)V1ESI_} RETURNING MATERIALS: Place in book drop to LIBRARJES remove this checkout from n your record. FINES will I be charged if book is returned after the date stamped beIow. MOI USE 6‘33? 1“" AL. 3 OPTIMAL AGGREGATION OF ELECTRIC POWER SYSTEM , DYNAMIC MODELS r By Heidar AIi Shayanfar A DISSERTATION Submitted to Michigan State University in partial fquiIIment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of EIectricaT Engineering and Systems Science 1982 ABSTRACT OPTIMAL AGGREGATION OF ELECTRIC POWER SYSTEM DYNAMIC MODELS By Heidar Ali Shayanfar Dynamic equivalents are required in power system analysis and design in order to reduce immense models and data bases down to reason- able sizes. Different types of models are required for simulation of contingencies, design of control systems, and for transmission planning applications. There are at present several methods of producing such equivalents; coherency, modal analysis, and singular perturbation. This research formulates an optimal aggregation problem that determines an optimal aggregation method based on the types of disturbance, the time interval over which the model must be accurate, and the specific appli- cation it will be used for. These optimal aggregations are derived based on the linearized model of the power system which is divided into two parts called the "study system", where the disturbances occur and whose detailed model is of interest and the "external system" whose detailed behavior is of no interest but whose effect on the “study system" must be accounted for. The optimality is derived using a parameter optimization for the reduced model where the performance index J, which is a function of the differences between each pair of generator bus angles in the "study system" Heidar Ali Shayanfar of the aggregated and unaggregated system is minimized. It is shown that the network aggregation method used in the EPRI dynamic equivalents package (Inertial Averaging), where inertia and power injections for the coherent generators to be aggregated are summed, is optimal (produced no error) for any disturbance restricted to the "study system" if the simulation interval is small (T1 = A). It is also shown that a singular perturbation like aggregation called "Optimal-Modal-Co- herent” method produces equivalents that are optimal for any disturbance restricted to the "study system” if the simulation interval is long (T1 = m). The specific applications and advantages for each of these aggregation techniques are also discussed. A computational algorithm extendable to the large scale systems is developed for aggregating each coherent group of gener— ators into a single generator based on this OMC aggregation, assuming that the coherent groups of generators are identified using the RMS coherency measures for a particular type of disturbance. The results derived in this thesis are satisfactorily verified by testing on the 39 bus New England System. To my Parents, my Wife, and my Son 11' ACKNOWLEDGMENTS I sincerely wish to express my thanks and appreciation to my major advisor, professor R.A. Schlueter, for his invaluable assistance in the planning and preparation of this dissertation. I am deeply indebted for the tremendous amount of time he devoted to all phases of this re- search from the beginning to the end. I also would like to gratefully acknowledge the contribution of the other members of my committee, Dr. G.L. Park, Dr. R.0. Barr, Dr. W.W. Symes, Dr. M.A. Shanblatt, and Dr. H.K. Khalil, for their suggestions to improve the quality of this study. I am particularly indebted to the Chairman of the Department of Electrical Engineering and Systems Science, Dr. J.B. Kreer, for the financial support of the department. Special acknowledgment is made of the contribution of Dr. J.F. Dorsey, for providing some of the simulation programs which have been used in this research. I am forever thankful to my parents for their unconditional love and sacrifices, and for getting me started right. Most of all, I would like to thank my wife, Fereshteh, and my _son, Ali, for their love, patience, encouragement, and understanding during the entire course of this study. Finally, my special thanks to Clara Hanna, whose fast and excellent typing of this dissertation is greatly appreciated. iii TABLE OF CONTENTS List of Tables .............................................. List of Figures ............................................. Chapter l INTRODUCTION ................................... l.l Review of Previous Work ............... l.l.l Modal Methods for Producing Dynamic Equivalents ........................... l.l.2 Identification Methods for Producing Dynamic Equivalents ................... l.l.3 Coherency Based Methods for Producing Dynamic Equivalents ................... l.l.3.l Coherent GroUp Identification ......... 1.1.3.2 Inertial Averaging Method of Aggregation ........................... l.l.4 Singular Perturbation Methods for Producing Dynamic Equivalents ......... l.l.4.l Coherent Group Identification Based on Fast and Slow Modes ................ l.l.4.2 Singular Perturbation Method of Aggregation ........................... l.2 Thesis Objectives ..................... 2 POWER SYSTEM LINEAR MODEL, GENERALIZED DISTURBANCE MODEL, THE RMS COHERENCY MEASURES FOR STEP, IMPULSE, AND PULSE INPUT DISTURBANCES, AND THE AGGREGATION TECHNIQUES ......................... 2. N NNN w “(JON-H d .3.3 .3.4 Linearized Power System Model ......... Disturbance Model ..................... Generalized RMS Coherency Measure ..... Infinite Interval RMS Coherency Measure for Step Input Disturbances ... Infinite Interval RMS Coherency Measure for Impulse Input Disturbances .......................... Infinite Interval RMS Coherency Measure for Pulse Input Disturbances .......................... Grouping Algorithm .................... iv Page vii 03010100“) l7 l7 I9 20 24 24 41 45 50 56 57 Chapter TABLE OF CONTENTS (continued) 2.4.1 Coherency Based Aggregation Technique ............................. 2.4.2 Singular Perturbation Aggregation Technique ............................. DERIVATION OF OPTIMAL AGGREGATION TECHNIQUES FOR DIFFERENT CONTINGENCY TYPES WHEN THE OBSERVATION INTERVAL IS INFINITE (T1 = w) ...... 3.l Optimal Problem Formulation ........... 3.2 Derivation of Optimal Aggregation for Step Input Disturbances when the Observation Interval is Infinite T = w .............................. l 3.3 Derivatimiof Optimal Aggregation for Impulse Input Disturbances when the Observation Interval is Infinite (T1 = 0°) .............................. 'COMPUTATIONAL RESULTS FOR THE OPTIMAL-MODAL- COHERENT EQUIVALENTS ........................... 4.1 Global Equivalents for Step Input Disturbances .......................... 4.2 Modal Disturbance of Generator 1 or 8 ..................................... 4.2.1 Modal Disturbance of Generator 1 ...... 4.2.2 Modal Disturbance of Generator 8 ...... 4.3 Modal Disturbance of Generators 1 and 8 ................................. DERIVATION 0F OPTIMAL AGGREGATION FOR STEP OR PULSE INPUT DISTURBANCES WHEN THE SIMULATION INTERVAL IS SHORT (T1 = A) ..................... 5.1 The Mean Square Transient Coherency Measure ............................... 5.2 ATaylor Series Expansion of the Transient Coherency Measure ........... 5.3 Derivation of Optimal Aggregation for Step or Pulse Input Disturbances when the Observation Interval is Small (T1 = A) .............................. V Page 58 65 68 69 73 77 81 88 105 105 110 121 131 132 134 140 TABLE OF CONTENTS (continued) Chapter Page 6 COMPARISON OF THE "OPTIMAL MODAL COHERENT" AGGREGATION WITH THE "INERTIAL AVERAGING" AGGREGATION FOR THE INFINITE SIMULATION INTERVAL (T1 = w) .............................. 149 6.1 Global Equivalents for Step Input Disturbances .......................... 152 6.2 Modal Disturbance of Generator 1 or 8 .................................. 156 6.2.1 Modal Disturbance of Generator 1 ...... 158 6.2.2 Modal Disturbance of Generator 8 ...... 164 6.3 Modal Disturbance of Generators 1 and 8 ................................. 167 6 4 Comparison of IA Aggregation and OMC Aggregation for Impulse Input Disturbances when the Simulation Interval is Infinite (T1 = 0°) ......... 178 7 CONCLUSIONS AND FUTURE INVESTIGATIONS .......... 186 7.1 Overview of Thesis .................... 186 7.2 Topics for Future Research ............ 193 BIBLIOGRAPHY ................................................ 195 vi LIST OF TABLES Table Page 4-1 Generator Data for the 39 Bus New England System ........................................... 83 4-2 Load and Bus Data for the 39 Bus New England System ........................................... 84 4-3 Line Data for the 39 Bus New England System ...... 85 4-4 The Synchronizing Torque Coefficient Matrix, 1, in the Generator 10 Reference Frame for the 39 Bus New England System ............................... 92 4-5 Ranking Table of the RMS Coherency Measures for the Modal Disturbance of all 10 Generators ....... 93 4-6 The Matrix [TI and the Optimal Reduced Order Structure Matrix MTOMC for the Generators 6 and 7 Aggregation .................................... 100 4-7 Eigenvalue Data for Modal Disturbance of all Ten Generators and Optimal Modal Coherent Aggregation ...................................... 101 4-8 Coherency Measure Data for Modal Disturbance of all Ten Generators and Optimal Modal Coherent Aggregation ...................................... 103 4-9 Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generator 1 ................. 106 4-10 Eigenvalue Data for Modal Disturbance of Generator 1 and Optimal Modal Coherent Aggregation ......... 107 4-11 Coherency Measure Data for Modal Disturbance of Generator 1 and Optimal Modal Coherent Aggregation ...................................... 109 4-12 Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generator 8 ................. 114 vii LIST OF TABLES (continued) Table Page 4-13 Eigenvalue Data for Modal Disturbance of Generator 8 and Optimal Modal Coherent Aggregation ...................................... 115 4-14 Coherency Measure Data for Modal Disturbance of Generator 8 and Optimal Modal Coherent Aggregation ...................................... 116 4-15 Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generators 1 and 8 ...... 123 4-16 Eigenvalue Data for Modal Disturbance of Generators 1 and 8 and Optimal Modal Coherent Aggregation ...................................... 124 4-17 Coherency Measure Data for Modal Disturbance of Generators 1 and 8 and Optimal Modal Coherent Aggregation ...................................... 125 6-1 The Inertial Averaging Reduced Order Structure Matrix for the Generators 6 and 7 Aggregation ...................................... 153 6-2 Eigenvalue Data for Modal Disturbance of a11 Ten Generators and Inertial Averaging Aggregation ...................................... 155 6-3 Coherency Measure Data for Modal Disturbance of a11 Ten Generators and Inertial Averaging Aggregation ...................................... 157 6-4 Eigenvalue Data for Modal Disturbance of Generator 1 and Inertial Averaging Aggregation ...................................... 159 6-5 Coherency Measure Data for Modal Disturbance of Generator 1 and Inertial Averaging Aggregation ...................................... 160 6-6 Eigenvalue Data for Modal Disturbance of Generator 8 and Inertial Averaging Aggregation ............. 165 6-7 Coherency Measure Data for Modal Disturbance of Generator 8 and Inertial Averaging Aggregation ...................................... 166 viii Table 6—8 6-12 6-14 LIST OF TABLES (continued) Eigenvalue Data for Modal Disturbance of Generators 1 and 8 and Inertial Averaging Aggregation ...................................... Coherency Measure Data for Modal Disturbance of Generators 1 and 8 and Inertial Averaging Aggregation ...................................... Ranking Table of the Impluse RMS Coherency Measures for the Modal Disturbance of all 10 Generators ....................................... Eigenvalue Data for Modal Disturbance of all 10 Generators (Impluse) and Inertial Averaging Aggregation ...................................... Eigenvalue Data for Modal Disturbance of all 10 Generators (Impulse) and Optimal Modal Coherent Aggregation ...................................... Coherency Measure Data for Modal Disturbance of all Ten Generators (Impulse) and Inertial Averaging Aggregation ...................................... Coherency Measure Data for Modal Disturbance of all Ten Generators (Impulse) and Optimal Modal Coherent Aggregation ............................. ix Page 172 173 179 181 182 183 184 Figure 4.1 4.2 4.3 4.4 6.1 6.2 6.3 LIST OF FIGURES One-Line Diagram of the 10 Generators, 39 Buses New England System ............................... Simulation Response of the Reduced Order Model Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 1 Based on the Optimal Modal Coherent Aggregation of the Coherent Groups .................................. Simulation Response of the Reduced Order Model Versus the Response of the Full System for a one Per Unit Step Disturbance on Generator 8 Based on the Optimal Modal Coherent Aggregation of the Coherent Groups .................................. Simulation Response of the Reduced Order Models Versus the Response of the Full System for the Coherent Groups Identified Based on the Modal Disturbance of Generators 1 and 8 and Optimal Modal Coherent Aggregation of the Coherent Groups ........................................... Simulation Response of the Reduced Order Models Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 1 Based on the Inertial Averaging Aggregation of the Coherent Groups .................................. Simulation Response of the Reduced Order Models Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 8 Based on the Inertial Averaging Aggregation of the Coherent Groups .................................. Simulation Response of the Reduced Order Models Versus the Response of the Full System for the Coherent Groups Identified Based on the Modal Disturbance of Generators 1 and 8 and Inertial Averaging and Optimal Modal Coherent Aggregations of the Coherent Groups ........................... X Page 82 111 118 127 161 168 174 CHAPTER 1 INTRODUCTION Electric power systems havel become probably the most extensive, intricate and imposing of all dynamical systems ever constructed by humans. Alongside the evolution of these systems have gone the devel- opment of concepts and techniques of representation, analysis and com- putation for system planning and operation became complicated. The dynamical simulation of electric power systems, as required for the design of transmission facilities and of generating plant controls, involves the numerical integration of a very large set of nonlinear differential equations which are coupled by the algebraic equations describing the transmission network. Because of computer size and speed limitations, it is frequently impractical or uneconomical to handle the complete set of differential and algebraic equations needed to describe a whole interconnected system in detail. It is, therefore, frequently necessary to restrict the use of the differential and algebraic equations describing each component in detail to those parts of the system where detailed results are required; and to use simplified representations, or "dynamic equivalents" to represent those parts of the system which influence its performance but whose internal performance is not under 'study. Thus, for the purpose of analysis, the interconnected power system is divided into a "study system", and one or more "external systems". These subsystems are defined below. Study System (or Internal Group): The study system is that part of the system in which the distur- bances occur and is of specific interest. All of its components are repre- sented in detail by their individual equations. For example a generating station and its local transmission system could be defined as the study area in order to analyze the stability of the station with respect to the rest of the system. The types of disturbances which may occur in power systems fall into four basic categories; generator dropping, load shedding, line switching, and electrical faults. External System (or External Group); The external system is that part of the interconnection which is sufficiently removed from the site of the disturbance under study to warrant the use of an equivalent, but which has such a significant influence on the study system that the effect of it on the study system must be accounted for. 1.1 Review of Previous Work Several techniques have been proposed in the literature for constructing dynamic equivalents of power systems. These techniques can be classified into four categories; modal methods [1-6], identificat- ion methodsE7,8], coherency based methods [9-21], and singular perturbation methods [22,23,24]. Among these four approaches those of the coherency based methods have been generally accepted, at least conceptually, by 'the power systems engineers because they are closest to the nature of power systems and also they can be computed for arbitrarily large systems at reasonable cost. Since the results in this thesis are based on the coherency methods, a brief summary of the modal and identification will be given, while a more detailed description of the coherency based approach will be discussed. The singular perturbation methods are based on aggregating coherent groups which are identified based on modal or coherent structure. A summary of these four methods is given in the following subsections. l.l.l Modal Methods for Producing Dynamic Equivalents: These methods are used in [1,2] to obtain reduced order dynamic model of the external system which is linearized about some base case conditions. Rules for mode elimination are given in [l] which would essentially eliminate modes whose effect on the "study system" transients is insignificant. In other words, the order reduction occurs in the external system, i.e., modes identified with the external system are discarded if they are not excited by the disturbances in the study system, decay quickly to zero, or are either uncontrollable or unob- servable. This method is theoretically sound and applies a well-estab- lished modal reduction technique which may be used to study any step input disturbances in the study system [31. The problems with this ap- proach are: l. Substantial computation effort is required to compute the eigen- values and eigenvectors of the external system. This in effect limits the order of the external system. '2. The rules of the mode elimination given in [1] may not be applicable. In other words, it may not be clear which modes should be eliminated [4]. 3. The reduced order model can not be represented in terms of equivalent power system components; hence they can not be used with the present transient stability programs. 4. The modal model reduction technique is restricted to loss of generation, loss of load, or line switching contingencies that can be modeled as step disturbances and can not be used for model reduction for faults that cause pulse or impulse like disturbances. 