,6),(3-7) 2 5 3 6-7 (3,5,6,7) 3 4 4 5-7 (3,5,6,7) 3 4 5 - (3,5,6,7) 3 4 6 3-5 (3,5,6,7) 3 4 7 2-7 (2,3,5,6,7) 4 3 8 3-4 (2,3,5,6,7) 4 3 9 - (2,3,4,5,6,7) 5 2 10 — (2,3,4,5,6,7) 5 2 ll - (2,3,4,5,6,7) 5 2 12 - (2,3,4,5,6,7) 5 2 13 - (2,3,4,5,6,7) 5 2 l4 - (2,3,4,5,6,7) 5 2 15 - (2,3,4,5,6,7) 5 2 l6 - l7 - 18 - l9 - 20 - N f.- | 117 group identification process were to terminate at rank r. Table 6-4 gives the same information for the transitive rule. The following observations are based on Tables 6-3 and 6-4. 1. The transitive rule tends to reduce the number of generators retained in the equivalent faster, that is at a lower rank, than the commutative rule. This result is not unexpected since more ranks must be examined to merge coherent groups under a commutative rule than a transitive rule. 2. The modal-coherent equivalents retaining 6,5,4 and 2 generators defined by the commutative and transitive rules are identical. 3. The two rules disagree on the coherent group which defines the 3-generator version of the modal- coherent equivalent. Corresponding to a 3-generator equivalent, the commutative rule identifies the co- herent group (3,4,5,6,7) at rank 12 in Table 6-3 whereas the transitive rule identifies the group (2,3,5,6,7) at rank 7 in Table 6-4. Each rule pre- viously identified a 4-generator equivalent con- taining the single coherent group (3,5,6,7). Since generators 2 and 7 are found to be a coherent pair at rank 7 in the ranking table, the transitive rule immediately merges generator 2 with the coherent group C3,5,6,7). The commutative rule also recognizes generators 2 and 7 as a coherent pair at rank 7, but must wait until generator 2 is also found to be cc- herent with generators 3,5 and 6 as well, before generator 2 can join the group (3,5,6,7). The com- mutative rule proceeds through the ranking table from rank 7, and generator 4 is found to be coherent with each member of (3,5,6,7) before generator 2 can meet that requirement. Thus, the commutative rule joins 118 generator 4 to the group (3,5,6,7) instead of gen- erator 2 as observed with the transitive rule. The disagreement over the coherent group which defines the 3-generator modal-coherent equivalent for the MECS model points out a fundamental differ- ence in the character of coherent groups which are likely to be identified by the transitive and com- mutative rules. The synchronizing torque coeffi- cient matrix for the MECS model given in step 1 shows that the synchronizing torque coefficients of the equivalent lines which connect generator 2 to the generators in the coherent group (3,5,6,7) are T23 = 1.82 T25 = 1.62 T26 = 1.73 T27 = 6.32 Since generators 3,5,6 and 7 have essentially the same inertia, the torque coefficients determine the relative stiffness of the connection between generator 2 and each member of the group (3,5,6,7). The synchronizing torque coefficients clearly in- dicate that generator 2 is tightly coupled to gen- erator 7 but not to the remaining generators in the group. Thus, the use of the transitive rule would be likely to identify coherent groups which contain "weak" members, that is a generator which is stiffly connected to just one member in the co- herent group. The commutative rule avoids this problem by requiring that a generator be coherent with all members of a coherent group before that generator can join the group. 4. In Chapter 4, the ZMIID disturbance identified a 4-generator equivalent of the MECS model contain- ing the single coherent group (3,5,6,7). Both the commutative and the transitive rules applied to the 119 ranking table cflf the modal disturbance identify the same 4-generator equivalent. Thus it is confirmed that there is no difference in the co- herent groups identified by the ZMIID and modal disturbances for the 4-generator modal-coherent equivalent of the MECS model. 5. The 3-generator version of the modal-coherent equivalent defined by the commutative rule in Table 6-3 contains the single coherent group (3,4,5,6,7). The same coherent group was aggre- gated in Chapter 4 to construct the coherent equivalent based on the UD disturbance. Thus, the 3-generator modal-coherent equivalent of the MECS model is the same equivalent as the "UD coherent equivalent" in Chapter 4. The perform- ance of the UD equivalent in preserving the rms coherency measure for various system disturbances was shown to be quite poor. It was also indicated that the eigenvalues of the UD coherent equiva- lent did not match the system eigenvalues retain- ed by the modal equivalent based on the UD disturbance. The fact that the 3-generator co- herent equivalents for the MECS model based on the modal and UD disturbances are identical is purely a coincidence, however, it does indicate that even if the modal-coherent approach is used to identify coherent groups there is the danger of over-aggregating and destroying the modal and coherent structure of the unreduced