5. The modal reduction methods in [1,2] can not be used to selectively eliminate modes in a particular part of the system or for specific disturbances making these modal method disturbance independent. Deficiencies 3 and 5 above are addressed in rules of mode elim- ination based on the RMS coherency measure [14]. These rules eliminate modes based on the differences in generator angles and not on the angle changes (referenced to a reference generator) for a disturbance. It is found that slowly varying modes may cause very small changes in angle differences within the coherent generator groups and very large changes in angle differences between the coherent groups but fast modes cause large changes in angle differences within the coherent groups and small changes in angle differences between generators in different groups. Observing absolute angle changes and not the difference in angle changes between all pairs of generators does not provide the information needed to eliminate the fast modes that cause oscillations in coherent groups. -Thus, for the first time mode elimination linked modal methods to coherency methods and rules of mode elimination were derived-h1[14J that were totally independent of any specific disturbance as well as rules of mode elimé- ination for specific loss of generation, loss of load, or line switching contingencies. Thus, modal methods can be disturbance independent or disturbance dependent. Two recent papers [5,6] proposed a new procedure for producing modal dynamic equivalents. This new approach is termed selective modal analysis (or SMA). It goes beyond the traditional modal methods in ways that are crucial to successful practical utilization in off-line planning and on-line operation for large scale systems. SMA can accurately and efficiently focus on selected portions of the structure and behavior of the system, i.e., the part of the model that is relevant to the dynamics of interest is singled out in a direct manner, and the remainder of the model is collapsed in a way that leaves the selected structure and behavior intact. 1.1.2 Identification Methods for Producing Dynamic Equivalents: These methods have been used to construct dynamic equivalents based on parameter identification of postulated models for the equivalents [7,8]. These efforts are still going through initial feasibility analysis and have little chance of achieving the desired properties for dynamic equivalents without better understanding of dynamic system structure. l.l.3 Coherency Based Methods for Producing Dynamic Equivalents: These methods are based on the observation that the post-disturbance behavior of certain groups of generators in a system tend to swing together, i.e., to maintain nearly constant angular differences among themselves. ‘Two or more generator buses are defined to be "coherent" if their voltage angle differences remain constant within a certain tolerance over a certain time interval. Such a group of generators is called "coherent group". The procedure to obtain a coherency based equivalent consists of two steps: 1. Identification of nearly coherent generators. 2. Aggregation of a coherent group into a simplified model. These two steps will be discussed separately in the next two subsections. 1.1.3.1 Coherent Group Identification Several approaches have been proposed to identify the coherent groups. In [10] pattern recognition techniques are used for coherent groups identification. Various definitions of electrical distancesare proposed in the literature and various clustering algorithms are developed for determining coherent groups. These techniques are heuristically based and suffer from the serious limitation that the only way of checking the accuracy of these methods is to run a full system stability program [11]. In [11] nearly coherent generators are identified by' a simplified linear model of the power system. This model is solved for single: or multiple disturbances using a fast trapezoidal intergration algorithm. The approximate swing curves produced by the linear simulation are then processed by a clustering algorithm to determine the coherent groups of generators. This clustering algorithm minimizes the number of data curve comparisons by recognizing that the coherency of the generators is a transitive process, i.e., if generator A is coherent with generator C, and generator B is coherent with generator C, then it follows that generators A and B are coherent. A reference generator is defined in .each group and other generators are always compared against this reference generator in order to determine whether they should fall in the same group. 111e first generator is arbitrarily defined as the reference unit for group one. The remainder of the generators are evaluated in turn with two alternative sequences; either the unit is combined with an existing group or the unit does not combine with any existing group and a new group is created with the unit defined as the reference. The criterion (coherengy measure) which is used for determining whether a generator should be added to an existing group is: |Adi(t) - A6r(t)| < e1 for all samples of time t 6 [0,1]] (l-l) where, A6(t) = rotor angle deviation i = index for generator being clustered r = index for the reference generator for the group under consideration s1 = specified tolerance T1 = simulation (or observation) interval The requirement of a priori simulation and the disturbance - dependent character of coherent groups are the basic disadvantages of this method especially for on-line applications. Another disadvantage of this method is that the coherent groups are reference dependent, i.e.,the coherent groups are sensitive to the arbitrary order in which generators may be processed by the algorithm that forms these coherent groups. A series of recent papers [13,14,15,l6] have shown that each of these criticisms can be resolved if the coherency analysis procedure in the previous method is modified to use root-mean-square (or RMS) coherency ‘measure and a probabilistic "modal disturbance" to identify the coherent groups. The RMS coherency measure between any pair of generators, say h and 2, is defined as: 1 1 1 2 12=12...N-l c (1): ——5( [A6 (t)-A6(t)]dt, ,, : (1—2) M 1 15’ o h K £=Iz+l,...,N where the expectation operator, E, is used to analyze random disturbances, p is a positive integer chosen so that the measure is finite but nonzero over the infinite observation interval (T1 = m), and N is the number of generators in the system. It is shown [15] that the RMS coherency measure evaluated over an infinite observation interval can be analytically related to generator inertias, synchronizing power coefficients of equivalent lines connecting internal generator buses, and the statistics of the disturbance. Moreover, it is shown that for the step disturbances in the mechanical input power of generators, probabilistic disturbances can be found which cause the RMS coherency measure to be a function of system structure alone. This is particularly important because it would identify coherent groups which are not disturbance-dependent, leading to what may be called "structural coherency” as opposed to "disturbance dependent coherency". This RMS coherency measure can thus produce either disturbance independent or dis- turbance dependent coherent groups. In an independent, but closely related study [20], another approach to identify coherent generators in terms of the system structure has been sought. A group of generators is shown to be coherent if all generators of the group have the same ratios of synchronizing power coefficients of .1ines connecting the group terminal buses to generators in the study system over the inertias of coherent generators. The synchronizing power coef- ficient between any pair of generator buses, say i and j, is defined as: = —‘_—J— cos(6. - a) (1-3) 13 where, VIZfi and Vjfl are the complex voltages at buses i and j, and xij is the reactance of the line connecting these two buses. Similar results to [15] and [20] were recently derived in [21]. This type of structural coherency is independent of the stiffness of the interconnections between members of coherent groups. By requiring re- latively stiff interconnections (Tij = w) between members of group, the group can also be structurally coherent. Although [20,21] never considered the effect of stiff interconnections of generators in a group, a recent doctoral thesis [19] showed that it is more important than the strict geometric coherency property of [20,211. Moreover, this thesis showed that there are five basic structural conditions that can cause a group of generators to be coherent or to appear coherent. These structural conditions are now given in terms of the linearized model. For a system of N generators of which the first m generators fOrm the study system and the last n = N-m generators form the external group whose structural coherency is to be investigated the linearized swing equations take the form: 51 = 51151 + i1212 ‘ 011 + E1-”- “'4“ 332 = E21331 + 52212 ‘ °12 (Mb) where 54 is the m-dimensional swing vector of the generators inside the study system with components di-Ge, i = 1,2,...,m, and N N D. a =({ dej)/(Z M),O=—J.- ,i=l,2,...,N e j=m+1 j=m+l 3 M1 10 where M. J Di the inertia constant of generator j the damping constant of generator 1 The (n-l)-dimensional vector 52 is the swing vector of the generators of the external group with components 52'5e’ 2 = m+l,...,N-l. The disturbance input u, represents step change in the mechanical input power of generators inside the study system. The external group remains coherent or appears coherent for all t 3 0, if the external group satisfies one of the following structural conditions: a) Strict Geometric Coherency(or SGC): ( + Q) 521 This condition was derived in [20] and requires that the ratio of synchronizing power coefficients of lines connecting a generator in the study group to a generator in the external group over the inertia of that generator in the external group be equal for all pairs of generators; one belonging to the study group and the other belonging to the external group, i.e. _.| pg§'= constant for all i = 1,2,...,m and J = m + 1,...,N J b) Strict Synchronizing Coherency (or SSC): (fé;‘+ 9) This condition requires that at least (n-l) lines in the external group connecting all n generators be infinitely stiff. It is shown in '[19] that if a system satisfies this condition, the system has a two time scale property. 11 c) Pseudo Coherency (or PC): (£42 +-9) This condition requires that: = constant for all i = 1,2,...,m and j = m + l,...,N ZZLJT uni. —h This condition does not cause the external group to be coherent but just to appear coherent to the study system. This condition does not hold in the nonlinear model as does SGC and SSC. 1 2 521* 9-) This condition is quite different and independent of SGC and SSC d) Strict Strong Linear Decoupling (or SSLD): (E; eventhough it is a combinition of the two conditions. A group of generators can be strongly linear decoupled when part of them are infinitely stiffly connected to generators that satisfy conditions similar to SGC but are modified since the infinitely stiffly connected generators are now one equivalent generator. . . , -1 -1 e) Weak L1near Decoupl1ng (or WLD). (5,2 £22 + Q, 5J2 F 2 £2] +-g) This condition is similar to SSLD but depends in part on PC which does not hold up in the nonlinear model and thus WLD is considered of little use at is PC. It is shown in [19] that these conditions are either controllability or observability conditions for mode elimination in the modal analysis aggregation technique. It is also shown that if the coherent groups that satisfy these conditions are aggregated to form a single equivalent generator ~using coherency based aggregation techniques (discussed later in this chapter), the reduced linearized model is identical to that produced by the modal analysis method that would eliminate modes based on the 12 controllability and observability conditions. Thus, the link between modal and coherent methods is theoretically based. The RMS coherency measure is shown to detect SSC, SGC, and SSLD coherent groups if different probabilistic disturbances are utilized [19]. Since the RMS coherency measure can detect such groups and since it can be used to derive the modal elimination rules that cause mode elimination for the controllability and observability conditions, that produce SGC, SSC, and SSLD coherent groups, the RMS coherency measure is an appropriate measure for producing equivalents that retain both fre- quency domain (modal) and time domain (coherent) dynamic structure. This RMS coherency measure will thus be utilized in this thesis for determining optimal modal aggregation methods that maintain modal and coherent dynamic structure. It is shown that the modal disturbance of all generators of the study system or a specified subregion would identify the coherent groups outside the disturbed area based on SGC and SSLD for that region [19]. These groups are appropriate for constructing parochial or local dynamic equivalents for a specific location or subregion. It is also shown that the modal disturbance of all generators of the system determines the strongly bound cOherent groups based on SSC property. These groups are appropriate for producing global dynamic equivalents. It should be noted that the classical modal analysis methods [1,2] produced only disturbance independent equivalents and that the 'classical coherent group determination methods [11] produced only dis- turbance dependent equivalents. The RMS coherency measure, on the other hand, can produce either disturbance independent or disturbance dependent modal or coherent equivalents. 13 It will now be shown that there are three distinct types of dynamic equivalents possible; a disturbance dependent equivalent (parochial), a partial disturbance independent equivalent (local), and a total distur- bance independent equivalent (global). These three types of dynamic equivalents have completely different applications and exploit different structural properties (SSC, SGC, SSLD) and can be produced by evaluating the RMS coherency measure for different probabilistic or deterministic disturbances. a) Parochial Dynamic Equivalent: This type of equivalent is designed for a single contingency type in a specified location. Some applications of this type of equivalents are: 1) For use within transient, midterm, and long term stability programs to reduce the computational requirements when the unreduced model is very large and the simulation interval is long. 2) For on-line transient stability for security assessment where a low order accurate model for a specific contingency is desired. b) Local Dynamic Equivalent: This type of equivalent is designed for investigation of transient or dynamic stability at a specific subregion. Some applications of this type of equivalent are: 1) For design of excitation systems including power system stabilizer where all eigenvalues, fast or slow, that are excited and must be appropriately damped by the excitation system for disturbances in a local region must be preserved in the local equivalent. C) 14 For design of discrete supplementary control. For on-line transient stability for security assessment. The local dynamic equivalent could be used for simulating all disturbances in a local area rather than only for a single disturbance. Since the equivalent must be computed off-line, an equivalent could not be calculated for each contingency to be investigated on-line by security assessment procedures. Global Dynamic Equivalent: This type of equivalent is designed for investigating global dynamic structure and stability. Some applications of this type of equivalent are: 1) For study of the weak boundaries and lines that would be vulnerable to security and stability problems for loss of generation contingencies. It has been shown that the boundaries between the strongly bound groups, which are detected by the RMS coherency measure and modal disturbance of all generators, are the vulnerable boundaries for loss of generation contingencies. The intermachine dynamics in these strongly bound groups could thus be eliminated for dynamic equivalents required to study these security and stability problems on very largeinter-regional data bases. In other words, this type of dynamic equivalents can be used for study of inter-area power transfer limit in transmission planning. For the study of the islands that naturally form when contingencies occur which lead to islanding. The islands would likely be formed of strongly bound groups if no relaying action causedseparation within 15 one or more strongly bound groups. The relaying action that could form islands should produce islands that should not have weak boundaries that would be vulnerable to the significant stresses on the trans- mission network when islanding occurs. Thus, the study of islanding may be assisted by the use of equivalents that only retain the weak transmission boundaries where islanding would or should occur. The second task in obtaining coherency based dynamic equivalents is to aggregate a coherent group into a simplified model that faithfully and accurately represents the effect of the group on the rest of the system. In [25] coherent generators are aggregated using an "inertial averaging” method of aggregation. A brief description of this aggregation method is given in the next subsection and a more detail discussion of this method will be presented in Chapter 2. In [13,14] a modal method derived based on the RMS coherency measure is used to obtain a simplified model of a group of coherent generators. The method establishes a significant link between modal and coherency based equivalents, but it has a computa- tional disadvantage similar to the modal method of [1]. To overcome the computational disadvantage, a modal-coherent equivalent is proposed in [15], where coherent generators identified by means of the RMS coherency measure, are aggregated using the inertial averaging method of [25]. Finally, an aggregation technique using singular perturbation is used in [23,24] to obtain a dynamic equivalent. A brief discussion of this method is given later in this chapter. 1.1.3.2 Inertial Averaging Method of Aggregation: This method of aggregation represents each coherent group of generators by an inertial averaged equivalent generator and reduces the 16 number of equivalent lines. For a system of N generators of which the first m generators form the study group and the last n = N-m generators form the coherent group the parameters which describe the equivalent generator and the reduced equivalent lines are: N N 1 Meq = 12ml Mi 2 D94 = 12m” Di N N 3 APMeq = iZm+l APM. 4 APGeq = 1§m+1APGi N 5 Tej = iZm+l Tij , J = 1,2, ,m where M. = inertia constant of generator i (p.u.) D. = damping constant of generator i (p.u.) APM. = deviation in mechanical input power of generator i (p.u.) APG. = deviation in electrical output power of generator i (p.u.) T . = The synchronizing power coefficient of the equivalent line which connects the equivalent generator to the generator j in the study system. In a case of SGC, SSLD, or SSC, the inertial averaging aggregation method would produce the following equivalent: 3;1 = 51111 ' 031 T 53 (1'5) A basic feature of this averaging aggregation is that the identity of the system outside the coherent group is retained. 17 1.1.4 Singular Perturbation Methods for Producing Dynamic Equivalents: Using the singular perturbation techniques, an N-th order system with r slow and (N-r) fast frequency oscillations is decomposed into two lower order subsystems, one containing only the slowly varying part and the other containing only the fast oscillatory part [22,23,24]. The only states used for long term studies are the slow states, while the differential equations for the fast states are reduced to algebraic equations and solved for the fast states and then substituted in the differential equations for the slow states to obtain a reduced order model of the original system. In the next two subsections the coherent group identification based on slow and fast modes and singular perturbation aggregation will be discussed separately. 