system. Intuitively, the less model aggregation that is done to construct a dynamic equivalent, the more accurate the equiva- lent should be in predicting the behavior of the unreduced system model. An indication of the gradual degradation of the modal structure of the system that occurs as the number 120 of generators retained by the modal-coherent equivalent de- creases, is shown in Table 6-5. Table 6-5 compares the modes of intermachine oscillations in the unreduced linear system model with the modes of oscillation computed for the linearized 6,5,4,3 and 2-generator versions of the modal- coherent equivalent defined by the ranking table and the commutative rule as shown in Table 6-3. Except for the 4- generator and 3-generator versions, the modes corresponding to the equivalents in Table 6-5 were computed from linear models obtained by directly aggregating the unreduced linear MECS model using the procedure for linear model aggregation described in Section 2.4. As indicated earlier in the dis- cussion, linear models of the 4-generator and 3-generator equivalents (and their modes) were previously derived in Chapter 4. The following comments are based on the results in Table 6-5. 1. The commutative rule identifies generators 5 and 6, at rank 1 in Table 6-3, as the most coher- ent pair of generators in the MECS model. The aggregation of these two generators to form the 6-generator modal-coherent equivalent removes the oscillation at 13.614 rad/sec observed in the unreduced system. Apparently, the oscillation at 13.614 rad/sec is associated exclusively with the intermachine behavior of generators 5 and 6 since the remaining modes of oscillation observed in the unreduced model are closely (and almost exactly) preserved in the 6-generator modal- coherent equivalent. 121 Table 6-5 Modes of Intermachine Oscillation for Modal- Coherent Equivalents Derived from the Commutative Rule MODES OF INTERMACHINE OSCILLATIONS (rad/sec) number of generators retained in the modal-coherent equivalent unreduced system 9 2 4 .3. 2 7.415 7.414 7.226 7.288 6.653 8.680 9.481 9.481 9.432 9.351 10.176 10.389 10.391 10.201 10.659 12.756 12.761 13.614 13.729 14.304 14.309 122 2. The S-generator modal-coherent equivalent is con- structed by aggregating two coherent groups, (5,6) and (3,7). The construction of the 5-generator equiva- lent may be viewed as a two step aggregation process in the unreduced system or a one step aggregation (of generators 3 and 7) in the 6-generator modal-coherent equivalent model. Taking the later view, the changes in the modes of oscillation from the 6-generator to the S-generator equivalent indicate that the modes at 12.761 and 14.309 rad/sec in the 6-generator equivalent are associated with the intermachine oscillations of generators 3 and 7, and the group to group oscillations of (5,6) and (3,7). When generators 3 and 7 in the 6- generator model are aggregated to form the 5-generator modal-coherent equivalent, the modes at 12.761 and 14.309 rad/sec are replaced by a "new" mode at 13.729 rad/sec. It is emphasized that the new mode at 13.729 rad/sec in the 5-generator equivalent does not corres- pond to the oscillation at 13.614 rad/sec in the unreduced model. Rather, the new mode represents the effort by the coherency aggregation procedure to com- pensate in the S-generator equivalent for the loss of one degree of freedom in representing the group to group behavior of (5,6) and (3,7), which was determined by two modes in the 6-generator equivalent and must be represented by a single "average" mode in the 5-gener- ator equivalent. Notice also that the remaining modes in the 5-generator equivalent have shifted below the corresponding modes in the 6-generator equivalent. 3. The 4-generator modal-coherent equivalent can be constructed by aggregating the generators in the 5- generator equivalent model which represent (5,6) and (3,7). The transition of the modes of the 5-generator equivalent to the modes of the 4-generator equivalent indicates that the mode at 13.729 in the 5-generator equivalent, the new mode, is strongly associated with 123 the group to group oscillations of (5,6) and (3,7), since the aggregation of (5,6) and (3,7) to form the 4-generator equivalent effectively eliminates that mode. The modes which remain in the 4-generator equivalent correspond quite well with the low frequency (10.389 rad/sec and below) modes in the unreduced model. This also indicates that the high frequency modes 12.756, 13.614 and 14.304 of the unreduced model are all associated with the intermachine behavior of the coherent group (3,5,6,7). 4. In each of the transitions from the 6-generator to the 4-generator equivalent there is a slight but steady deterioration of the agreement between the low frequency modes of the modal-coherent equiva- lents and the corresponding modes of the unreduced system. 5. The modes of the 3- and 2-generator equivalents bear little resemblance to any of the modes of the unreduced system model. The above observations indicate that the user of modal- coherent equivalents should not specify unnecessarily "small" equivalents. Unfortunately, there isn't an obvious way to tell whether or not a dynamic equivalent is over-aggregated. However, for the MECS model, the ranking table approach for identifying coherent groups does provide some indication that there might be a reasonably dramatic drop in performance be- tween the 4-generator and 3-generator modal-coherent equivalents. Tables 6-3 and 6-4 indicate that for both the commutative and transitive rules that large jumps occur in the ranking table between the rank at which the 4-generator and 3-generator equivalents are identified. This suggests that the ranking table approach for identifying coherent 124 groups may be able to provide information to suggest where the model aggregation process should be cut off in order to guarantee that modal and coherent system structure are ade- quately preserved. Further research is planned to investi- gate the uses of the ranking table in identifying coherent groups and in controlling the amount of model aggregation for large scale systems. Summary of Chapter 6 In this chapter an efficient algorithm for constructing modal-coherent equivalents for large scale systems was developed. The feasibility of the algorithm was confirmed by applying it to two relatively small example systems. Pro- gramming of the algorithm for large scale systems is already underway. CHAPTER 7 CONCLUSIONS AND FUTURE INVESTIGATIONS The major results of this thesis are now summarized on a chapter by chapter basis and related topics for future research are proposed. 7.1 Thesis Review In the first chapter, the properties of power system dynamic equivalents derived from the present modal and coher- ency analysis approaches are discussed. Both approaches are indicated to have considerable merit and at the same time significant drawbacks. The criticisms that have limited the acceptance of these two techniques are used to deduce the properties of an "ideal" dynamic equivalent which would be found highly desirable by the power industry. It is argued that an ideal equivalent should be 1. Suitable for studying any contingency that might occur inside the study area for which the equivalent is derived. 2. Simultaneously a modal and a coherent equivalent. 3. Derived from an efficient computational technique. 4. Expressed in terms of normal power system components. The above properties are a composite of the most desirable features of present modal and coherent equivalents. This 125 126 suggests that it may be possible to develop an ideal equiva- lent by properly linking the modal and coherency analysis techniques. A historical perspective for this research is provided by briefly reviewing the modal analysis approach based on the rms coherency measure [8,9], the so-called coherent-modal approach, which was an early effort to link modal and coherency analysis. It is indicated that the coherent-modal approach does not lead directly to an ideal dynamic equivalent but it does suggest that the rms coherency measure is the key to the development of a nearly ideal dy- namic equivalent. The first chapter closes with a statement of the thesis objective which is to develop the justification and the means for constructing a modal-coherent equivalent whose properties closely approximate those of an ideal dynamic equivalent. In the second chapter the mathematical models used in the development of the modal-coherent equivalent are defined. The justification for the modal-coherent approach is de- veloped in Chapters 3 and 4. In Chapter 3, the expected value of the rms coherency measure, evaluated over an infinite interval, is algebraically related to the parameters of the power system state model and the statistics of the system disturbance. For random initial conditions and pulse type disturbances (faults), an implicit relationship is developed which, in each case, takes the form of a Lyapunov equation. For step disturbances in mechanical input power or load bus power injection, an explicit formula is derived, relating the rms measure directly to system parameters and the statistics 127 of the step disturbance. Two probabilistic step disturbances, the ZMIID disturbance and the modal disturbance, are shown to cause the rms coherency measure to depend solely on system parameters. Coherent groups based on the ZMIID disturbance are shown to depend strictly on line stiffnesses, while the coherent groups determined by the modal disturbance are deter- mined by relative line stiffnesses. Using an example system,iJ:is shown in Chapter 4 that the eigenvalues of the coherent equivalent constructed by aggre- gating the coherent groups identified by the rms coherency measure and the ZMIID disturbance closely approximate the sys- tem eigenvalues retained by the modal