1.1.4.1 Coherent Group Identification based on Fast and Slow Modes The coherent groups of generators are identified based on the r slowest modes of the system given in [23,24]. A summary of their clustering algorithm is as follows. Step 1) For the N-th order linearized system model of g_= Ax_+ Bu, find the largest gap between eigenvalues of the plant matrix A, and divide them into r small and (N-r) large eigenvalues, and choose the number of coherent areas (r) and the modes or = {Ai’ 1,2,...,r}. V1 Step 2) Compute an (er) basis matrix V_= [}:Le] of the Or- 12 eigenspace for a given ordering of the state variables. Step 3) Apply Gaussian elimination with complete pivoting to V_ and obtain the set of reference generators. This is done to find a set of 18 the r most linearly independent row vectors to be used as the reference row vectors. During the elimination, the rows and columns of V. are permuted such that the (1,1) entry of the resulting V_ is the largest in magnitude. Note that permuting the rows of V_ is equivalent to changing the ordering of the generators. This (1,1) entry of V_ is the pivot for performing the first step of the Gaussian elimination. Then the largest entry from the remaining (N-l)x(r-1) submatrix is used as the pivot for the next elimination step. The elimination terminates in r steps and the generators corresponding to the first r rows of the final reduced V, are designated as the reference generators. Step 4) For the set of reference generators found in step 3, find the (N-r)xr matrix Ed from V} LT = vg, using the L-U decom- position of 31 already obtained from the Gaussian elimination. Step 5) Construct the (N-r)xr grouping matrix Lg from L To do ~d' this, each row of Ed will be examined. If the largest positive entry in row i is the j-th entry, then in the matrix [Q entry (i,j) is l and all other entries in the i-th row are zero. From this grouping matrix, the coherent groups are identified by checking each column of Lg matrix and adding the generators corresponding to l, in the area designated to that column. It is obvious that this clustering algorithm is very complicated, especially for large scale systems, and can be only applied to a system with two time scale property. Recent work by Schlueter-Dorsey [19] showed that the structural condition that the singular perturbation can be applied 19 in power system is exactly the condition for strict synchronizing coherency (SSC). Thus, similar coherent groups may be found by using an RMS coherency measure evaluated for a modal disturbance of all generators with much less computation. Once the coherent groups are identified, the coherent groups are aggregated based on the singular perturbation aggregation. 1.1.4.2 Singular Perturbation Method of Aggregation: Singular perturbation aggregation technique is appropriate if the system possesses a two time scale property with the relative motion of the members of the "external group” faster than the relative motion of those inside the ”study group". A system has a two time scale property if it can be expressed in the form: 54 = 94154 + £q2§2 ‘ 054 + B42. (1'63) 23£=Cx+Cx-d$< (1-6b) “—2 —21—1 —22—2 “—2 where, u is a sufficiently small positive scalar. In the singular perturbation method the reduced order model is obtained by neglecting the effect of the group intermachine fast oscillations on the relative motion of the generators inside the study system. Mathematically this is done by setting u = O, in the equations (1-6) to produce the following reduced order model for the system: " _ -1 . 51 ' (911'912922921)11'°-x—1 + g1-ll (1'7) Thus, singular perturbation (or SP) aggregation represents the "external group" by a single equivalent generator and changes the "study system" 20 to properly represent the steady state contribution of the neglected fast modes. Although there exist different aggregation techniques, there is no theory at present on which is best for different contingency type and time interval combinations. This is the objective of this research. This research is focussed on a complete methodology for dynamic equivalents which include proper modelling, grouping, and aggregation. The techniques developed will utilize RMS coherency measures for step, impulse, and pulse input disturbances, developed under modal-coherent equivalent. The optimal aggregation methods for these disturbances will be derived in this research. The equivalents produced by an RMS coherency measure, the modified grouping algorithm, and the new aggregation method that will be derived in this thesis (termed, optimal-modal-coherent or OMC) will be called "optimal modal coherent equivalents". The equivalents produced by using the RMS coherency measure, the modified grouping algorithm, and the ”inertial averaging" aggregation will be called averaging equivalent. 1.2 Thesis Objectives: The objective of this research is to find optimal aggregation techniques for different classes of structurally coherent generators which are best for different contingency type, time interval combinations, and the specific applications they will be used for. The objectives of this research can be summarized as follows: ‘1. To derive an optimal aggregation method for fault and step disturbances in the mechanical input power of the generators of the "study system" when the observation interval is infinite (T1 = 00) based on the performance measure which is the coherency measure used to determine the coherent groups. 21 To show that the OMC aggregation is similar to SP aggregation but is not identical since SP aggregation can not be derived for fault dis- turbances and for disturbance dependent coherent structure formed based on SGC and SSLD coherent groups. SP aggregation can only be applied to SSC coherent groups that cause the system to have a two time scale property. Thus, OMC aggregation can be used for producing local and parochial dynamic equivalents but SP aggregation can be only used for producing global dynamic equivalents. To indicate that the OMC aggregation better preserves the coherency measures, the eigenvalues, and the simulation responses for long simulation intervals than the inertial averaging aggregation. Thus, OMC aggregation would be better for mid-term and long term stability simulations as well as for design of excitation controls, design of power system stabilizer, and discrete supplementary control, where eigenvalues and load flow information must be preserved in the dynamic equivalents. To show that the "Inertial Averaging” aggregation is optimal for step, or pulse input disturbances in the mechanical input power of generators of the "study system", when the simulation interval is small (T1 = A) based on the coherency measure that preserves system structure for short intervals. This coherency measure correctly predicts the initial angles and their accelerations on the machines and does not measure the steady state inertial load flow structure when (T1 = m). To indicate that the "Inertial Averaging" aggregation is better than OMC aggregation at the peak first swing after the contingency occured and thus is a better aggregation method for transient stability simu- lations. 22 Since the derivation of optimal aggregations are based on the linearized model of the power system, Chapter 2 is devoted to developing the linearized power system model, the generalized disturbance model, and the RMS coherency measures for different types of disturbances. An efficient algorithm for identifying the coherent groups of generators, based on the RMS coherency measures for different types of disturbances, is also defined. Finally, theinertial averaging(IAO and singular perturbation (SP) aggre- gations by which coherent groups can be aggregated to form an equivalent are discussed. In Chapter 3, it is shown that a SP like aggregation method, called OMC aggregation, produces equivalents that are optimal for either any step or impulse disturbances restricted to the study system if the simulation interval is infinite. Using an example system, it is shown in Chapter 4, that the reduced order models obtained based on the RMS coherency measure and the OMC aggregation closely approximate the eigen- values (modal behavior), the RMS coherency measures (coherent behavior), and the simulation response of the unreduced system in producing parochial, local, and global dynamic equivalents. In Chapter 5, it is shown that the IA aggregation is optimal for any type of disturbance restricted to the study system if the simulation interval is short. Chapter 6 compares the behavior of the full example system with the reduced order models obtained based on the IA and OMC aggregations for long simulations intervals. The results in Chapter 6 show that the reduced order model obtained based on the OMC aggregation better preserves the coherency measures, the eigen- ‘values, and the simulation responses (for long simulation intervals) than the IA aggregation. It is also shown that the IA aggregation performs better for short simulation intervals. Finally, Chapter 7, summarizes 23 the contribuiton of this research and proposes topics for future investi- gations based on this research. CHAPTER 2 POWER SYSTEM LINEAR MODEL, GENERALIZED DISTURBANCE MODEL, THE RMS COHERENCY MEASURES FOR STEP, IMPULSE, AND PULSE INPUT DISTURBANCES, AND THE AGGREGATION TECHNIQUES The principal objective of this chapter is to develop a linearized power system state model, a generalized disturbance model, and the root mean square (RMS) coherency measure. These models and the generalized coherency measure is used to derive algebraic expressions which relate the RMS coherency measure, evaluated over an infinite observation interval for step, impulse, and pulse disturbances in mechanical input power, to the parameters of the power system state model and probabilistic description of the disturbance vector. Finally different aggregation techniques is discussed. 2.1 Linearized Power System Model A system of linearized state equations is derived for a power system which is composed of classical synchronous machine models, voltage dependent load models and a transmission network model. The justification for assuming that the simplified linear model is sufficient for the co- herency behavior of the system is based on the recent work on coherency based dynamic equivalents at System Control Incorporated which has shown .that a simplified model which is suited for coherency analysis can be derived by making the following assumptions: 1. The coherent groups of generators are independent of the size of the disturbance. Therefore, coherency can be determined 24 25 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 syn- chronous machine model is considered and the excitation and turbine-governor systems are ignored. 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 radically affect the more basic characteristics such as the natural frequencies and mode shapes. The third assumption recognizes that the generator accelerating powers are approximately constant during faults with typical clearing times. The above assumptions and their justifications are quoted from [12]., A linear model can be derived from the nonlinear differential equations for the electromechanical motion of the classical synchronous generators plus a set of algebraic equations for the power flows between the generators and the load buses of the system. The electromechanical equations for the motion of each synchronous generator are: 6.(t) = w.(t) (2-16) d M]. d—fdim = PM1.(t) - PGi(t)-Dlwl(t) (2-lb) 26 where i subscript for generator i M. inertia constant of generator i (in p.u.) D. damping constant of generator i (in p.u.) 5. rotor angle of generator i (in radians) w. speed of generator i (in rad/sec) PMi mechanical input power of generator i (in p.u.) PG. electrical output power of generator 1 (in p.u.) N number of generators in the system In some literatures equation (2-la) is given as: 61(1'.) = 2nf0w1(t) = wowi(t) (2-2) QQ d- where, f0 is the synchronous frequency of the system in Hertz, and mi is in per unit (p.u.) instead of rad/sec . The reason that equations (2-1) are nonlinear is because of nonlinear relationship between PG, and the bus angles in the interconnected network. The system network equations for the transmission network can be written as [26]: N 2 . PG. = E. no + o I r I. I- o + o- u- a G jZl IE1| IEJ|_GUcos(- at) = _Ayt) + [32M (2-12) A§ APM .)_(_ = "" 9 _U_ = "'"' (2-123) A__ AEL_ - 1 — I '- 9(N-1)x(N-1) 5 —I-N-l L 9 E 9 I I """"""" T"“"' , g = "”1“” (2-12b) _ ’W— {TO-I-N-l _ _ U- : ALL- J 35 M1 -1 -Me] -M ..... -Me M2 -Me = -1_ -1 _ l1 -1 N. Mm+1Me Me ..... . . . . 'Me (Z-IZC) _M-1 M-1 _M-1 e "1+2 e -m'] e -1 -l -1 -1 .-1 -M - . . . -- - _ e Me MN-l Me Me 4 Equation (2-12) shows that, even though a fictitious reference angle, de, has been used, the linear model for the N generator system has only 2N-2 states. This is possible because the angles of the external group of n generators are dependent. This angle dependence results from the fact that the sum of the power changes generated by the generators of the system must be equal to zero [29]. The dependence of the n generator angles of the external group means that one of these angles can be eliminated as a state. A second state can be eliminated, namely the .speed of one of the generators, under the assumption of uniform damping. In our model the two states 5N and 5N have been eliminated. The next task in this chapter is to derive the disturbance model which can be used for deterministic as well as probabilistic system 36 disturbances. The disturbance model has been developed in [13] and the presentation here follows that development. 2.2 Disturbance Model The initial conditions of the linear differential equations (2-12) are assumed random with E{3<_(0)} = 0 (2-13a) E{§(0)§_ (0)} = v (o) (2-13b) since the expected deviations from any operating state is zero but the variance of such deviations is nonzero. The coherency measure to be developed in the following subsections will be shown to depend on this !x(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 external operating conditions. The input .u(t) composed of the deviations in the mechanical input power AEM_ on the generators and the deviations in load power APE, can be used to model —J 0 loss of generation due to generator dropping 2 loss of load due to load shedding 3. changes in load injections due to line switching 4 electrical faults These contingencies can be modeled by an input u(t) that has the following form em = 2m) + ego) (2-14) The vector function , u1 t 3 O 940:) =< (2-151 ( g t < 0 where 94(t) is a vector step function with amplitude 24- Thus, the non-zero entries in 94(t) can model the first three types of disturbances. The modeling of the first three disturbances requires determination of u, and possible modification of the network before determination of matrices A_ and B, The procedure used in [12] for each disturbance type is discussed below: Generator dropping - the transient reactance of the generator dropped is omitted from the network and the deviation in the generator input APMi of the generator dropped is set equal to the loss of generation. Load shedding - the load deviation PLh for all buses h where load is shed should be set equal to the change in load caused by the load shedding operation. Line switching - the network is modified to represent the system after the line switching operation is performed. The load deviations, PL,2 and PLm, at buses to which this line is con- nected, are set equal to the changes at that bus which occur due to the particular line switching operation. Note that in each case above all variables in 94 are zero unless otherwise specified and the operating point used to obtain matrices .5 38 and B_ is that obtained from the base case load flow even if network changes are made. The results obtained without finding the post disturbance load flow conditions is apparently satisfactory because the effects due to changes in the load flow are assumed to be confined to the study system and thus should not effect the coherency of the external system being equivalenced. The uncertainty due to a generator dropping, load shedding, and line switching disturbance could be modelled by A5”— 1 9111 Eth} = E l ------ >= ------ = EH (2-16a) A-P—L— 9112 L J F R E o ' T _ ‘” : "" - E{[y_'l - m'IJEE] - m1] } - ....... : ........ " _R_-l (2-]6b) I _ 9 ' E221 where (1) ‘-”—11 and 311 magnitude of generation changes due to generator dropping when the can describe the uncertainty in the location and particular station, the generator in the station, and the power produced on the generator are unknown. (2) £52 and 522 describe the uncertainty in the location and magnitude of the load being dropped by any manual or automatic load shedding operation. (3) £52 and 322 can describe the uncertainty in the location and the change in injections on buses due to any line switching operation. 39 The uncertain model of E, can handle the case of a specific deterministic disturbance by setting Eh = Q_ and EN = 94 for the particular disturbance. The function u,(t) can only model disturbances that resemble step changes. To model electrical faults, we define the vector function 9_ t > T2 32(t) = ( 92 O f t 5 T2 (2-17) 9_ t < 0 that is g2(t) represents a pulse of duration T2 and amplitude g2. This vector function can represent the effects of electrical faults where T represents the fault clearing time and 2 represents the step change in generation output equivalent to the accelerating powers due to a particular fault. This change of mechanical powers, ABM, which is equal to the accelerating powers on generators due to a particular fault is calculated by an ACCEL program [12], 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 Iconditions are assumed to be confined to the study system and should not effect the coherency of the external system being equivalenced. 40 The above model can be generalized to model the uncertainty of any particular disturbance and yet handle specific deterministic disturbance as a special case. If the size and location of an electrical fault is not known and if the clearing time T2 for this fault is known, then a pro- babilistic description of this electrical fault is , _ -"121 E{uel= ----- = m2 (2-18a) 9. T 521 : 9 E{[_u_2 - szuz - m2] } = "'"T"" = 32 (2-18b) 9. 1 9. where m2] and 32] describe the uncertainty in accelerating power on all generators due to this electrical fault. This mean and variance should be determined based on observed historical records or hypothesized based on the present network and present internal and external conditions. If 52 = Q, and me] = ABM_ for a specific fault, this generalized model then reverts to the deterministic model of a specific electrical fault. It should be noted that AEM_ and AEL_ are assumed uncorrelated because this model is to represent only one specific type of contingency at a time. For the same reason u, and ye are assumed uncorrelated with initial conditions, i.e. Hamel} = g (2-19a) name; = 9 (2-19b) The probabilistic descriptions of generator dropping and line switching are made assuming the network changes associated with the 41 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 the study system and should not seriously effect coherency of the external system being equivalenced. 2.3 Generalized RMS Coherency Measure The RMS measure of coherency between generator internal buses h and 2 based on the uncertain description of disturbances is [14] c (1 ) = -l—-E{ TIAB (t) - A6 (t)]2dt} ht 1 T1P 0 h 2 _ 1 11 2d - E15.540 (A6h(t) - A6N(t))-(A6£(t) - A6N(t))] t} yearns... 1....) where s (1 ) = 1 (T1 E{x(t)xT(t)}dt (2-21) _ 1 {[5 0 _ '— is a (2N-2)x(2N-2) symmetric matrix which is defined in terms of the state vector x(t) of the linear model of the power system. The integer P 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 one or zero so that the above integral will be finite and non-zero for an infinite observation interval (T1 w). 