equivalent based on the same coherency measure and disturbance. It is further shown that both the modal and the coherent equivalent based on the ZMIID disturbance are useful for studying any disturbance that might occur outside the areas of the example system ag- gregated to form these equivalents. These results indicate that a coherent equivalent which closely approximates a gen- eral purpose modal equivalent can be derived when the rms co- herency measure and an appropriate probabilistic disturbance are used to identify coherent groups. The example system is a special case composed of many generating units with similar inertias and there is little difference between line stiff- nesses and relative line stiffnesses. Consequently, the same set of coherent groups are identified using either the ZMIID or the modal disturbance. In general, this would not be the case and to identify a general purpose dynamic equivalent the modal disturbance would normally be preferred over the ZMIID 128 disturbance since the coherent groups identified by the modal disturbance are determined by relative line stiffnesses which are a more complete description of dynamic system structure than line stiffnesses which are the basis for aggregation when the ZMIID disturbance is used to identify coherent groups. Chapter 5 proposes that a modal-coherent equivalent can be derived by usingtflmacoherency-based aggregation technique to aggregate the coherent groups identified by the rms coher- ency measure and the modal disturbance. The justification for the approach rests on the analytical developments in Chapter 3 and the observations based on the example system in Chapter 4 which indicate that the rms coherency measure and the modal disturbance can identify coherent groups which prop- erly reflect dynamic system structure. The construction of the modal-coherent equivalent is a 3 step procedure which includes the evaluation of the expected value of the rms coherency measure for the probabilistic modal disturbance, the identification of coherent groups based on the computed coherency measure and finally, the aggregation of the coherent groups using the coherency-based aggregation technique [2]. Since the coherency-based aggregation techni- que is well established, the problem of developing the means for constructing the modal-coherent equivalent reduces to finding a procedure for computing the rms coherency measure and identifying coherent groups. In Chapter 6, an efficient computational algorithm, ap- plicable to large scale systems, is developed for computing the rms coherency measure for the modal disturbance. The 129 algorithm expands the coherency measure for the probabilis- tic modal disturbance in terms of the coherency measures observed for a sequence of deterministic step disturbances known as the modal disturbance sequence. For each distur- bance in the modal disturbance sequence the coherency mea- sure is shown to depend strictly on the steady state response of the generator angles to that disturbance. It is shown that steady state generator angles for any determinis- tic step disturbance can be efficiently computed from the steady state network equations using a triangular factoriza- tion technique. The coherent groups identified by the generator cluster- ing algorithm, which is presently used to identify coherent groups based on the max-min coherency measure, were shown to be sensitive to the arbitrary order in which generators may be processed by the algorithm. To eliminate this sensitiv- ity, a new approach for identifying coherent groups was pro- posed based on a ranking table in which the N(N-l)/2 possible coherency measures between the various pairs of generators are ordered from the smallest to the largest. Coherent groups are identified using the ranking table by proceeding down the ranks in the table and using either a transitive or a commu- tative coherency rule to assign generators to coherent groups. The ranking table approach has an additional advantage over the generator clustering algorithm since the size of the de- sired equivalent can be specified a priori and coherent groups may be identified to conform to the prespecified number of generators to be retained by the equivalent. 130 A rough comparison was drawn between the computational effort required to construct the general purpose modal- coherent equivalent and the effort required to construct a coherent equivalent based on the max-min coherency measure. It was shown that the computer time needed to construct the modal-coherent equivalent was approximately equal to the total time required to construct nine coherent equivalents, based on a system containing 250 generators. Thus, the modal-coherent approach will be computationally competitive with the coherency analysis approach based on the max-min coherency measure when a relatively modest number of distur- bances are to be examined in a transient stability study. 