42 eh, is a (2N-2) x 1 vector defined by -1 j=£ for th,£7‘N 0 i f h,£ {9&2} = l j = h (2-22) —- j for h f N, 2 = N 0 j i h 1 j = 2 for h = N, 2 f N 0 i f 2 Since the objective is to compute the N x N coherency matrix C_ where {thfi = ch£(11), t,£ = 1,2,...,N we can easily calculate the elements of this matrix provided that we know the upper left (N-l) x (N-l) submatrix of §x(Tl)’ because the coherency measure between any pair of generators depends only on the generator angles. Denoting the upper left (N-l) x (N-l) submatrix of §x1111 by §X(T]) the coherency measure Ch£(Tl) can be found as _ T _ T Ch£(l]) 7V//ék£§x(Tl)§h£ — Tr{eE£eE£§X(T1)} 01" , . x A h ? \//1§x(11)1hh T L§X(T])}££ ' {§x(T1)}£h ' 1§x(T1)}h£’ K f chem)= /{§x(1111et I: ,1 N, 1: = N (2-23) \/{§X(T,)}a 12 = N, I. r N [in 43 where The matrix 5x111) can be easily computed by substituting x(t) in equation (2-21). For the input function g(t) = gq(t) + g2(t), x(t) has the following form: ' At Av e—-x(O) + 10 e— de(u u1 +g2) for t < T2 At) = < (2-24) t T 1 eAFEIO) + I Avdeu1 + eflflt'Tz) I 2 eA-vdeu2 for t > T2 0 0 Substituting x(t) into equation (2-21) and taking expectation term by term using equations (l3,l6,18,19) we obtain 5 1 )= 1 T14w _(0) d —X ( :r—‘FO e— T 1 T l 2 Av T T T m1 Av + -—E11 ([1;&dVB][R1 + R2 + m1m1 + @292 + mlm2 + m2m 1][[Oe——de]T )dT T T + 1 1([ eflydv31[m_m1 + 3_1[ Avdv311)d 171' T o T‘ 1 1 1 o 1 2 1 T T + 1 1([e511’121 2 AVde][mmT +R 1te511'121 2 AVde]1 )dt 1EF’ 1 0 T2 m2 T2 0 1 2 1 11 A(T-1 )12 Av T Av 1 ‘+ (£e—- 2 rde][mm ][ e—-deJ )d 171' T o sz T1 0 1 2 1 11 Av m1 A(T-T ) 12 Av 1 + -p-( ([1 %de][m1_2][e— 2 1 deB] )dT (2-25) 1 T o 0 1 2 dis the sinc the as we Dyna” 01 an 18 an this 1 COherg 1111M 44 If a specific load shedding, loss of generation or line switching disturbance occurs since in this case P = 1,13 the matrix Sx(Tl) has the form T 'r T ( 1([( eflydvgqugltf eflyde])dt (2-26) 1 S (T ) = .... x l T O 0 O -- 1 If the specific deterministic disturbance is an electrical fault since in this case P=0,m1=9am2=£2:31=R =B,and V(0)=Q. —2 the matrix §x1111 becomes T T T s (T ) = I 2([1 eflydv§]g_g1[( eflydv§J1)dt x 1 2 2 o 0 0 T1 A(T-1 ) T2 Av T A(T-T ) T2 Av T + I ([e—- 2 I e—-de]2222[e- 2 I e-de] )dT (2-27) 1 o o 2 This generalized RMS coherency measure can handle both deterministic as well as probabilistic descriptions of power system disturbances. Dynamic equivalents determined based on this measure would be independent of any specific contingency of a particular type and thus could be used as an equivalent for any disturbance which could be described in terms of this probabilistic disturbance model. It is shown in [15] that the RMS coherency measure evaluated over an infinite interval (T1 = 00) can be analytically related to generator inertias, synchronizing power coefficients of equivalent lines connecting internal generator buses, and the statistics 45 of the disturbances. Moreover, it is shown that, for disturbances in mechanical input power, probabilistic disturbances can be found which cause the RMS coherency measure to be a function of system structure alone, allowing coherent equivalents to be constructed which do not depend on the location of any particular disturbance. This is particularly important because it would identify coherent groups which are not disturbance dependent, leading to what may be called structural coherency as opposed to disturbance dependent coherency. In the next three subsections an infiniete interval RMS coherency measure will be derived for step, impulse, and pulse disturbances in the mechanical input power of the generators. 2.3.1 Infinite Interval RMS Coherency Measure for Step Input Disturbances In this subsection an algeraic formula is derived which relates the RMS coherency measure, evaluated over an infinite interval for step input disturbance to the parameters of the power system state model and the statistics of the step input [18]. An RMS coherency measure can be found from the general equation (2-23) provided we derive §x(w) for the step input disturbance. The appropriate form of §x (T1) for a step 5 input disturbance can be found by substituting P = 1, m2 = O, R _2 = 0, 1x10) = 0 into equation (2-25) and the resulted expression has the form T '1.‘ 1; 5x (T1) = -1—( 1([1 e—A-Vdvgth1 + 2151qu eAVdv_3_11)dT (2-28) —5 1100 0 Since the matrix A_ of the linear model is nonsingular, the interior an 111 am 46 integrals may be evaluated as T I eflvdv = [WeAT - I) (2‘29) Defining T T V] = gig] + m1 31.119. (2'30) and evaluating the interior integrals of equation (2-28) using (2-29) the following expression is obtained T T T S (11) = T" E1 J 1[efll_v]e5 '1 - eflT11-v]efi T+y_13th’ ——s l O 1 (2-31) Assuming that the system is asymptotically stable, i.e. all the eigenvalues of matrix A_ have strictly negative real parts, then as observation interval T1 approaches infinity, the first three terms in equation (2-31) vanishes and (2-31) becomes 1 T 1 T 1 T rm _Sx (T1) = §x (...) = A LA = [A mg] + gfllng B] (2-32) Considering the fact that for the matrix A_ given in (2—12b) -om:)1 5 -(m_)'1 5'1 = ----------- f ----------- (2-33) I I l o — — I _ d and using matrix B_ also given in (2-12b), AF1B_ becomes ............. (2-34) up up II I I I I I I I I I I I ---*--- For the step disturbances in the mechanical input power, m1 and R, as defined by (2-16) have the forms ....... (2-35) 13 u—J II I I I I I U I) c—J II I I I I I -—--‘--- where 5W1 and 3,] are the mean and variance of the step disturbances in the mechanical input power, respectively. Substituting (2-34) and (2-35) into (2-32), §x1m1 for step disturbances in mechanical input power becomes -1 T -1 [(M1) ENE.” '1' ."lnflnJHL’111 L1] ----- (2-36) (I! A 8 V II I I I I I I l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I -—--L-—- The upper left quadrant of Sx (m) which determines the coherency measure -—s between any pair of generators is defined as Sx (w) and is given by -—s § (w) = [(M )‘1MJER + m mT ][(M )‘1M1T (2-37) x —— — 41 ~4L41 —— — __5 This expression shows that for the step input disturbances Sx (w) and therefore, infinite interval RMS coherency measure are relatEdSto the parameters of the linear system model and the statistics of the disturbance. This expression also eliminates the need to process the swing equations to 48 compute the coherency measure as is required to determine the max-min coherency measure. Inspection of equation (2-37) shows that the RMS coherency measure will be a function of only system structure for any disturbance which satisfies the following relation. + m mT - I (2-38) 51 -4r41'— This condition is clearly met when the step disturbance in the mechanical input power at each generator bus is zero mean, independent of the dis- turbance at all other generator buses, and identically distributed (ZMIID), that is Eh1 T 0 : 5h] T 1- (2‘39) Substituting (2-39) into (2-37), Sx (w) becomes ——s T (...) = [(m1'1m1um1'1m (2-40) Since (BI)-1BI_= I, equation (2-40) shows that when the disturbance in mechanical input power is ZMIID the RMS coherency measure is a generalized inverse function of the synchronizing torque coefficients, thus the coherent groups are determined by line stiffness [15]. Another disturbance of interest is the disturbance which satisfies T T _ M154] + Eqim411M. T l. (2’41) In this case (2-42) 49 Thus, the coherency measure is determined by generator inertias 11 and synchronizing torque coefficients 1_ and the coherent groups identified for aggregation are determined by line stiffnesses weighted by the inertias of the generators at the ends of the lines. The disturbance which satisfies (2-41) is M2 M2 _ - 2 - D1ag(M 2,. , N-l’ =0 9 _R_ 19 1] 0) (2-43) m11 This disturbance is clearly dependent on the choice of the reference generator used to establish the state model. A reference independent disturbance can be found using the following disturbance 2 M2 M2) =o,3 2,....,N_], N _ - 2 1] - D1ag(M], M m“ (2’44) This disturbance is zero mean, independent over all generators of the system, and inertially weighted (ZMIIW). For this disturbance equation (2-41) becomes 2 for i = j T T = . , _ {M(B'” + m11fl111fl11'j 1 13.] '1,2,...,N 1 for i f j L Since the inertially weighted synchronizing torque coefficients also determine the modal structure of the system, the modal and coherent equivalents derived from the RMS coherency measure and the disturbance (2-44) will be nearly identical. They will be exactly the same if the coherent generators are so tightly tied to each other that the coherency measure between them is zero. Thus, the RMS coherency measure and the ZMIIW disturbance can capture both modal and coherent structure of the 50 system. The disturbance defined by (2-44) will be called the modal- disturbance of all generators. No deterministic disturbance can be found which satisfy (2-38) or (2-41) but it is shown in [15] that the effect of stochastic ZMIIW disturbance can be obtained as the summation of N deterministic dis- turbances. That is, let Sx.(”) be the matrix that results from a step -—4 disturbance proportional to M? on generator i only. Then N S (w) = 2 S (m) (2-4561 where (...) = (1 E{2<_,-(t)_>511(t)}dt (2-45b) 0 and x,(t) is the solution of the linear state model for the disturbance on the generator i. A similar sequence of N deterministic disturbances where each generator, in turn, experiences a step disturbance of one per unit would duplicate the effects of the ZMIID disturbance. Thus, to construct a coherent equivalent based solely on system structure using a single disturbance to identify coherent groups, a probabilistic disturbance is required. 2.3.2 Infinite Interval RMS Coherency Measure for Impulse Input Disturbances In this subsection an algebraic formula is derived which relates the RMS coherency measure, evaluated over an infinite interval for impulse input disturbance, to the parameters of the power system state model and the statistics of the impulse input. The RMS coherency measure matrix for the impulse can be found from the general equation (2-23) provided 51 we derive §x(m) for impulse input disturbances, using the following relation 1 s (...) = lim I 1 E{_x_(t)5 (t)}dt (2-45) where x(t) is the solution of 3(t) = Ax(t) + Buo(t) for the impulse input go(t) 6(t) and 6(t) is the Dirac delta function. Assuming =90 zero initial conditions i.e., t0 = O, and 5(0) = 9_ we can find x(t) as _ t A(t-T) At x(t) — 0 Bgod( )dT - Buo (2-47) substituting (2-47) into (2-46), §x (w) for impulse becomes I T T T T _ 1 At T T A t _ 1 At T T A t SX (m) - 11m I E{e—-Bu0BOB_ —- ldt - 11m I (e B EIBOB01B_e—- )dt -—4 T1 0 T1+m 0 (2-48) Am. where, B0 = --6-— represents the step change in generation output equivalent to the accel- erating powers due to this impulse. The statistics of this disturbance is defined to be _ _ 9-01 E01 m0 - E{BO} - E ____ _____ (2-498) 9. Q T .30] E Q 30 — E{[go - gojtgo - go] 1 - ------ 7 ----- (2'49b1 Q : Q 52 thus, .4 I l 1 T moml = .............. é ...... (2-49C) l I d T . . . where 301 - E{[BO] - 90]][B01 - 90]] } 15 the var1ance matr1x and £51 is the mean vector of the uncertainty in accelerating power on all generators due to this impulse (electrical fault). Letting yo = B( )BT equation (2-48) becomes e—-V e—- )dt (2'50) A closed form solution for §x(m) can be found by approximating the impulse as a pulse of very short duration. To clarify this let start with the pulse and its statistics and try to relate it to the impulse and its statistics approximately. We have already defined the pulse of duration T2 to be r B_ t > T2 3.2.”) = ( B2 0 f t 5 T2 , O t < 0 where g, = [AEB_E .QJT, and its statistics are m - = —21 — I T 1321 E 9- 32 ' E{[E£ ' EQJEEZ ' m2] } = """" 1 ---- O ' O _. l .— The impulse can be approximated by the above pulse as follows ' B_ t > T2 _ - __l Bo(t) - 905(t) _ 11m0 B2(t) ~ , B2 - T220 0 g t 5 T2 2 L B_ t < 0 This approximation shows that the statistics of pulse and impulse may be related as m = 1- m (2-51a) —2 T2 —0 R = 1L- R (2-51b) -2 T 2 —o 2 It is shown in [18] that for a pulse of very short duration (impulse), Sx (m) can be determined from the following equation -—I 5,100) = TS U (2~52> Where B_ is the solution of the following Lyapunov equation M + LAT = - 12 (2-53.) where _ T T y, ' —I32 T E29219- and T T T -T T w = lim 1 1(e51v eA-t)dt = 1im ( 1 2(ea-1v e5 t)dt (2-53b) —- —2 —2 T o T1+w 0 Comparing equations (2-50) and (2-53b) it is obvious that B_= T£2§x1(w), because V, = T2 Xo’ where subscript I means impulse. Substituting 54 matrices B_ and V. into equation (2-53a) the following Lyapunov equation 2 is obtained AS (e) + s (m)AT = - v (2-54) This equation shows that the §x (w) for impulse satisfies the Lyapunov I equation. To find Ex (m) we have to solve Lyapunov equation (2-54). I This Lyapunov equation can be easily solved by considering the fact that §_ (m) is a symmetric matrix [33]. Thus, partitioning §x (m) as I x1 I §1 5 §2 s (e) = ...... I ..... (2-55) —X I 1 T 1 _ §2 1 §3 _ and substituting matrices A_ and B_ as defined in (2-12) into equation (2-54) the following algebraic relation is obtained - l - F - '- I - (- I _ _- Q. E l. §q E §2 §q E §2 Q. 5 (MI) I I I I __ TTTTTTT 1"TTTTT T'T'lT'TT + TT'TlTTT T'TlTTT'TT T T 0 -(MT) 1 - I 51 1 s 51 1 s I 1 - I I.“ _ TT' 1 OT' .4 L T2 5 ‘_31 ..72 1 ‘31 {T 1 0" A (2-56) yo can be put in partitioned form by substituting matrices B, 30 and vector Eb in the following equation I T I T I T T T T 9- g 9- 301 1 9019015 9- 9- E 15 -0 = em, + momom = ---------------- 1 ------- 1 ----- 1' d EAL. 2 i 9. _q i (_L) After carrying out the multiplication yo becomes 55 _ . - Q i 9 yo = ------- i ----------------- imi' (2-57) h ‘9- i MB01 T E101-"301111” - Substituting yo into equation (2-56) we get — T i s 1 ’5 i -s (MT)1-ds - 110 i o - §2 : —3 —2 g —1 -- —2 -— g -— --------------- -ll------—-------- + -----:---—--------- = -----}------- T . . T T g , :(fl)_5_.l T 022 I T (fl)_5_2 T 0.5.3— _§3 I T._S_2(MT) -053] _9. l .11 ,1 (2-58) where H = -M(R + m m1 )MT —- --ol —ol-ol — From equation (2-58) one can derive four equations and four unknown, that is T _ §§_ + 22.- Q. (2'59) -s (MT)T - US + s = 0 (2-50) .;L-‘ —2 —3 -— -(MT)S - 031 + s = 0 (2-61) -——-4 —2 —3 -— T T _ T T -(|‘_’|l)_5_2 - _S_2(M_T_) - 2d_S_3 - - M0301 + 30190115 (2-62) Since -§x(m) is symmetric matrix, that is B2 = B; from equation (2-59) B2 =.g (2-63) Substituting B2 = 9_ into equation (2-62), B3 becomes - QL. T T - §a‘%[m%1+%flmm1 95“ Substituting B2 = 9_ into equations (2-60) and (2—61) the following expressions are obtained 56 (2-65) I or, equivalently (MD‘1S (595 = s s _3 —1 —3 -1 (2'65) I Adding both sides of equations (2-65), (2-66) and substituting B3 given in (2-64), B, becomes = . .. = 1 -1 _ -5-1 5511 1 40 (Ml) M- (301 —01'-"-011M—T 1 M1301 + —01—o11MT(M—-1—1T1 (2 671 S§{(m) is the upper left quadrant of S§{(m) which determines the coherency measure between any pair of generators for the impulse disturbances. This expression shows that for the impulse input disturbances the matrix 551(w), which defines the infiniteinterval RMS coherency measure is related algebraically to the parameters of the linear system model and disturbance statistics. 2.3.3 Infinite Interval RMS Coherency Measure for Pulse Input Disturbances The appropriate form of SX (T1) for a pulse input disturbance ——P can be found by substituting P=O,m_]=B,R ...] = 9.’ lx(0) = into equation (2-25). Sx (00 ) may be obtained as the limit of Sx (T ) ——P when T1 approaches to infinity. Sx (m) for a pulse disturbance of ——P duration T2 has been derived in [18] and is shown to be m T" sX (...) = 2 -§,— [An-211+ A_(A"'2)11 (2-68) ——P n=2 ' where B_ is the solution of the Lyapunov equation (2-53a). The solution of this Lyapunov equation is similar to the solution of Lyapunov equation 57 for the impulse disturbance. The solution of B_ is exactly the same as the solution of Sx (w), except that we have to use the statistics of —-I the pulse input disturbance instead of the impulse statistics. This solution has the form 1 ‘1 T T T T -T $111511) MKiev-1321921111 +fl(321+m21m21)11 (III) 1 9 fl = 0 -1-[M(R +m mT )MTJ (2-69) - - zd --21 —21—21 — j where m2] and 32] are the mean vector and variance matrix of the un- certainty in accelerating power on all generators due to the pulse input disturbance, respectively (electrical fault). If the pulse duration (fault clearing) time T2 is very short, the first term in the series will be required, and under this assumption _ 2 equation (2-68) shows that for the pulse input disturbance, §x (w), which p defines infinite interval RMS coherency measure is related algebraically to system structure and disturbance statistics 2.3.4 Grouping Algorithm: The first step in producing dynamic equivalents is the identification of coherent, or nearly coherent generators. Several approaches have been proposed for identifying coherent generators. In this research we will identify nearly coherent generators based on the RMS coherency measure. This prcedure is as follows: Once the matrix §x(w) is computed for a particular disturbance, the coherency measure between any pair of generators can be computed using 58 (2-23). Based on the computed coherency measures, a ranking table of the coherency measures between each pair of generators, ordered from the most coherent pair of generators to the least coherent ones will be found. The coherent groups of generators are identified by applying a commutative or transitive rule to this ranking table. The commutative rule means that a generator must be coherent with all generators in an existing group before it can be added to that group. The transitive rule means that if generator G1 is coherent with generator G2 and generator G2 is coherent with generator 63 it implies that generator G1 is coherent with generator GB. Once the coherent groups of generators are identified, the second step in producing dynamic equivalents is to aggregate each coherent group of generators into a single equivalent generator. In the next section a brief discussion of the aggregation techniques will be presented. 2.4.1 Coherency Based Aggregation Technique The objective of coherency based aggregation technique is to represent each group of coherent generators by a single equivalent generator that faithfully and accurately represents the effect of the coherent group on the rest of the system and also reduce the number of equivalent lines in the network to correspond to the number of generators retained in the aggregated system. The aggregation problem is to determine the parameters which characterize the equivalent generators and the reduced set of equivalent lines. The procedure for aggregating the linear system model using the coherency-based aggregation technique is now presented. The linear model that will be used in the following discussion 59 is a slight modification of the model developed in (2-12). That model was a state space representation of order 2N-2. Half of the equations in that model are simply defining equations that relate the generator angle deviations A3, to the generator speed deviations A8,, i = 1,2,...,N-l. For the present analysis it is more convenient to rewrite the 2N—2 state equations (2-12) as N-l second order equations A ’5 ~ A—N-I = (-M_T_)A6N_] - 0A6N_-I 4' _B (2'70) where A6 = [A6 A8 A6 A6 A3 JT N-1 1’ 2"°°’ m’ m+1""’ N-1 Di 0 = MT- : I = 192, IN 1 E_= [m_ ML] E - [APM-I,APM2,...,APMms-o-,APMN : APL'I IAPLZI-ooaAPI-KJ and matrices I, L, and B_ are defined by (2-lla) and (2-12c), respectively. Now consider a power system of N = m+n generators of which the first m generators form an internal group, or "study group" where the disturbances can occur and the last n = N-m generators form an "external group" whose behavior is being investigated for the purpose of finding some property which will allow the group to be replaced by a single generator. Partitioning the vector ASN-l = [5h : I matrices in (2—70) in such a way to match the dimensions of the two vectors 5,], and also all of the 5H and x, equation (2-70) can be written as £11 ”(‘M1111 “M1112. 151- F0—1-m 9-1311 _MT (M1111 FARM- ---1---- I I I I I I I I I I + I I I I l I I I I I I I I I I I I II I I I I I I I I ---|------ T (1111122, 521 9 where (2-71) -1 I -1 -1 -1- M] E-Me -Me ........ -Me -1 I -1 -1 -1 M2 9 I-Me -Me ........ -Me 5 O I I I I I ' I ‘ I An: -. -. : M I-M -11 M = ..-..-J. ...... = .................. T-:.-_§ ............................ €- — I I I I -1 -1 -1 1 LT ' T224 1 -1 -1 -1 -1 _0_ E-Me Mm+2-Me “I'I I I o l -1 2 “Me I -1 -1 -1 -1 -1 L 5 'Me 'Me ' MN-1_Me -Me.J x, = [A6],A62,...,A6m]T is the m-dimensional swing vector of the generators inside the "study group" with components 81 = a, - 6e, 1 = l,2,3,...,m. _,.. «.T. .. . 52 - [A6m+1’A6m+2"°"’A5N-1J 15 the (n-l) d1men510nal sw1ng vector of the generators inside the "external group" with components 8. = a. - a j= m+l,m+2,...,N-l. JJe’ 61 (M ) = ....... .1 ....... L = 6 = 1=m+] i i _ : ( a _ i , e E] _21 . —-—22 _—21 : 22‘ MM T Im and ln-l are identity matrices of dimension m and n-1, respectively. Eh = [Eh] E MHZ] is me matrix consists of the first m rows of matrix B, E, = [Q_E M22] is (n-l)xN matrix consists of the last (n-l) rows of matrix M, (_M_L)1 = mug“ + Miz-L-ZT 5 MULTZ + MTZBZZJ is me matrix consists of the first m rows of (MB) matrix. (ML) M L E M22L22] is (n-1)xK matrix consists of the last 2 T 1—22—21 (n-l) rows of matrix (ML). K is the number of load buses in the system. Assuming that the disturbance can only occur in the mechanical input power of the generators inside the I'study group" that is T I I T g: [APM],APM2,...,APMm ; g] T1E1 :9] equation (2-71) can be written as X, = (214.1qu + (211312222 - 05s] + 2,2, (2-72a) £2 = (-M_T_)21_x_1 + (@1521, - 032 (2-72b) where B1 = B1] is a (mxm) matrix of the first quadrant of the B_ matrix. 62 g, = [APM],APM2,...,APMm]T represents change in the mechanical input power of generators inside the "study group“. It is now required to find a reduced order model to represent the relative motion of the m generators inside the "study group" by aggregating the n generators of the external group. In order to determine the parameters which describe the equivalent generator for the coherent group of n generators, consider summing the synchronous machine equations of the members of the external group, that is N APM. - Z APG, - N I M' TT'Awi T ._ . - m+l 1-m+1 1 1 t DiAw. (2-73) 1 IIMZ m+l Since coherent generators swing together, thus each Ami in (2-73) can be replaced by an equivalent speed deviation, Aweq, and in this case equation(2-73) becomes N N N N d - - 244) M. ———A - APM. - APG. ( X D.)Aw ( (1=m+1 1) dt weq iZm+l 1 iZm+l 1 i=m+l 1 eq Or, in more compact form 51—— = - - (2-75) Meq dt Ameq APMeq APGeq DeAweq Comparing equations (2-73) and (2-75) it is obvious that the parameters which describe the equivalent generator are 63 Meq = igm+lM1 (2-76a) N Deq = iZm+101 (2-76b) N APMeq = iZm+1APM1 (2—76c) N APGeq = 1§m+1APGI (2-76d) The parameters which describe the reduced set of equivalent lines can be determined from equation (2-76d). Each equivalent line in the un- reduced linear model is characterized by a synchronizing torque coefficient which is determined from (2-lla). The synchronizing torque coefficients may be used to compute the deviation in power flow from any internal generator bus i to another internal generator bus j, which is defined as [18] APij = Tij1A51 - Adj) (2-77) The total deviation in the electrical output power produced by generator i can be found by summing equation (2-77) over all possible choices of j, that is . - A6.) (2-78) Subs! Sinc: rotOI lngh wherl term whEr 91 CORD to t Whic C0he (2.7 Drgd 64 Substituting (2-78) into (2-76d) i i APG = T..(A6. - A6.) (2-79) eq i=m+l i=1 11 1 1 - iii Since the coherent generators are assumed to have the same deviation in rotor angle, each A61 in (2-79) can be replaced by an equivalent rotor angle deviation, Aaeq and APGe becomes q N 111 APG = T..(A6 - A6.) (2-80) eq iZm+l jZl ‘3 eq 3 where the summation interval on the index j has been changed to drop terms which are zero. Equation (2-80) can be written as m APGeq= .ZlTej1A5eqT Adj) (2-81a) where N T . = I T (2-81b) 93 i=m+1 ‘3 Tej is the synchronizing torque coefficient of the equivalent line which connects the equivalent generator to the generator j which is external to the coherent group. Equation (2-81b) shows that, Tej is the sum of the synchronizing torque coefficients in the unreduced linear model which connects external generator j to the individual generators of the coherent group. This aggregation technique is called averaging because of equation (2-76), and has the advantage of preserving the network structure in producing dynamic equivalents. Thus, a basic feature of this averaging 65 aggregation is that the identity of the system outside the coherent group is retained. In the original work of [25], there is no theoretical justification for developing averaging equivalents based on a coherency measure, which depends on the voltage angle and is independent of voltage magnitude, but the recent study of [20] provided this theoretical justi- fication. If the n generators of the "external group" constitute an "ideal coherent group", i.e. x, = 0 then the coherent group of n generators can be replaced by an inertial averaged equivalent generator, and in this case the transient 51(t) of the generators inside the "study group" in response to any disturbance internal to the "study group" can be found by substituting 52 = 32 = 9_ into equation (2-72), and the reduced order model has the form 2, = (2111),,2, - oi, + 2,2, (2-82) The validity of this aggregation technique is shown in [20]. 2.4.2 Singular Perturbation Aggregation Technique: Singular perturbation in the usual sense means that a system con- tains a set of states that are highly damped and decay rapidly to zero in a short boundary layer of time after a disturbance has been applied to the system. It is also shown in [22] that the singular perturbation can be applied to systems with lightly damped oscillation where some modes of oscillation are fast and others slow. Thus, singular perturbation ag- gregation technique is appropriate if the system possesses a two time scale property with the relative motion of members of the "external group" faster than the relative motion of those outside the "external group", i.e. 66 inside the ”study group". System (2-72) has a two time scale property if it can be expressed in the form 51 T iT1151 T E1252 T “51 T 31! (2‘83“ 2" _ . “ 52 T 521351 T 52212 T 1‘ 01‘2 ”’83” where u is a sufficiently small positive scalar. The two time scale property of (2-72) can be expressed as norm conditions on the matrices of (2-72) [22]. It is shown in [19] that these norm conditions can be satisfied if the system satisfies the strict synchronizing coherency (SSC) property. In a two time scale system of the form (2—83) the relative motion of the n machines of the external group comprises slow oscillation and fast oscillation. The effect of the fast oscillation on the machines outside the external group is of order p. This suggests that the singular perturbation method would be an appropriate aggregation technique. In the singular perturbation method the reduced order model is obtained by neglecting the effect of the group intermachine fast oscillation on the relative motion outside the external group. Mathematically this is done by setting p = O in (2-83b) and eliminating 52 from (2-83a) to obtain the reduced order model " _ _ -1 , _ 1‘1 T (511 512522521) 35-1 ' 0911 T 13.19.} (2 84) Notice that the aggregation here is not equivalent to solely replacing the external group of n machines by one equivalent machine because there is change in the equations describing the relative motion of the machines outside the external group. Therefore, singular perturbation aggregation represents "external group” by a single equivalent generator 67 and changes "study group“ to properly represent steady state of the ex- ternal group. Thus, a basic feature of the singular perturbation method is that the structure of the system outside the external group is changed to preserve the steady state contribution of the neglected fast modes. The validity of the reduced order model (2-84) to represent x, for sufficiently small p is provided by the singular perturbation theory [24]. Note that the theory at present only allows singular perturbation aggregation of a power system model when the eigenvalues within the external group are fast (high frequency) compared to those outside the external group which is the SSC condition. It is obvious that if strict synchronizing coherency (5:; = 9) holds the singular perturbation equivalent (2-84) is identical to averaged equivalent (2-82). It can be seen that the form of the singular perturbation equivalent (2-84) and the averaged equivalent (2-82) are also identical under strict strong linear decoupling (f5; [2, + Q) and strict geometric coherency (£2, + 9) but the present singular perturbation theory will not permit aggregation under these conditons. Although there exist different competing aggregation techniques, there is no theory at present on which is best for different contingency type and time interval combinations. This is the objective of the future chapters. Moreover, a new aggregation method is developed that is similar to singular perturbation but allows aggregation for SGC and SSLD coherent structural conditions and for impulse disturbances. Singular perturbation can not be applied under either of these conditions. CHAPTER 3 DERIVATION OF OPTIMAL AGGREGATION TECHNIQUES FOR DIFFERENT CONTINGENCY TYPES WHEN THE OBSERVATION INTERVAL IS INFINITE (T1 = 0°) Once the coherent groups of generators are identified for a particular type of disturbance (impulse, pulse, or step), the next step in producing dynamic equivalents is to aggregate each coherent group of generators by a single equivalent generator which accurately represents the effect of the coherent group on the rest of the system. The principal objective of this chapter is to find optimal ag- gregation techniques for different classes of structurally coherent gen- erators which is best for different contingency type and time interval combinations. The derivation of optimal aggregation is based on the linear model of the system. To motivate the discussion, consider the system represented by equations (2-72a) and (2-72b) where the contingency is change in the mechanical input power of generators inside the "study system". It is required to find an optimal reduced order model to represent the relative motion of the m = N-n generators of the "study system" by optimally aggregating the n generators of the external group. Aggregation techniques derived based upon the structural properties of the system are to be developed that would preserve the effect of the relative motion of the generators inside the "study system". There are two types of struc- tural conditions which can be used as a basis for aggregation. First, 68 69 the structural conditions for the coherency of the n generators of the coherent group. These conditions may belong to any of the five conditions defined in Chapter 1, namely (SSC, SGC, SSLD, PC, WLD). Satisfaction of any of these structural conditions on the power system causes the external group of generators to remain or appear coherent in response to distur- bances confined to the "study system". A second type of structural con- dition for aggregation is the two time scale property. It is shown in [19] that if a system satisfies SSC condition then the system possesses the two time scale property. It is also shown that the structural con- ditions SSC, SGC, and SSLD apply to both the nonlinear and linear models of the power system while PC and WLD apply only to the linear models. The optimal aggregation methods derived will only exploit SGC and SSLD because the disturbances are restricted to the "study system". Thus, although the optimal aggregation are derived based on the linear models the resultant aggregation methods could be extended to the nonlinear models if the optimal linear model aggregation methods could be extended to the nonlinear models. The optimality of aggregation methods will now be formulated based on both approximate satisfaction of the above structural conditions (SGC, SSLD) and based on each contingency type and time interval combinat- ions. 3.l Optimal Problem Formulation The optimal aggregation methods for each contingency type and time interval combination will be derived using the model simplification method of [30]. The method can be described as follows. Given a linear time 7O invariant power system model described by the state equation (2-12), i.e. gm = Am) + gem (34a) and the output equation x(t) = 9 zit) (3-lb) where the parameters of this model are defined in equations (2-11a) and (2-12). Find a reduced order model of the form :(t) = E 2(t) + E _u_(t) (3-2a) flm=§zu1 mew where dim(§_) = 2m )ET} —-—xs -— xS -— , T T +w 1 l 1 E{_>_<_(t)gT(t)}dt]§T} 0 T I 1 E{X(t)x_T(t)}dt]BT} (3—7) .1. 1 o 1 Tri§£1im T+oo 1 It was shown in Chapter 2 that §x (m) is a (2N-2)x(2N-2) s symmetric matrix of the form T s (...) = lim 1— 1 E{x(t)xT(t)}dt = [[1313 + mgTJTAT1§11 -x T —- - —- l l l s T,+w l O T -1 T -1 T . T [(141) MIA” +[n_,,m,,][(M_T_) M1 5 g l = ------------------------------------------ ,L ------- (3-8) I 74 where, the (N+K)xl mean vector m4 and (N+K)(N+K) variance matrix 34 are defined in equation (2-l6). The (le) vector m1] and (NxN) matrix 54] are the mean and variance of the step disturbances in the mechanical input power of the generators, respectively. Assuming that the disturbances can only occur in the mechanical input power of the generators of the "study system", £51 and 551 be- COMES -Jl m1] = """ a R = """" ‘E """" (3‘9) N where, the (mxl) vector E5] and (mxm) matrix 34] are the mean and the variance of the step disturbances in the mechanical input power of the generators in the ”study system", respectively. Since in the derivation of optimal aggregation the upper left (mxm) submatrix of §xs(w) is only needed, partition the (N-l)x(N-l) matrix flI_ and the (N-l)xN matrix M_ as <_M_T_)n 5 (MT)12 Fun 5 u], M = ........ J ......... ’ M: ........ J ....... — E “ E Now considering the fact that [(241)11- s11" : -g(t)g<_ (twang TI T o 0 l (3-l8) T T ~ 4 T—1,|Tr{g§x(T1)g} - mg §X(T1)£ ll T For impulse type disturbances the performance measure 3 can be obtained from equation (3-l8) by substituting p = 0 and taking the limit of the integrals when T1 + m, i.e. J = |Tr{§__S__X (w) I 2T} - m: s; («JETTI (349) I It was shown in Chapter 2 that §x (m) is a (2N-2)x(2N-2) matrix of the I form _l -l T T T T -T T ETTT—T-T-T MTBOTTTOTEOTTM- T-MTBOTTMOTEOTTM-(MD T 9- _s_X (...) = T o —T-—[M(R +m mT M] L - o -—OI —OI-OI - where, the (le) vector m01 and (NxN) matrix R are the mean and —Ol variance of the impulse type disturbances in the mechanical input power of all generators in the system. Assuming that the disturbance can only occur in the mechanical input power of the generators of the "study system”, and partitioning the matrices MI, M, @01, and 301 to match the dimension after carrying out the multiplications and defining V0 = E01 + fio1fig1, 31 = Q-éx (”)ET I becomes 79 ~ - T 4 411T T ‘TT ‘ Egg-ICTM—TTH ' TLTTT2T —T-MT22(MT—T21T TMTTMoMTTTC-T l T -T T T 4—oCTTM 4T-TToMTTTTTL TTTT ' TLTTTzTLTT22TLTTzTT 9T (3'20) where _~ _~ 1 _ TMT 3m 5 9. — __ I MOT """ ’ MOT """" 1' “““ 9 9 i 2 and m01 and 301 are the mean and var1ance of the fault deturbances in the mechanical input power of the generators in the "study system", respectively. Similarly,the (2m)x(2m) matrix §§ (m) can be shown to be I .1. 4o [(LVTM V MT + M V MT )‘TT ll—O—Jl ‘ (:1 T ~ T L 9- 2_<:TMTT—o—V MTTT _T becomes 3 =—T-—C[('”T)'TM VMT +M VMT(~T)T T —2 4o 4 -— —ll—0—-ll —ll—O—ll-—-— TCT (3‘2” Substituting equations (3-20), and (3-2l) into equation (3—19) the following expression can be obtained for the performance index 3. 80 -_T 4~TT ~ _T_O 1 MT - l -T T + 4 Tr{_C_][MnyO_n][(l__4T)H (L_1T)]2(_T1_T)22(L1T)213 £1} _me [MUM ‘ MT TcT T- 1 OTr{C [M MT (M_T_)’T:T_c_ TT }| (3 22) " 40 —l —— —TT—oM TT —l —ll—O —ll ' ~ This expression shows that this error criterion is minimized (J = 0) if M: = (MT) MT)5;(_M_T_) —— TT ‘ (MDT2(—— 2T This result indicates that the "Optimal Modal Coherent" aggregation for the impulse has the same form as the step case, but the coherent groups are different from the coherent groups determined based on the step, because the coherent groups are determined based on the RMS coherency measures for a particular contingency type. This result also indicates that OMC aggregation can be used for fault type contingencies approximated by impulse, while present singular perturbation method is not valid for impulse inputs either restricted to the "study system" or permitted in both the study and external systems. Singular perturbation aggregation of fast dynamics for impulse inputs is not possible because the impulse disturbance contains both low and high frequency variations which generally excites the fast modes and does not permit their elimination. In this chapter we have shown the procedure for deriving aggregation techniques when the contingency is a change in the mechanical input power. For other types of contingencies that can be represented as step changes, e.g., load shedding or line switching, the procedure is the same except that the equations for deriving optimal aggregation will be different. To justify the ideas formalized in this chapter these results will be tested on the 39 bus New England System. These results are given in the next chapter. CHAPTER 4 COMPUTATIONAL RESULTS FOR THE OPTIMAL-MODAL-COHERENT EQUIVALENTS To verify what has been derived in Chapter 3 as the "Optimal- Modal-Coherent" aggregation, extensive testing of the procedures on the l0 Generators, 39 Buses New England System has been conducted. This system is frequently used as an example system in the literature. The one-line diagram of this system is shown in Figure 4-l. The generator data, bus data, and line data are given in Tables 4-l, 4-2, and 4—3, respectively. The main objective of this chapter is to compare the eigenvalues, the coherency measures, and the simulation results of the full 39-bus New England System with the reduced order model obtained based on the "Optimal-Modal-Coherent" aggregation for the following disturbances: 1. Global Disturbances: Modal disturbance of all ten generators. 2. Parochial Disturbance: (i) Modal disturbance of generator l. (ii) Modal disturbance of generator 8. 3. Local Disturbance: Modal disturbance of generators l and 8. The results given in this thesis are based on a linearized model of the New England System, because for the determination of the 81 82 .Empmxm ucmpmcm zmz mmmzm mm .mcowmcmcmu o_ any we Emcmmwo mcw4-mco ._.¢ mgsmwm m mm m or m mm F_ am my Fm T. , om . $, n _ mm o_ NP 9 a" TTTTTTWPI_ \\\\\\ .TT _ _ S TU». mm mm LN .. q\\ 55. m m_ Emcm>w qumMme m _ .. _ L F \ T TIII:IITTIII -_ .HIIIILI \\)I \\ F 11;.111 «N am mp _ .. /IT..T......IT/ \Tlllll N am os//\\ , om mm .4 . WW L c . 6 mN 2mhm>m >o=pm . h ei- f0 si sh wT' 83 eigenvalues and coherency measures a linear model is needed. However, for the simulations, a nonlinear model is more proper to use. But, since a nonlinear simulation is much more expensive and in many cases show minor differences from the linear simulation [l2], linear model will be considered adequate throughout these analysis. Moreover, one would expect the same simulation results if a nonlinear model were used instead of a linear model, because the "Optimal-Modal-Coherent" ag- gregation only exploits SGC and SSLD and these structural conditions apply to both the nonlinear and linear models of the power system. Thus, although the optimal aggregations are derived based on the linear models the resultant aggregation methods could be extended to the non- linear models if the optimal linear model aggregation methods could be extended to the nonlinear models. Table 4—l. Generator Data for the 39 Bus New England System. Generator Kinetic Energy RA Xé Number (MWS) (p.u.) (p.u.) l 4200. 0. .031 2 3030. 0. .070 3 3580. 0. .053 4 2860. 0. .044 5 2600. 0. .132 6 3480. 0. .050 7 2640. 0. .049 8 2430. 0. .057 9 3450. 0. .057 l0 50000. 