7.2 Future Research Based on the developments in the first six chapters it is concluded that the modal-coherent approach to power sys- tem dynamic equivalents represents a viable alternative to the present modal and coherency analysis techniques. However, further research is required to fully explore the relative merits of the approach. Some areas where further investiga— tion is indicated are now discussed. The procedure for evaluating the rms coherency measure for the probabilistic modal disturbance must be programmed for large scale systems. No major difficulties are expected in this task since the triangular factorization technique which is the heart of the computational procedure is an often used tool in power system analysis. This work is already underway. 131 There is also a need to investigate the suitability of the generator clustering algorithm and the ranking table methods for identifying coherent groups. Based on experi- ence with the MECS example system it is believed that the ranking table procedure using the commutative rule will identify the most meaningful set of coherent groups for use with the modal-coherent approach. The use of the ranking table to limit the amount of aggregation in order to avoid constructing over-aggregated equivalents which do not pre- serve modal or coherent dynamic system structure is also a topic for future investigation. Another item for future research which was briefly des- cribed in Chapter 6 is the possibility of reducing the number of disturbances in the modal disturbance sequence while re- taining the essential character of the modal-coherent equivalent. The computational attractiveness of the modal- coherent approach would be significantly enhanced if it can be shown that only those disturbances corresponding to gener- ators in the internal system and a few generators in the external system which are near neighbors to the internal sys- tem need be included in the modal disturbance sequence in order to preserve the integrity of the modal-coherent equivalent. The performance of the dynamic equivalents derived in this research were judged on their ability to reproduce the coherency measure observed with the unreduced system model. Since these equivalents are intended for use in transient stability studies, there is a definite need to compare the 132 time domain properties of the equivalents derived from the infinite interval rms coherency measure to the properties of equivalents derived from the max-min coherency measure which is evaluated over a short interval. Because the modal- coherent equivalent closely preserves system modes it is ex- pected that the time response observed with the modal- coherent equivalent will closely match the time response of the unreduced system. BI BLIOGRAPHY 10. BIBLIOGRAPHY J. M. Undrill and A. E. Turner, "Power System Equivalents," Final Report on ERC Project RP904, January 1971. "Coherency Based Dynamic Equivalents for Transient Stability Studies," Final Report on EPRI Project RP904, December 1974. W. W. Price, E. M. Gulachenski, P. Kundur, F. J. Lange, G. C. Loehr, B. A. Roth, R. F. 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Modir, "An RMS Coher- ency Measure: A Basis for Unification of Coherency and Modal Analysis Model Aggregation Techniques," 1978 IEEE PES Summer Power Meeting. R. A. Schlueter, U. Ahn, "Modal Analysis Equivalents De- rived Based on the RMS Coherency Measure," 1979 IEEE PES Winter Power Meeting, Paper No. A-79-06l-3. J. Meisel, "Reference Frames and Emergency State Control for Bulk Electric Power Systems," Proceedings of the 1977 Joint Automatic Control Conference, Vol. 2, pp. 747-754. 133 11. 12. 13. 14. 15. 134 "Development of Dynamic Equivalents for Transient Stability Studies," Report on EPRI Project 763, May 1977. A. Germond, R. Podmore, "Dynamic Aggregation of Generat- ing Unit Models," IEEE Trans., Vol. PAS-97, July/August 1978, pp. 1060-1068. D. L. Hackett, "Coherency Based Dynamic Equivelant Application Experience - Eastern U. S. Data Base," 1978 IEEE PES Summer Power Meeting. R. A. Schlueter, J. Preminger, G. L. Park, R. VanWieren, P. A. Rusche, D. L. Hackett, "A Nonlinear Dynamic Model of the Michigan Electrical Coordinated System," 1975 IEEE PES Winter Power Meeting. N. Gastinel, Linear Numerical Analysis, Academic Press, Inc., New York, NY, 1970. General References 16. 17. 18. 19. 20. O. I. Elgerd, Electric Energy Systems Theory; An Introduction, McGraw-Hill, New York, NY, 1971. F. C. Scheweppe, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973. P. Anderson, A. Fouad, Power System Control and Stability, Iowa State University Press, Ames, IA, 1977. R. L. Sullivan, Power System Planning, McGraw-Hill, New York, NY, 1977. P. Anderson, Analysis of Faulted Power Systems, Iowa State University Press, Ames, IA, 1973. MICHIGAN STATE N u H 1 V.L I IBRRRIES WIllHllHlllllllllmll ca (55311 037' I )II IHI (3 312932