0. .006 Note l. The inertia constant M can be found from the kinetic energy using the following formula: M = kinetic energy_ = KE [ MJ-secJ n.f l88.496 rad where, f is the frequency of the system in Hertz Not la Bu NL 84 Note 2. The per unit (p.u.) value of the inertia constant M based on the loo MVA base power can be obtained from the following formula: KE 3 I Table 4-2. Load and Bus Data for the 39 Bus New England System. Bus Voltage Angle Pgen Qgen Pload Qload Number (p.u.) (Deg) (MN) (MVAR) (MW) (MVAR) 1 1.048 -9.37 0.0 0.0 0.0 0.0 2 1.049 -6.80 0.0 0.0 0.0 0.0 3 1.030 -9.65 0.0 0.0 322.0 2.4 4 1.004 -10.47 0.0 0.0 500.0 184.0 5 1.005 -9.31 0.0 0.0 0.0 0.0 6 1.007 -8.62 0.0 0.0 0.0 0.0 7 .997 -10.81 0.0 0.0 233.8 84.0 8 .996 -11.32 0.0 0.0 522.0 176.6 9 1.028 -11.12 0.0 0.0 0.0 0.0 10 1.017 -6.21 0.0 0.0 0.0 0.0 11 1.013 -7.03 0.0 0.0 0.0 0.0 12 1.000 -7.04 0.0 0.0 8.5 88.0 13 1.014 -6.92 0.0 0.0 0.0 0.0 14 1.012 -8.58 0.0 0.0 0.0 0.0 15 1.016 -8.97 0.0 0.0 320.0 153.0 16 1.032 -7.55 0.0 0.0 329.4 32.3 17 1.034 -8.55 0.0 0.0 0.0 0.0 18 1.031 -9.40 0.0 0.0 158.0 30.0 19 1.050 -2.92 0.0 0.0 0.0 0.0 20 .991 -4.34 0.0 0.0 680.0 103.0 21 1.032 -5.14 0.0 0.0 274.0 115.0 22 1.050 -.69 0.0 0.0 0.0 0.0 23 1.045 —.89 0.0 0.0 247.5 84.6 24 1.038 -7.43 0.0 0.0 308.6 -92.2 25 1.058 -5.43 0.0 0.0 224.0 47.2 26 1.052 -6.68 0.0 0.0 139.0 17.0 27 1.038 -8.70 0.0 0.0 281.0 75.5 28 1.050 -3.17 0.0 0.0 206.0 27.6 29 1.050 -.41 0.0 0.0 283.5 26.9 30 1.048 -4.38 250.0 145.1 0.0 0.0 31 .982 0.00 563.3 205.5 9.2 4.6 32 .983 1.79 650.0 205.7 0.0 0.0 33 .997 2.29 632.0 109.1 0.0 0.0 34 l.012 .85 508.0 167.0 0.0 0.0 35 1.049 4.27 650.0 211.3 0.0 0.0 36 1.064 6.96 560.0 100.5 0.0 0.0 37 1.028 1.35 540.0 .7 0.0 0.0 38 1.027 6.65 830.0 22.8 0.0 0.0 39 1.030 -10.92 1000.0 88 0 1104.0 250.0 m (p.u.) (4-1) Table 4-3. From Bus LoomNOSO‘U'lm-h-hwwNNN—Jd To Bus 2 39 3 25 30 4 18 T4 Line Data for the 39-Bus New England System. A U23 C OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 85 x (p.u.) .041 (p.u.) d OOOOOOOOOO B .699 .750 .257 .146 .000 .221 .214 .134 .138 .043 .148 .113 .139 .078 .380 .073 .200 .073 .172 .366 .171 .134 .304 .255 .068 .132 .322 .257 .185 .361 .513 .000 .240 .780 .029 .249 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 d-d-d—l-d—J-J—‘d-JOOOO—‘OOOOOOOCOOOOOOOOOOOOOOOOOO—‘OOOO 86 The overall procedure for forming an “Optimal-Modal-Coherent" dynamic equivalent is composed of three steps: 1. Definition of the "study system”. 2. Identification of groups of nearly coherent generators which are valid for disturbances in the "study system". 3. Dynamic aggregation of a coherent group into a simplified model. The study and external systems of the New England System is shown in Figure 4-l. This figure shows that the "study system" consists of generators l, 8, 9, and l0 and the rest of the generators form the "external system". The identification of the coherent groups would be based on the procedures proposed in [l8], which rank order the infinite interval RMS coherency measures between each pair of generators for a particular disturbance (step, impulse, pulse) from smallest to largest. This RMS coherency measure between each pair of the Eigill- possible generator paris can be computed using the equation (2-23) for a system of N generators once the matrix §x(m) is known for a particular contingency (step, impulse, pulse). Then, the coherent group selection procedures begin aggregating the most coherent generator pair first and continue aggregation of less coherent pairs moving down the ranking table using the commutative or transitive rule. It has been shown that serious loss of model accuracy is predicted when the rate of change of coherency measures required to obtain additional levels of aggregation suddenly changes [l8]. This technique can be used to assess the minimum order equivalent with satisfactory performance. 87 Transitive Rule: This rule requires that: (i) A generator be added to an existing coherent group if it is coherent with one of the generators of that group. (ii) Two coherent groups be combined if one member of the first group is coherent with one member of the second group. Commutative Rule: This rule requires that: (i) A generator be added to an existing coherent group if it is coherent with all generators of that group. (ii) Two coherent groups be combined if every member of the first group is coherent with every member of the second group. Comparison of these rules obviously indicates that the com- mutative rule is too conservative, while the transitive rule is too liberal. Preliminary results showed that the transitive rule works better at the beginning of the ranking table in some cases, while the commutative rule works better going further down the ranking table. Further research is needed to determine which of the two rules performs best on the performance of the equivalents in reproducing the behavior of the unreduced system for different levels of aggregation and different types of contingencies. Based on the above discussion the commutative rule will be used throughout this research in determining the coherent groups. Once the coherent groups are identified, the next step in con- structing dynamic equivalents is to aggregate each coherent group by a 88 single equivalent generator. In this chapter the performance of the "Optimal-Modal-Coherent" aggregation will be examined for different disturbances. 4.1 Global Equivalents for Step Input Disturbances: To construct a global dynamic equivalent which is accurate for any disturbance it is shown that an infinite interval RMS coherency measure evaluated based on the modal disturbance of all generators of the system can be used to identify the coherent groups [19]. These coherent groups has been shown to depend on the synchronizing coherency property of the system that are characterized by a tree of (Pj-l) stiff interconnections between Pj generators in each group j for each level of aggregation [19]. Thus, these groups represent the ”strongly bound coherent groups", or tightly interconnected groups of the system for different levels of aggregation. These tightly interconnected generators tend to remain coherent in the face of very strong disturbances in the overall system. Equivalents produced based on these strongly bound groups are appropriate for investigating global dynamic structure and stability. Therefore, this type of global equivalents can be used for: l) Study of the weak boundaries and lines that would be vulnerable to security and stability problems for loss of generation contingencies. It has been shown that the boundaries between the strongly bound groups, which are detected by the RMS coherency measure and modal disturbance of all generators, are the vulnerable boundaries for loss of generation contingencies. The intermachine dynamics in these 89 strongly bound groups could thus be eliminated for dynamic equivalents required to study these security and stability problems on very large inter-regional data bases. In other words, this type of equivalents can be used for study of inter-area power transfer limit in trans- mission planning. 2) Study of the islands that naturally form when contingencies occur which lead to islanding. The islands would likely be formed of strongly bound groups if no relaying action caused separation within one or more strongly bound groups. The relaying action that could form islands should produce islands that should not have weak boundaries that would be vulnerable to the significant stresses on the trans- mission network when islanding occurs. Thus, the study of islanding may be assisted by the use of equivalents that only retain the weak transmission boundaries where islanding would or should occur. The modal disturbance of all ten generators of the New England System is a zero mean, independent, inertially weighted (ZMIIW) disturbance in the mechanical input power of all generators with _ _ . 2 2 2 2 Ehl - 0, 54] — DTag{M],M2,...,M9,M]0} (4-2) where, Ehl is the mean vector and 34] is the variance matrix of the uncertainty in the mechanical input power of all generators of the system and Mi's are the generator inertia constants of the system. These inertia constants in (p.u.) may be found from the generator data table using the equation (4-l), i.e. 90 M1 = 0.2228 M2 = 0.1607 M3 = 0.1899 M4 = 0.1517 M5 = 0.1379 M6 = 0.1846 M7 = 0.1401 M8 = 0.1289 M9 = 0.1830 M10 = 2.6526 The coherent groups can be formed based on the RMS coherency measures between any pair of system generators using equation (2-23) A once the matrix S“ (w) is known. As shown in Chapter 2, fix (w) has 5 s the form (...) = (MTV‘M T )mTLMD'T —x — ll + m115111 (4'3) To compute §x5(w), it is obvious that the inertia matrix E, the synchronizing torque coefficient matrix I, and the statistics of the disturbance for the system must be known. All the computations of the New England System were made using the generator l0 as the system reference. Thus, the (9x10) dimensional inertia matrix M_ for the New England System has the form fl = ' . (4-3a) IO 91 The (l0x9) dimensional synchronizing torque coefficient matrix I, in the generator 10 reference frame is computed using the modified version of Podmore's linear simulation (LINSIM) program for the New England System, with all load and generator terminal buses eliminated, and is given in Table 4-4. Now that the matrices M,I, and 34] are known, §x5(w) may be easily obtained and then the coherency measures between any pair of gen- erators can be found using the equation (2-23). Table 4-5 is the ranking table of the coherency measures between each pair of generators, ordered from the most coherent pair of generators to the least coherent ones for the modal disturbance of all l0 generators. Coherent groups of generators were formed by applying the com- mutative rule to this ranking table. As one proceeds down this ranking table individual generators are included in groups and later groups are merged to form larger groups. Once the coherent groups are identified for all levels of aggrega- tion, the next step is to aggregate each coherent group at each level of aggregation into a single equivalent generator. Each level of ag- gregation reduces the number of generators by one and the order of the system state model by two. The general procedure for aggregating coherent groups based on the "Optimal-Modal-Coherent" aggregation is now presented. Given a linearized system state model of the form shown in equation (2-70),i.e. AS —N-l '0 A = - (mm —N-l + MIAEM.+ L.AELJ (4-4) 9N4 92 fiNwmm._u mwm_.m mmmm.pu vmmo.u moom.- comm.u mnpm.u wwwm.u ommv.n cowo.Fu womm.mu mmom._u mmoo.o_ opom.n onmm.n mnmm.u cumm.u mmmo.u womm.u muop.Mu wsmgd mucmcmmmm mmwN.Fu vmmm.u opmm.u Nnmm.op —wow.mn memo.u «pum._- mmmn.u wonm.n mmpo.Fn Nwmm.Pn meow.u ommn.n Fwow.mu mmpo.- mowm.n Nmmm.Fu m—mm.u mmon.- oovm.~u vam.u #Omm.u mmm~.u memo.n mowm.u voom.m mm—¢.N- Nmmm.u mNmN.u oopm.u moom.Pu mmpw.u vmmm.u epmm.Pn Nmmm.~u NmP¢.Nu w~¢o.- mmwm.a mmpm.u mmmN.Pa omeo.mu ewmm.n mmmo.n mmwn.u mpnm.u Nwmm.n mmmm.a upom.o~ mmpo.mn Nmmm.pu mnpm.mu omm¢.- womm.u momm.u NNON.- mumm.u mmpn.u mmpo.mn mo¢~.m unmo.pa .1 Empmxm ucmpmcm 3oz mam mm on» com op Loumgwcmu mg“ cw .h .chpm: ucmwuwmwmou mzcgoh mchwcoczocxm ms» Nmm~.mu cowo.pu mno_.mn mmpo.Pu oovN.Fu oopm.u mmmN.Pu Nmmm.pu unmo.—n mmpv.om1 .euv mpawh F4 Table 4-5. Ranking Table of the RMS Coherency Measures for the Modal Disturbance of all l0 Generators Rank Generator Coherency Coherent Aggregation Order Pair Measure Groups Level l. C( 6, 7) = .0l4533 6,7) l 2. C( l, 8) = .014595 (l ,8), (6, 7) 2 3. C( 4, 7) = .0l66l6 4. C( 4, 6) = .0l7503 (l,8), (4, 6 ,7) 3 5. C( 4, 8) = .020268 6. C( 2, 3) = .020658 (1,8), (2 3),(4,6,7) 4 7. C( 3, 8) = .020766 8. C( 2, 8) = .0208l2 9. C 7, 8) = .020840 10. C l, 2) = .02ll36 ll C( l, 3) = .021217 (l,2,3,8),(4,6,7) 5 l2. C( l, 4) = .021339 l3. C( 4, 5) = .02l663 14. C( l, 7) = .02l922 l5. C( 6, 8) = .022063 l6. C( 3, 4) = .022211 l7. C( 3, 7) = .022705 l8. C( l, 6) = .023l47 l9. C( 2, 4) = .023249 20. C( 2, 7) = .02375l 21. C( 3, 6) = .023796 22. C( 8, 9) = .024318 23. C( 4, 9) = .024655 24. C( 2, 6) = .024868 (l,2,3,4,6,7,8) 6 25. C( 7, 9) = .024989 26. C( 5, 7) = .025437 27. C( l, 9) = .025679 28. C( 6, 9) = .025826 29. C( 5, 6) = .025968 30. C( 3, 9) = .028275 3l. C( 2, 9) = .028965 (l,2,3,4,6,7,8,9) 7 32. CE 5, 8) = .030493 33. C 3, 5) = .03l460 34. CE 1, 5) = .03l663 35. ‘ C 5, 9) = .03l8l0 36. cg 2, 5; = .032569 (l,2,3,4,5,6,7,8,9) 8 37. C l, 10 = .046012 38. C( 8, l0) = .047294 39. C( 2, l0) = .048929 40. C( 3, l0) = .050724 4l. C( 4, log = .0566l3 42. C 7, l0 = .057l46 43. C( 6, 10) = .058l46 44. C( 9, 10) = .059228 45. C( 5, l0) = .065850 93 94 Define a new state variable vector 5.: g-ééN-l’ where g_ is a transfor- mation matrix which transforms the system state variables into the center of inertia variable vector, 54 and the fast variable vector 52. This transformation matrix is similar to the transformation matrix used in [243 for separating slow and fast states. For a system of N generators, the (N-l)x(N-l) dimensional matrix g_ can be constructed as follows: a) Forming the Center of Inertia Rows: In forming g_ matrix it is assumed that the reference generator N is a very "large" generator in the sense thatits inertia is much larger than any other generator in the system (approximately l0 times in the New England System), and thus it will be aggregated in a very high level of aggregation (the last generator to be aggregated in the New England System). Based on this assumption let Nu be the number of unaggregated generators excluding the reference generator and NC be the number of coherent groups, then the first (Nu + Nc) rows of the (N-l)x(N-l) matrix g_ will be formed using the following procedure: Represent each coherent group by the smallest generator number in that group and then order the numbers associated to all unaggregated generators and these coherent groups in ascending order and assign new numbers 2 = l,2,3,...,g,...,h,...,(Nu + Nc) to these ordered generator numbers. Where, it is assumed in general that g The new number assigned to the unaggregated generator 1 h The new number assigned to the coherent group j Now, the first (Nu + No) can be formed as follows: 95 l. Form one row for each unaggregated generator i by putting a (l) in the i th_ element of the g th_ row and zero elsewhere. 2. For each coherent group j containing Pj generators form the h th_ row with Pj nonzero elements of the following form M =——r; = r th_ element of row h M . , r a1j,a2j,...,a _ e3 P.J J where P. M - {J M 63 q=l “qj r = Preassigned generator numbers in the coherent group j. Mr = The inertia constant of generator r. N = The number of generators in the system. Mej = The sum of the inertia constants of all generators in the coherent group j. Example: Suppose that the coherent group j consists of generators l,3,4,6, then the corresponding row for this group has the form where, Mej = M1 + M3 + M4 + M6 b) Forming the Fast Rows: The last (N-Nu-Nc-l) rows of the matrix Q_ will be formed based on the following procedure: For each coherent group j containing pj generators, form (pj-l) rows assuming that the highest generator number in that group is 96 the reference generator. Then, form one row for each generator of this group excluding the reference generator (starting from the smallest gen- erator number and continuing in ascending order) by putting a (l) in the element which corresponds to the generator number and a (-l) in the element which corresponds to the reference generator (a .) and zero elsewhere. For the previous example the following 3 roagawill be formed based on this procedure: 1 o o o o -1 o ..... o— o o 1 o o -1 o ..... 0 Lo 0 o 1 o -1 o ..... o _ Once the matrix g_ is formed the new system has the form 3. = 'iLUi 5 ' 03 + QWAEM + L 43L.) (4-5a) A? —' V II [Ca A 3 —{ V IL; Defining (r , and assuming that éEL_= 9_ equation (4-5a) becomes i = -(:T_)x -og°<_ + 111101811) (4-5b) Partitioning the vector x_= [54 E 523T, and also all of the other vectors and matrices in such a way to match the dimensions of the two vectors 5, and 52 equation (4-5b) can be written as: 1. - F N ' ~ '- F - F ' - F. - F ' - -1 11 (W11 HEY-’12 3E1 °lm§ 9- 11 (El-”D11 E (QM-’12 E‘P—Mm I I I --- = - ------- q ------- --- - ----|---- --- + ------ «I» ------------ : : : N ' ~ I O I 52 (M21 : (”1’22 izj 9- : “Ln x 1 (M21 1 (M22) _AEMnJ 97 where m = Nu + Nc , n = N-m _ T APMm - [APM1,APM2,...,APMm] _ T ABM” - [APMm+],APMm+2,...,APMn] The state vector 54 represents the center of inertia variables, that is component wise: P I.- q=l “qa qJ - h th_ element of vector 54 = p 4—- for all pj generators 1n the I q group j. Where, a for q = 1,2,...,pj represent the preassinged OJ generator numbers of the coherent group j containing pj generators. The center of inertia variables are the weighted sums of the generator angles in the coherent groups and can be regarded as the angles of equivalent generators for the coherent groups [ll,3l,32]. The fast states 52 represent the fast intermachine oscillations (5a-6b) of generators a and b within a coherent group, where (b > a) is the reference generator of that coherent group. Now, assuming that the disturbances can only occur in the mechanical input power of the generators of the study system,i.e. (ABMn = 9), based on the derivation in Chapter 3 the optimal reduced order structure matrix (MTOMC) for the aggregated system has the form - ~' ~ '~ -1 'v m " (Ml)11 ' (fl)12(fll)22(fl)21 (4'7) Thus, the reduced order state model has the form: 2+EE (PM 12% 1x2- 98 where i = [x x x 1T 6 = [APM APM APM 1T 1923' 9m 9 _ ”I, 2""9 m+‘l- _ 1 _ _ 9 5 1m , (9. 21: --------- 4 --------- 3= 1 -(MTOMC) , -olm J 551 m = Nu + NC Eh] is a matrix of the form (4-3a), where the equivalent generators are represented by the sum of the inertia constants of the generators of the coherent groups. All of the results for the New England System is based on this linear state model. To show how this aggregation procedure works consider the first level of aggregation where generators 6 and 7 are combined into a single generator. The (9x9) matrix g_ in this case has the form: , _ 1 o o o o o o o o o 1 o o o o o o o o o 1 o o o o o o o o o 1 o o o o o 3 3 2 3 ‘ w2 19 °° 0 M6+M7 M6+M7 0 0 o o o o o o o 1 o o o o o o o o o 1 o o o o o 1 -1 o o 99 Substituting M6 and M7 into the above matrix, after carrying out the multiplications the matrix (Mi) = gfim1)gf1 can be found. This matrix is given in Table 4-6. Now, partitioning the matrix (ME), and using equation (4-7) the (8x8) optimal reduced structur matrix (MTOMC) may be found. The result is MTOMC, shown in Table 4-6. Once the structure matrix (MTOMC) for the reduced system is found the eigenvalues and the coherency measures between each pair of generators (or equivalent generators) can be found. It is shown in [l9] that the system eigenvalues are complex conjugate pairs of the form ,/02+4yi , 1 = 1,2,...,m (4-9) * ~ where Ai’xi’ i = 1,2,...,m, are the 2m eigenvalues of A_ matrix and rupfi O' . _ = - —+ >"I’Al 2 ‘ yi, i = 1,2,...,m, are the m eigenvalues of (:MIQMC) matrix. In our analysis 0 is assumed to be (c = 0.275). Since - g— is small and the same for all eigenvalues, the magnitude of the imaginary part of the eigenvalues are given in this chapter. Table 4-7 shows the magnitude of the imaginary part of the system eigenvalues for different levels of aggregation. Table 4-7 clearly in- dicates that as one go down this table the largest eigenvalue pair is discarded and those with small imaginary parts are retained. This makes sense, since the modal disturbance of all generators detects the tightly interconnected groups and what should be discarded are the high frequency oscillations within the tightly interconnected groups. For example, the eigenvalues at level I are very close to the eigenvalues at level 0 to the left, and the eigenvalue pair with the imaginary part 9.90l which is the highest eigenvalue pair in level 0 is discarded at level 1. 100 1wm©~.m¢ momm.mu ovom.mu mpmm.Fu comm.mu memo.mn monm.Fu mmmw.nu NmFN.mu momm.mm wo¢~.mn mmmo.mu mmpp.mn mmm~.vu mmnp.mu mmpo.mms nmpm.mu mvwo.m- mmmw.m¢ ompp.e1 momp.opl “www.cu mmpw.mu Nmmm.¢u mwom.Fu nmwF.P1 mmwp.m1 wemm.me opvm.mpu mmmn.Pn ommm.1 omvu.P1 u.wmmHm nvom.¢u momm.mu mopo.mmu ween.mpu vmnm.mn mmem.m1 comm.mu wmmm.on mFFm.N- mmmm.mu Nmmp.ma «mmm.~u owwm.¢n wmmo.mm mmpm.mu mmom.¢1 mmmm.mu mvmv.~1 mem.mv anm.Fu womm.mu wmwm.ppu mwmo.wm coom.vn [wowm.on oovm.mpn memo.mu mwvo.mu mono.mn mwmm.¢u Pwmm.mn emmw.mfiL mqmw.wm W nmpm. mmmm. Nmov.m1 mmom. .mmmn. anm. mmmm. mvmc. _ pmpp. m wwop.mv Pomm.mu Nwom.ou m—mm.F1 mmow.mu vomo.mu ooum._u mmmm.nu momp. m mmpm.mu opmm.mw moqp.mu mmoo.mu mnpp.mn m¢NN.Vu wmup.mn nmpm.mmu Nmmw.1 m empm.mu wnmo.mu mmmm.m¢ nm~_.¢u Nmmp.o_u omnm.¢1 mFNw.Nu mpoo.mu Onmp. m Pmow._1 mmwF.Pu mum—.mu mmmm.m¢ mmmm.opu mmmx.Fu omwm.u mecm._u ".mw comm. m mmow.¢u wwww.m1 mmNO.NN1 omom.m_u mmmm.mm «Nem.m1 mmmm.m- ovum.on peep. m oppm.mn mm~o.~- _wm_.m1 ommw.Fu omwm.e1 memo.wm —Npm.mu wmow.¢u on—F. m FmNN.Nu ¢¢m¢.N1 vaw.ou novm.~- mmmw.mu mmwm.F~1 ommo.mm wmmv.en mvmp. m mmum.on mmvm.m_u Nomo.m- wmwo.mu mmoo.mu mmmm.eu mmmm.mn Nemm.mm .cowpmmeMm< n ucm m mgopmcmcmw ms» com QEOHz chpmz weapoacum Lmuco twosnwm Fms?pao any ucm r: chumz mg» .mue mpamp 101 mom.¢ Am.w.m.o.m.¢.m.m.pv w mam.o mke.e Am.m.~.o.e.m.m._v A 0mm.“ mAN.o mxe.e Aw.~.o.e.m.m._v 0 04¢.“ mom.“ Noo.o Ame.¢ A~.o.ev.fim.m.N.Fv m omm.m mm~.~ mmo.n mmo.o Ame.a AA.©.eV.Am.Nv.Am._V a 0mm.m mm~.m .mA.A ems.“ mmo.© Nm¢.e A~.©.4V.Aw.FV m omm.m m_m.w om~.w mxm.u one.“ oem.o Ame.e Ax.ov.fim._v N mmm.m omm.m mpm.m omm.m mum.“ mmo.~ mmo.w ome.¢ A~.©v P _om.m mmm.m ©_m.m mpm.m mmm.m New.“ omo.N mmo.o om¢.e acoz o covummwcmm< $0 mFm>m4 _Fm Lek mm:_m>:mmwm Emumxw Go magma ammcwmeH mo mmuzuwcmmz cowpmmmumm< Fm>m4 Acowpmmwcmm< pcmcocou Fave: pmsmpaov mcoumcmcmu cob ppm Go mucmagspmwo Paco: com mama mapm>cmmwm .Niv m_nmh 102 This eigenvalue pair (- g-: j 9.90l) can be associated with the inter- machine oscillations between generators 6 and 7. Similarly, the eigen- value pair (- %t 3' 9.859) discarded atlevel 2 represents the oscillation between generators l and 8 and the pair (- g-t j 9.520) discarded at level 3 represents the oscillations between generator 4 and the aggregated generator (6,7). Examining level 4 of aggregation, it can be seen that the same phenomena happens, but in going from level 4 to 5 there is no clear elimination of specific eigenvalue pair. This indicates that there is no single eigenvalue pair which can be associated with the oscillation between aggregated generators (l,8), and (2,3). Below level 4 the . eigenvalues in each row are a sort of averages of the eigenvalues in the row above. As will be seen later in this chapter, when an eigenvalue pair is discarded rather than averaged in going from level (h-l) to level h, the resulting model at level h will be almost as good as the model at level (h-l). Thus, in this case one would expect the model to be very good through aggregation level 4. It will be shown that the eigenvalue information given in Table 4-7 is essentially contained in the RMS co- herency measure matrix S (...) = (MTOMC)'1MH_RHM_-]r](MTOMC)'T, for dif- ferent levels of aggregatizn. Table 4-8 shows the RMS coherency measures between the reference generator 10 and each of the other generators (or equivalent generators) for different levels of aggregation. Since generator 10 is the last generator that will be aggregated with the rest of the generators, the RMS coherency measures between this generator and each of the other equivalent generators will be available for all levels of aggregation. 103 Romeo.o Am.m.n.o.m.v.m.m._v w oomko.o commo.o Am.w.N.o.¢.m.N._v A Ammoo.o mopxo.o mommo.o Aw.k.o.e.m.~._v o “memo.o upwoo.o mammo.o cummo.o AA.0.4V.Am.m.N._V m Nmomo.o mokoo.o wm_oo.o mmmmo.o _Nmeo.o Am.m.ev.fim.mv.fiw._v 4 _Nooo.o wmuoo.o proo.o mm_mo.o Romeo.o ommeo.o Au.m.¢v.flw.Fv m mammo.o mmooo.o Amoco.o amumo.o o_Fmo.o omaeo.o momeo.o A“.ov.fim._v N aemmo.o omNeo.o moooo.o ommoo.o “Pumo.o wmomo.o m_m¢o.o omoeo.o AA.©V _ mmmmo.o mmueo.o m_~mo.o m_mmo.o mmmoo.o _oomo.o «Nomo.o mmmeo.o _ooeo.o acoz o ~o_.mv aoF.mq Ao_.mv ~o_.ov ~o_.mm Ao_.qv Ao_.mv Ao_.Nw ~o_.Fv :o_pmmmcmm< mo Pm>m4 FFm com op Loumcwcmw ucm coumcmcmu comm cwmzuwm mczmmmz mocmcmnou cowpmmmxmm< Fm>m4 Acowpmmmcmm< ucmcmgou choz pmswuaov weepmcmcmw so» _Fm to mocmnczumvo Paco: cow mums mgammwz zu:mcm;ou .m1e mpamh 104 The general rule for constructing this coherency measure table is to aggregate each coherent group to the lowest generator number in that group and list from left to right the RMS coherency measures between the highest numbered generator and the rest of the equivalent generators for each level of aggregation. As an example consider the 4 th_level of aggregation where the reduced system generators are the aggregate (l,8), the aggregate (2,3), the aggregate (4,6,7), and the generators 5,9, and lO. The five entries in this row, from left to right correspond to the coherency measures between generator l0 and the generators (l,8), (2,3), (4,6,7), 5, and 9. The variance matrix Bl] used for this level of aggregation has the form 2 2 2 2 2 8), ,M M M B- 5’ 9’ 10} ll = Diag{(M1 + M (M +113)2,(M + M + M 2 4 6 7) A careful loOk at Table 4-8 indicates that the RMS coherency measure has essentially the same information as the eigenvalues in Table 4-7. For example, the coherency measures at level l are very close to the corresponding coherency measure values at level 0, except the coherency measure between generator l0 and the aggregated generator (6,7) which has a larger coherency measure than either generator 6 or 7 at level 0. As can be seen from the coherency measure Table 4—8, at lower levels of aggregation there is a sort of averaging of the coherency measures, but it is difficult to recognize because the entries in Table 4-8 are not rank ordered. 105 4.2 Modal Disturbance of Generator l or 8: This type of parochial disturbances can be used to construct a reduced order dynamic equivalents for investigating disturbances which may occur on a pre-selected generator of the study system. Some applica- tions of this type of parochial equivalents are: l) For use within transient, midterm, long term stability programs to reduce the computational requirements when the unreduced model is very large and the simulation interval is long. 2) For on-line transient stability for security assessment where a low order accurate model for a specific contingency is desired. These parochial equivalents can be very low order, because only one generator is disturbed and the remainder of the system often splits into only a few coherent groups. 4.2.1 Modal Disturbance of Generator 1: Modal disturbance of generator I is a probabilistic disturbance with zero mean and the variance matrix of the form 3_ = Diag{M2,O,O,O,O,O,O,O,O,O} (4-10) 11 Table 4-9 gives the ranking table of inter generator coherency measures that result from applying a modal disturbance of generator I, using equations (4-3), and (2-23). The coherent groups are identified again by applying the commutative rule to this ranking table. Table 4-lO shows the eigenvalues for the linear models (formed based on the optimal aggregation of the generators of the coherent groups) that represent the reduced system at each level of aggregation. Table 4-10 indicates that the eigenvalue pair (- g-: j 9.901) is discarded 106 Table 4-9. Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generator l. Coherent Rank Generator Coherency Aggregation Order Pair Measure Groups l. C( 4, 7) = .000034 (4,7) l 2. C( 6, 7) = .OOOO38 3. C( 3, 5) = .OOOO65 (3,5),(4,7) 2 4. C( 4, 6) = .OOOO73 (3,5),(4,6,7) 3 5. C( 2, 3) = .OOOl96 6. C( 2, 5) = .00026l (2,3,5),(4,6,7) 4 7. C( 5, 6) = .OOO492 8. C( 5, 7) = .000530 9. C( 3, 6) = .000557 10. C( 4, 5) = .000564 11. C( 3, 7) = .OOOS95 12. C( 3, 4) = .000630 13. C( 4, 9) = .000703 14. C( 7, 9) = .OOO737 15. C( 2, 6) = .OOO753 16. C( 6, 9) = .OOO775 (2,3,5),(4,6,7,9) 5 17. C( 2, 7) = .090791 18. C( 2, 4) = .000826 19. C( 8, 9) = .001247 20. C( 5, 9) = .001267 21. C( 3, 9) = .001333 22. C( 2, 9) = .001529 (2,3,4,5,6,7,9) 6 23. C( 4, 8) = .001949 24. C( 7, 8) = .001984 25. C( 6, 8) = .002022 26. CE 5, 8) = .002514 27. C 3, 8) = .002579 28. C( 2, 8) = .002775 (2,3,4,5,6,7,8,9) 7 29. C( 2, 10) = .OO3045 30. C( 3, 10) = .OO3241 31. C( 5, 10) = .003307 32. C( 6, 10) = .003799 33. C( 7, 10) = .003837 34. C( 4, 10) = .003871 35. C( 9. 10) = .004574 36. C( 8. 10) = .005820 (2,3,4,5,6,7,8,9,10) 8 37. C( 1, 8) = .009994 38. C( l, 9) = .011240 39. C( 1, 4) = .011943 40. C( 1, 7) = .011977 41. C( 1, 6) = .012015 42. C( 1, 5) = .012507 43. C( 1, 3) = .012573 44. C( 1, 2) = .012769 45. C( 1, 10) = .015814 107 mew.m ©w¢.¢ mmm.m Npm.m mnv.v omw.m mum.w mmv.u _n¢.¢ mmm.m mmN.w mvu.n www.m Pm¢.¢ mmw.m mvm.m Nwm.n Pmo.m ¢_m.o mm¢.¢ mmm.m mum.m “mm.m mcm.m ovo.n vpm.o No¢.¢ mmw.m om~.m «mm.w vmm.w nom.m mvo.k emo.© mm¢.¢ Pom.m mmw.m ©Fm.m mpm.w mmm.m mmm.m mmo.n mmo.o om¢.¢ cowummmcmm< we m_m>m4 cw>mm com mmzpm>cmmwm Emumxw we magma xgmcwmoEH mo mmuzuwcmez Am.w.n.m.m.v.m.mv N 8.N.e.m.a.m.Nv e 3.N.e.:.$.m.Nv m :.e.$.$.m.Nv a $93.33 N tiéi N Ci _ mcoz o cowpmmmgmm< Fm>m4 Acowpmmmcmm< pcmcmzou _muoz _mewpqov _ LONNmemw mo mocmnczgmwo pace: com mums mapw>cmmwu .opiq mpamh 108 at level 1, but the frequency 9.859 persists up to level 4 and it stays almost the same in levels 5, and 6. This indicates that a somewhat dif- ferent pattern of eigenvalue elimination (not highest frequency of oscillation) is at work. The frequencies which are eliminated are those not excited by the parochial disturbance, which are not necessarily the high frequency inter generator oscillations of tightly connected groups. A careful look at Table 4-10 clearly shows that the frequencies 4.476, 8.212, and 9.855 at aggregation level 6 are very close to the values 4.436, 8.289, and 9.859 at level 0. Note also that the values near 4.4, 8.2, and 9.8 are preserved through all 6 levels of aggregation. This indicates that the frequencies 4.476, 8.212, and 9.855 represent the intermachine oscillation between generators l, 8, and the aggregated generator (2,3,4,5,6,7,9), which are very close to the oscillations between generators 1,8, and 5 at level 0. This means that conditions for geometric coherency, or strong linear decoupling or a combination of them are satisfied for the aggregated group (2,3,4,5,6,7,9), for a disturbance on generator 1 [19]. As was mentioned in the previous section the same information of Table 4-10 is contained in the RMS coherency measure matrix _s.xs(e). , Table 4-11 gives the RMS coherency measures between generator 1, and the individual (or equivalent) generators at each level of aggregation. A look at Table 4-11 indicates that the coherency measures at level 1 are exactly the same to the corresponding values in level 0, with the exception of aggregated generator (4,7) which has a slightly larger coherency measure. Table 4-11 also indicates that the coherency measures between generators (1,10), (1,8) are preserved through all six levels $.23: Nmzee $.e.N.e.m.e.m.N: N 523: momeee maze $.N.e.m.e.m.N: e 52: @885 e285 NeNse 3.N.e.:.$.m.N: m 5290 NE; 2385 emzee NeNse t.e.:.$.m.N: a 5386 «NE; @885 e285 mmNse RN85 25.3.3.3 m 520.0 3285 8255 N885 e285 mmNSe RNse Sigma: N m 5205 NE; 8255 N855 _mNse e285 NmNse RNse :4: _ 535 NE; 88: 8:2 NeNse SNse 2:85 $23. 585 aeaz e 8:: 8.: 3.: 2.: 3.: 3.: 3.: 8.: 8.: ce_pemecmm< we m_e>ee ee>em Lem F Lepegmcew ece Leweseeew seem ceezuee megzmeez zuceceseu :ewpemmpmm< _e>ee Aeewpemecem< ucececeo _eeez _eswaeov F seuececeo we eeeeeceumwe Feeez Lem epeo eczema: secegeceu ..Pie epeeh 110 of aggregation and the coherency measure between generator 1 and the aggregated generator (2,3,4,5,6,7,9) is close to the coherency measure between generator 1 and 5 or 1 and 6 at level 0. This shows that the eigenvalues and RMS coherency measures, are in fact carrying the same informations. To verify the conclusion reached by analyzing the data of Tables 4-10, and 4-11, simulations were made to compare the response of the full 39 bus New England System to the reduced system models formed based on the optimal modal coherent aggregation for the first 6 levels of aggregation. The disturbance used in these simulations is a one p.u. step disturbance on generator 1. These simulation results are given in Figures 4-2d and 4-2e. Note that the curves designated by D represent the full New England System and the curves designated by <3» represent the reduced order models. These simulation results clearly verify the conclusion reached by analyzing the eigenvalue data of Table 4-10, and the RMS coherency measure data of Table 4-11. 4.2.2 Modal Disturbance of Generator 8: Modal disturbance of generator 8 is a probabilistic disturbance with zero mean and the variance matrix 2 3 8. 1] = Diag{0,0,0,0,0,0,0,M 0’0} (4‘11) Tables 4-12, 4-13, and 4-14 show the ranking table, the eigen- value data, and the RMS coherency measure data for the modal disturbance of generator 8, respectively. Figure 4.2 Figure 4.2 (d) (e) 111 Simulation Response of the Reduced Order Model Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 1 Based on the Optimal Modal Coherent Aggregation of the Coherent Groups. Aggregation Level System Generators 4 l,(2,3,5),(4.6,7),8.9 5 l,(2,3,5).(4,6,7,9),8 Generator 10 is the reference generator TIME IN SECONDS 1 j 1.. I [DEOJEWONU 80108 S'N30 112 TIME IN SECONDS r LDfi'mN ITII 1'11 F' O F‘ N I I Iljll (030)310NU 80108 8'N30 ./ /‘ :I‘ ) :I I/ m i: 5 _ 1// :4 - \ ’1‘ 3)) 3‘. '.1 1 2" \> :2 02‘ :‘J : w 9: :2 J 1:) 1' ..v I “ wN ~\--‘ _ 7‘\ 1- "7 fl 11ifi11111fi1111‘ l‘ to ID v- 0) cu —- c: (030)310NU 80108 I'NBO TIME IN SECONDS (d) 113 \ .' 1— “\ ., «a ..D ,/‘ F v. I (D _ (D C O z ' Z O O Q _ Q m = u (D \\ '- (O ’ A Z .4 2 H H a) - 1— V w l.lJ Z I: H I—a .— I— ll r \911 :1 1 1 1\9{1 1 1 /1 -'- L. = 1- F‘I 1 l 1 l 1 1 1‘fi W 1 1 1 1*T 1 j #141 .41 1 7 1':T 1 0“? K) <1 W1 01 F1 C) 11 ?3 c~ to ID V 6) cu __ (D (930,319NU 30103 8°N30 (0301310NU 30103 I-Nae 114 Table 4-12. Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generator 8. Rank Generator Coherency Coherent Aggregation Order Pair Measure Groups Level 1. C( 4, 7) = .000020 (4,7) l 2. C( 6, 7) = .000022 3. C( 4, 6) = .000042 (4,6,7) 2 4. C( 2, 3) = .000125 (2 3),(4,6,7) 3 5. C( 3, 5) = .000135 6. C( 2, 5) = .000260 (2, 3,5), ( ,6 , 7) 4 7. C( 1, 9) = .000272 (1, 9).(2 .3,5 ), (4,6,7) 5 8. C( 5, 6) = .000285 9. C( 5, 7) = .000307 10. C( 4, 5) = .000327 11. C( 3, 6) = .000419 12. C( 3, 7) = .000441 13. C( 3, 4) = .OOO461 14. C( 2, 6) = .000544 15. C( 2, 7) = .000566 16. C( 2, 4) = .000586 (1,9),(2,3,4,5,6,7) 6 17. C( 4, 9) = .000745 18. C( 7, 9) = .000765 19. C( 6, 9) = .000787 20. C( 1, 4) = .001018 21. C( l, 7) = .001037 22. C( l, 6) = .001060 23. C( 5, 9) = .001072 24. C( 3, 9) = .001206 25. C( 2, 9) = .001332 26. C( l, 5) = .001344 27. C( 1, 3) = .001479 28. C( l, 2) = .001604 (l,2,3,4,5,6,7,9) 7 29. C( 2, 10) = .001845 30. C( 3, 10) = .001970 ‘ 31. C( 5, 10) = .002105 32. C( 6, 10) = .002389 _ 33. C( 7, 10) = .002411 34. C( 4, 10) = .002431 35. C( 9, 10) = .003177 36. C( l, 10) = .003449 (l,2,3,4,5,6,7,9,10) 8 37. C( l, 8) = .010482 38. C( 8, 9) = .010754 39. C( 4, 8) = .011500 40. CE 7, 8) = .011520 41. C 6, 8) = .011542 42. C( 5, 8) = .011826 43. C( 3, 8) = .011961 44. C( 2, 8) = .012086 45. C( 8, 10) = .013931 115 eow.m wm¢.e Am.w.c.m.e.m.mxpv NNa.m meN.N ema.e AN.e.m.a.m.N:.Ao._: NNa.m _eN.N Nm_.N mme.a AN.e.e:.Am.m.N:.Am._: mme.m NaN.N meN.N NNe.e FNe.a AN.e.e:.Am.m.Nv mme.m emm.e meN.N ame.N eme.e ema.e AN.e.e:.Am.N: mmm.m emm.e maN.e _mN.N mme.N eme.e ema.e AN.e.a: mme.m eNN.m amm.e emN.m NON.N mee.N ame.e eme.e AN.e: _om.m mme.m e_m.m epm.m mNN.e NNm.N emo.N mme.e eme.e aeoz :ewuemecmm< we mwe>ee ce>mm Lew meewe>cemwm Eepmxm we mused xgecwmeEH we meeeuwcmez :ewuemegmm< Acewuemecmm< peecegeu peeez _eswpeov Fm>e4 m Leueceeeo we eeeeecepmwo Peeez Lew ewes espe>eemwu .mpie eweew 116 mam_e.e _e__e.e Am.N.e.m.a.m.N.F: N mmmpe.e mNFPe.e eeepe.e AN.e.m.a.m.N:.Aa..: e mmmpe.e Nmp_e.e em__e.e eee_o.e AN.e.e:.Am.m.N:.Am._: m mmm_o.o mNo_o.e Nmppe.e em_1e.e eao_o.e AN.e.e:.Am.m.N: a mmm_e.e mNePe.e me__e.e Nm__e.e NeNFe.e eaepe.o AN.e.¢:.Am.N : m mmmpe.e mNeFe.e me_Fe.e Nm_Fo.e emFPe.o meN_e.e meo_e.o AN.e.e: N mmm_e.e mNe_o.e amppe.o NN__e.o _mp_e.e em_Po.e meN_e.e Neepe.e AN.a: _ _ mmmpe.e mNe_o.e Nm,_e.o em,_e.e NNP_o.e em__e.o em_~o.e meN_e.e wee.e.o aeaz o Nep.e: ~m.N: ~N.N: Ne.e: ~N.m: Ae.a: Ae.m: AN.N: Ne._: :ewpemegmm< we mwe>ee ce>em Lew w Lepececeu ece Leueceeeu seem cmezuee meczmeez xecesezeu :ewpemegmm< Fe>e4 Acewwemecmm< “emcegeu Feeez Feswueov m Leueceeew we weeeeceamwo _eeez sew eye: egemeez xecegeceu .epie eweew 117 A careful look at Table 4-13 shows that the eigenvalues in levels 1,2, and 3 are close to the corresponding eigenvalues at level 0, except for some entries which correspond to the aggregated generators. But, this is not the case below level 3, where the eigenvalues are a sort of averaging of the eigenvalues at level above. Especially the eigenvalue imaginary parts at level 5 are truely a sort of averages of those at level 4. Thus, one would expect a loss of model accuracy below level 4. The same conclusion may be drawn from the coherency measure Table 4-14. An examination of Table 4-14 indicates that the coherency measure between generator 8 and the rest of individual (or equivalent) generators in levels 1,2, and 3, are almost the same as the corresponding coherency measures in level 0, except for some entries which correspond to the coherency measures between generator 8 and the aggregated generators. Especially the coherency measure between generator 1 and 8 is preserved up to level 4, but in going from level 4 to 5 this coherency measure changes. This phenomena happens when generators l and 9 are aggregated. Recall that the modal disturbance of all 10 generators showed that the generators 1 and 8 are tightly interconnected. If generator 8 is consid- ered to be the "study system", then the coherent group (1,9) breaks the strongly bound coherent group (1,8). The way to solve this problem is to avoid disturbing only part of a group of generators which are tightly interconnected. This is the subject of the next section. Figures 4-3d and 4-3e show the simulation results of the full 39 bus New England System versus the reduced order model formed based on the optimal modal coherent aggregation for a l p.u. step disturbance on generator 8. Note again that the curves designated by D represent Figure 4.3 Figure 4.3 (d) (e) 118 Simulation Response of the Reduced Order Model Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 8 Based on the Optimal Modal Coherent Aggregation of the Coherent Groups. Aggregation Level System Generators 4 19(293’5)9(49697)$899 5 (l,9),(2,3,5),(4,6,7),8 Generator 10 is the reference generator TIME IN SECONDS 119 ‘I H '1 (DBOJBWONU 80108 6°N30 (030)310NU 80108 8‘N30 "CD 1"? TIME IN SECONDS / F0) '.1 \.~ ‘ ./'" /-I J a (4 : (D ‘ '10 D 3 z D Q .‘ u ._ 1.. m ' A 7 2 H '0 3 V U z “V I—l .— (OBOIBTONU 80108 I'N30 120 Am: mozouum z_ mznw mrhpm..fm.__ fir.rmw.z.pv___ .4 : 4 ... : . 1. : N. s, s. 3, .1 2 5 x = ,z 4 V A .» ,m‘v : : N. .I .J .J x. = . . > : . : . 5 : mozouww ZH u::. 1m m e _ e w e m a m N a _ _ _L w r _ _ . _ _ P _ . w r r w _ w _ _ . _ _ _ A P _ r O T 1N :1 1¢ fi- Tm i 1m T. 10g r- (0301310NU 80108 I‘N30 (030)310NU 80108 Q‘N30 121 the full New England System and the curves designated by <> represent the reduced order model. These simulation results clearly verify the conclusions reached from analyzing the eigenvalues and the RMS coherency measures. 4.3 Modal Disturbance of Generators 1 and 8: To produce a local dynamic equivalent for the power system, a modal disturbance of a pre-selected subset of generators can be used to identify the coherent groups. This type of disturbance detects a more parochial kind of coherent behavior, that is, generators which are coherent in response to the distrubances located in a certain region of the system. In other words, the coherent groups are dictated based on the SGC or SSLD coherent structure. Some applications of this type of local dynamic equivalent are: 1) For design of excitation systems including power system stabilizer where all eigenvalues, fast or slow, that are excited and must be appropriately damped by the excitation system for disturbances in a local region must be preserved in the local equivalent. 2) For design of discrete supplementary control. 3) For on-line transient stability for security assessment.1 The local dynamic equivalent could be used for simulating all disturbances in a local area rather than only for a single disturbance. Since the equivalent must be computed off-line, an equivalent could not be calculated for each contingency to be investigated on-line by security assessment procedures. To identify the coherent groups, a modal disturbance of generators l and 8 has been conducted. Modal disturbance of generators l and 8 is 122 a probabilistic disturbance with zero mean and the variance matrix of the form: 2 'l! 2 B 89 = Diag{M 0,0,0,0,0,0,M 0,0} (4-12) 11 Tables 4-15, 4-16, and 4-17 show the ranking table, the eigen- value data, and the RMS coherency measure data for the modal disturbance of generators l and 8, respectively. Therefore, in this case all gener- ators of strongly bound group (1,8) are disturbed. Table 4-16 shows that the eigenvalue pair (- %t 3' 9.901) is essentially discarded at level 1. This eigenvalue pair can be associated with the oscillation between generators 4 and 7. The eigenvalue pair (- %-1 j 9.720) discarded at level 2 represent the oscillation between generator 6 and the aggregated generator (4,7). Note that the frequency (9.720) is not the highest frequency in level 1. An examination of Table 4-16 indicates that in going from level 4 to 5, the eigenvalue pair (- %-i j 7.745) is discarded. This eigenvalue pair can be asso- ciated with the oscillation between the aggregated generators (2,3,5), and (4,6,7). Again this frequency (7.745) is not the highest frequency in level 4. Note, however, that the imaginary part (9.859) persists up to level 5, and even approximately in level 6. A careful look at Table 4-16 clearly shows that the frequencies 4.476, 8.212, and 9.855 at level 6 are very close to the values 4.436, 8.289, and 9.859 at level 0. This idicates that the frequencies 4.476, 8.212, and 9.855 represent the oscillation between generators 1,8, and the aggregated generator (2,3,4,5,6,7,9). An examination of Table 4-17 shows that this coherency measure 123 Table 4-15. Ranking Table of the RMS Coherency Measures for the Modal Disturbance of Generators 1 and 8. Rank Generator Coherency Coherent Aggregation Order Pair Measure Groups Level 1. C( 4, 7) = .000040 (4,7) 1 2. C( 6, 7) = .000044 3. C( 4, 6) = .000084 (4,6,7) 2 4. C( 3, 5) = .000150 (3,5),(4,6,7) 3 5. C( 2, 3) = .000232 6. C( 2, 5) = .000368 (2,3,5),(4,6,7) 4 7. C( 5, 6) = .000568 8. C( 5, 7) = .000612 9. C( 4, 5) = .000652 10. C( 3, 6) = .000697 11. C( 3, 7) = .000741 12. C( 3, 4) = .000781 13. C( 2, 6) = .000929 14. C( 2, 7) = .000973 15. C( 2, 4) = .001013 (2,3,4,5,6,7) 5 16. C( 4, 9) = .001024 17. C( 7, 9) = .001063 18. C( 6, 9) = .001105 19. C( 5, 9) = .001660 20. C( 3, 9) = .001798 21. C( 2, 9) = .002027 (2,3,4,5,6,7,9) 6 22. C( 2, 10) = .003561 23. C( 3, 10) = .003793 24. C( 5, 10) = .003920 25. C( 6, 10) = .004487 26. C( 7, 10) = .004532 '27. C( 4, 10) = .004571 28. C( 9, 10) = .005569 (2,3,4,5,6,7,9,10) 7 29. C( 8, 9) = .010826 30. C( l, 9) = .011243 31. C( 4, 8) = .011664 32. C( 7, 8) = .011689 33. C( 6, 8) = .011718 34. C( l, 4) = .011986 35. C( 1, 7) = .012022 36. C( l, 6) = .012062 37. C( 5, 8) = .012091 38. C( 3, 8) = .012236 39. C( 2, 8) = .012400 40. CE 1, 5) = .012579 41. C l, 3) = .012659 42. CE 1, 2) = .012869 43. C l, 8) = .014483 (l,8),(2,3,4,5,6,7,9,10) 8 44. C( 8, 10) = .015098 45. C( l, 10) = .016186 124 mmm.m Npm.m one.¢ mmm.m mmm.m Pew.o on¢.¢ mmm.m mmm.w mem.~ mmm.o Pm¢.e mmw.m mem.m Nmm.w Fno.m www.o mo¢.¢ mmw.m omm.m mmm.w Paw.“ mmo.w wmo.m om¢.¢ mmm.m omw.m «mm.w cmm.m non.n meo.w cmo.o mm¢.¢ Pom.m mmw.m cwm.m wpm.w mmm.w num.w omo.m mmo.o mm¢.e :ewuemegmm< we mwe>e4 xwm Lew meewe>cem3m Eepmxm we mused xceawmeeH we meeepwmmez Am.w.m.m.e.m.~v c 33343.: 3 33.5.3.3: a 3.3.3.3.: m 3.3.: N ti 1 ecez o cewpemecmm< Fe>me Aeewuemecmm< peececeu Feeez _eswpeov m use F mcepeceeeu we eeeeeceumwo Peeez Lew ewes ezwe>cemwu .epie eweew 125 0385 3285 @386 333.333.: 3 3385 e385 2255 2385 33.3.3.3.N: 3 338.0 2355 3805 £83. 3385 3.35.33: 3 Smeee e365 2385 338.0 emmeee 2355 3.3.5.3.: m 3385 2385 N385 3:58 Emcee 8255 2355 3.3.: N 338.0 238.0 238.0 Nomeee 3385 280.0 33805 @385 3.: _ 3386 2385 330.0 9385 N385 3.588 Emcee emmeee Emcee aeoz e 3:3 3:3 3:: 33: 3?: 3:: 3:: 31$ 3:: :ewyemecmm< we mwe>m4 xwm Lew ow Lepececeo ece Lepececew some eeezuee mecemeez xecegmgeu :ewpemegmm< we>ee Acewpemecmm< peececeu peeez Peewpeo: m eee _ mseaeceeew we eeeeeszpmwo _eeez Lew euee mesmeez xeemceceo .wwie eFeew 126 table yields exactly the same information as the eigenvalue Table 4-16. For example, this table indicates that the coherency measures between generators (1,10), and (8,10) are preserved through all six levels of aggregation and the coherency measure between generator 10 and the aggregated generator (2,3,4,5,6,7,9) is very close to the coherency measure between generators 6 and 10 at level 0. Figures 4-4d and 4-4e give the simulation results comparing the response of the Full New England System to the reduced model obtained based on the optimal modal coherent aggregations for a l p.u. step disturbance on generator 8. Note, again that the curves designated by D Represent the Full 39 bus New England System and the curves designated by 0 represent the reduced order model. These simulation results verify the conclusions reached from analyzing the eigenvalues and the RMS coherency measures. The results given in this chapter suggest that the RMS coherency measure seem to provide selective eigenvalue retention, based on the location of the partial modal disturbance of generators. That is, the eigenvalue retention seems to be based on the frequencies most excited by all the disturbances located in a certain region of the power system. This is why these local equivalents can be used for the design of excitation system and power system stabilizer or discrete supplementary control for on line transient stability for security assessment. Aggrega- ting all coherent groups except the one where the contingencies occur is shown to reduce the model order, by eliminating frequencies that describe intermachine oscillations in these coherent groups but retain the eigenvalues that describe the inter-generator oscillations in the 127 Lepececem eececewes esp mw o_ Lepecmeeu 3.3.AN.3.3.e.m.N:._ m toeaeaeae :0 aaem .=.a _ 3 Am: m.N.AN.e.e:.Am.m.N:.F N toeataeae ea eaem .=.e P e fie: mcepececew Eeumxm eeeeeceumwo :ewpeweewm Pe>ee :ewuemegmm< e.e egemww .meeecu ucmceceu on» we :ewpemecmm< ucecmzeu Peeez Feewueo ece m use _ mseuegeeeo we eeeeegaumwo Feeez age so eemem eewwwuceeH meeegw ucecezeu on» Low sepmxm __=d one we emceemem ecu memge> mweeez Leeco eeeeeem es“ we emceemem :ewuewzswm e.e eszmww 128 a“: :‘ .1 .- L. :9 'i ”U3 Pa) 4 " .“ - - \N I “1" ‘1‘ \ 1° "N: 4 co 0’) U) '10 C) ”1.0 D D ‘ z 2 Z - o o O _ u L) c.) 1.1 “J LJ LIJ \ 03 (D (O ‘ 2‘ ,- _ A ‘ Z Z \ E H H U . v ‘ In DJ in ./ x: 1: '3 L" 5 "fl' b-l H / I— +— :- FWIIDI—rIfiTl D 03 (D ‘1' N CD 1.0 fi' 0) N r" O r" N F' I I (DBOIBWONU 80108 S'NB‘J (030)310NU 80108 8'N30 (0301310148 80108 1‘N30 1 2 9 1’0) . F' 5 °° \\ ~ 3 P- r’:’ I— -/ 7’. ”CD 9..' ”co / (‘ 1- ?: . )— (D (f) S ' (0 c: 10 C: k fin CD 2 Z t 2 D O C) L) (.1 . L) LLJ LLJ 3‘ “J (D 0'.) 0'3 - .: ” A Z Z ' Z a) 0—0 u—a _ 0—4 v I! IJJ hJ , U I: Z , -, 1: H -¢ P-l " _fl’ 0—4 1— 1— / 1— ..., m D "N "N ~ >> -~ _v—l )— 1— 1711T1711 r311111111111 l 1 1 O 03 (D fi' N O U) '3‘ m N v-a D H (Tl H l (030)3'IONU 80108 6‘N30 (OEOIBTONU 80108 B'NBD (030)3'I‘JNU 80108 I'N30 130 disturbed (Xiherent group of generators. The eigenvalues that describe the motion of all other coherent groups against the disturbed coherent group are also retained. One other important result obtained was that despite the good results for modal disturbance of generator 1, the best rule is to dis- turb all generators of a tightly interconnected group for determining a local dynamic equivalent for a particular generator within the group. CHAPTER 5 DERIVATION OF OPTIMAL AGGREGATION FOR STEP OR PULSE INPUT DISTURBANCES WHEN THE SIMULATION INTERVAL IS SHORT (11:11) Since the derivation of optimal aggregation in this research is based on the coherency measure for a particular contingency type and sim- ulation interval, the first step in deriving the optimal aggregation is to obtain the appropriate coherency measure for a short interval. A transient coherency measure for short interval is derived in [17] which is a Taylor series approximation of a mean square coherency measure where the number of terms increases as the observation interval increases in order to keep the approximation error within sone given bound. This transient coherency measure is used to analyze the ripple effect of a disturbance through a power system and the contribution of particular components in the transient response at any location and time. The transient coherency measure can also be used to analyze the transient response of a power system and thus develop a better understanding of the dynamic structure and changes in that structure during the transient interval after a disturbance [17]. The RMS coherency measure evaluated over an infinite observation interval can not be used for the short in- tervals because it indicates the steady state power system dynamic struc- ture and may be called dynamic coherency measure. The principal objective of this chapter is to indicate the tran- sient coherency measure for a short interval and then use this transient 131 132 coherency measure to derive an optimal aggregation method for aggregating coherent groups formed based on this coherency measure when the simulation interval is small. The definition of the root mean square (RMS) coherency measures for the generalized disturbance model and the linearized power system 2N-2 model are given in Chapter 2. A mean square coherency measure is defined and used in [17] rather than a RMS coherency measure because the transient coherency measure based on the mean square coherency measure is easier to define and analyze. The transient coherency measure was developed in [17], and the Presentation here follows that development. 5.1. The Mean Square Transient Coherengy Measure In this section the transient coherency measure for the linea- rized power system 2N model and the generalized disturbance model are quickly reviewed. A detail derivation of transient coherency may be found in [17]. The linearized power system model used here is slightly different than the model which has been used in the previous chapters in the sense that no reference generator is selected as the system ref- erence. Thus, this new linearized model has 2N state variables, while the model used in the previous chapters had (ZN-2) state variables. This new linearized model has the following form £0) = 11. 2<_(t) + _B_ gm (5-1) where . 1 , - 99 A2111 9. : l 0. . 1(- = .....- y— : ..-..- , ..... .:l..-.... MJ- , 1:1,2,...,N e9 AEL fl i-o_I_ ‘ l. .1 b 133 N = The number of generators in the system. I O ' O -1 -1 -1 _ _ . _ .11 {M1,MN , ..... ,MN 1, p ____J____ u : _L I T_9119 _ aP_G 9P1 " 98L. — as @1391 39 P -.I L=-§_G_ 3_P_L. —- 00 39_ T = T ; T = 6 [51,52,...,5N] 9 g [61,62,...,6k] 9 LL)- [w19w230oong] K = The number of load buses in the system T _ T _ E§_ — [PG], PGZ,...,PGN] , EL_ - [PL], PL ,...,PL 2 KJ The input p(t) composed of the deviations in the mechanical input power AEM_ and the deviation in the load power 03;, can be used to model 1. loss of generation due to generator dropping 2. loss of load due to load shedding 3. line switching 4. electrical faults These contingencies can be modeled by an input p(t) that has the form 11.“) = 21“) 1' E2”) where, the vector functions _1_1_1 and p2 are defined in Chapter 2 and the probabilistic model for these contingencies have been indicated in Chapter 2 and will not be repeated here. Now that the linearized model has been developed the next step is to briefly indicate the procedure for deriving the transient coherency 134 measure. The transient coherency measure between generator internal buses h and 2 based on the uncertain description of disturbances is defined as T ch£(T]) E{( 1E00h(t) — A6£(t)]2dt} O T = g-h£§x (T where, the (2N) dimensional square matrix §x(Tl) and the (2N) dimensional vector E02 are defined to be T1 T §X(T]) = J E{x(t)x_(t)}dt (5-3) 0 r l j = h {Eh/Ch = 1 '1 j = ’8 (5'4) 0 otherwise 5.2 A Taylor Series Expansion of the Transient Coherency Measure The purpose of this section is to derive a Taylor series expression for the transient coherency measure as a function of the observation interval T1 over which the measure is evaluated. The Taylor series expression for the coherency measure matrix C(11) will be derived by first deriving a Taylor series expression for §x(Tl)‘ The matrix §x(Tl) may be found from equation (5-3) by substituting x(t) in that equation. Assuming that the observation interval 11 is less than the fault clearing time T2, the vector x(t) has the form t eydvym+9y 03) 0 am=fiymfi and upon substitution of equation (5-5) into equation (5-3), §x(Tl) 135 becomes T T _ 1 AT A T —S—x(Tl) - (0 Vx(0) e— d5 Tl T Av T T Av T + J [J e—-dv]§_Eu(0)§_[J e—-dv] dT (5-6) 0 0 0 where yx(0) = E{x(0)xT(0)} (5-7) Bum) = EI9<019T(0)1 — 3, + R2 + (9., + 11,0011 + 112) T 1 311* E11 m11+ R21+ 4219121” E111$ 21 + r1121411; 1‘111-"112 l m215112 = ................................................... .1. .................. T : T Lflhz EH1 + m12 E21 5 322 l E12 912 (5-8) p(0) = p,] + p2 for t < T2 The variance matrices R1 and 32 and the mean vectors m1 and me are defined in Chapter 2. Now, substituting the following integral 4411 w h h t m h k+l eAv = X 2:1. , J eflydv = A-T (5-9) h=0 ' 0 h=0 (h+l)! into equation (5-6) and integrating, the matrix §x(Tl) becomes 5 (T ) = Z Z . T1 .... [AI v (O)ATJ] —x 1 i=0 j=0 (1+j+l) 1.3. - —x -— + E E T11+j+3 [A1 B P (O)BTATj] (5 10) i=0 i=0 (i+j+3)li+1)!(j+1)! —- —-—u —-—— ' 136 Letting h = i + j, and multiplying numerator and denominator of each term in the expression by appropriate factors, the matrix §x(T]) can be ex- pressed as m [T]h+l T112+3 —S—x”1) = 121-lo 1m):— 91. + W352 (541) where h . h-i 9,, = ,1 (If)_1 130151 (5-12) 1-0 1 t 15.1- h+2 1 T T L = Z ( )AEP (083.6 (5-13) ‘5 i=0 i+1 ‘“ 1- mm It is shown in [17] that the matrices Eh and Eh satisfy the follow- ing recursive formulas _ T _ 942+] - 59,, + (59-12) 5 90 - 1,30) (5-14a) _ 12 L (2 L T _ T Lh51-A(Lh+A:20)+£A(Lh+A—20)J .LO-ZEEJNOE. h = O,1,2,.... (5-14b) To find a specific form for §x(Tl)’ the following two assumptions have been made [17]. First the initial condition yx(0) is assumed to have zero initial frequency deviations because the initial conditions are intended to represent some disturbed condition in the network and is not intended to represent the state of the system after any specific 137 contingency. Second, the damping contribution from the generator, load, and governor turbine energy system is neglected i.e. (0 = O). This assumption is justified because the length of the observation interval T1 is so small that the effects of damping will be small and the spread of the disturbance depends principally on the generator inertias and transmission line synchronizing coefficients of the power system. Thus, the system matrix A, has the following form 9. g .L 1 A= ----- -_T_: _Q L .. where N T.. . z—ii , h=l i = 4 INDU- T.. L '_;4;L 9 1%,] Tij is the synchronizing torque coefficient of the equivalent line connecting internal generator buses i and j for i,j = l,2,3,...,N. For the generator dropping contingencies the matrix Eu(0) has the form T 1 311* 41117111 : ............... 1---- (5-15) t A O v II Thus, based on the above two assumptions and assuming the generator 138 dropping contingencies after substituting the matrices A, B, and Eu(0) into equation (5-13) and carrying out the multiplications, §x(Tl) becomes T (2+3 _ 1 §x(T1) _ 1.20 1153)? H. (5'15) 8 where, the first seven matrices Lh’ h = O,l,...,6 can be obtained from the recursive equations (5-l4b) and have the form 0 : 1 _ : _ L = ------- J ----- V = R + m mT —0 : : — —11 —11—11 9 i ZMIE . 9 1:991 14 = """"" Tr -------- LBMIEE 9 F I “ 6996i 9 I 1:2: """" T ------------------------------ 9 i-NNDNIE-4mv491f L .1 9 i-Muvmxfl-MNINRMDT L3: ------------------------- 1 -------------------------------- LdmnymxfléntflMIf i 9 4HMDNMEJSNEEQDT 9 LI" emlfmxfl+2M9DNIEMIN+69190u3j IO 139 T 9 70112.0me + 21 099101.112 + 3501110900111 L5 = T. 2101. 1.120 .1. E + 7.0 I. 14.10 112 + 3501. 119 9 001 1.11 9 280 1120 9 E + 28.0 .11 9‘01. 112 + 7001 I10 9 010 :11 9 L6 = 3 T T 3T 2 T T T 2T L 9 801..) 991 + 800 0111 +5609) 90 (0106011019 (_T_) J A Taylor series expansion of the transient coherency measure can now be derived based on the Taylor series expansion of §x(Tl)’ provided that the upper left (NxN) submatrix of §x(Tl) is known. The reason is that the coherent groups can be identified by the coherency measures between each pair of generator internal bus angles and only the upper left (NxN) submatrix of §x(T]) is needed for identifying coherent groups, i.e. NT N NT N N _ N ‘3 §x(T1)942/a ’ TM—e-ltfi Etc §x(T1)} in! It = {§x(l])}hh + {§x(T])}££ - {§x(11)}h£ -{§x(T1)}£h (5-17) .1‘ where, §x(Tl) is the upper left (NxN) submatrix of §x(Tl) and £02 is the upper (N) vector of £02' The matrix -§x(Tl) can be expressed as m T 2h+3 N _ 1 N - §x(T1) ’ go Tit—+9):— L21. (5 ‘8) where the matrices {Eeklm are the upper left (NxN) submatrices of h=0 {L 17 . The upper left submatrices of {L_ }m have been shown to ~2h h=0 2h+1 h=0 be zero, and thus are omitted [17]. 140 5.3 Derivation of Optimal Aggregation for Step or Pulse Ipput Disturbances when the Observation Interval is Small (T1 = A) The principal objective of this section is to derive an optimal aggregation method for step or pulse type disturbances in the mechanical input power of the generators of the "study system" when the observation interval is short (T1 = A), based on the performance measure which is the transient coherency measure used to identify the coherent groups. The derivation of optimal aggregation is based on the linear model of the system developed in this chapter. The method can be described as follows. Given a linear time invariant power system 3(t) = 950) + B u(t) (5-19a) and the output equation x(t) = 9 z<_(t) (5-191)) where the parameters of this model are defined in section 5.1. Find a reduced order model 30) = 93.0) + 990.) (MM 20:) =99g (t)1dt10T1 - TrICtJ E{x(t )_T(t)}dt10T} 0 0 = Tr1g§x(a)gT} + Trisfi 5(a )0} A AT T - TrIETJ E{x(t )_ (t )}dt]CT } - Tr{C[J E{x(t )_ (t )}dt]CT 1 (5-24) 0 0 To minimize this performance index, the first step is to determine each of the above terms as functions of the system structure matrix, the inertia matrix, and the statistics of the disturbance. It is shown in Section 5.2 that §X(A) is a (2Nx2N) matrix of the form w 9+3 _ A 9.9) 3,20 (_‘1—12+3:L12 9959 where ’2 Th-TI 9+2 1 T L = ( A 99019 A (s-zsb) _h 120 i+1) —u Since the observation interval is short (T1 = A), the first seven terms of the Taylor series expression for §x(d) will be used in ..l' 143 in the derivation of optimal aggregation. Assuming that the disturbance can only occur in the mechanical input power of the generators of the "study system", @119 34], and _M becomes 1 J I I I I I I A . . T B“Tm-11311 IO l<> IO 341 ' : 541 = I ----L—--- lo I I l I U |< 1 _———__..|___.-____ 11 —-——-—’——--——- I IO where, the (mxl) vector Mn] and the (mxm) matrix M1] are the mean and variance of the step or pulse type input disturbances in the mechanical input power of the generators of the "study system", respectively. Now, since in the derivation of optimal aggregation only the upper left (mxm) submatrix of §X(A) is needed, after partitioning the (NxN) matrix M_I, and the (NxN) matrix .M as _ - - ' - (M I)“ E (T111112 I!“ E _0— I l I I 01= --------- IL --------- . _M.= ........ 1 ........ I I 1 1 1 LgT(t)1dt = 1.20 9+3 : L1; (5-29a) (O)B A (5-29b) Substituting M,§, M,§, and EuTOT into equations (5-29), after carrying out the multiplications, A A J3 = Tr{§_[T E{M(t)MT(t)}dtJ§T} has the form 0 J = 6A5 Tr{CM v MT cT1- T5A7 Tr{CM T1TM T)E 3 5: 4 41 —11 4 7: 4 -111 7 9 21 l5A MT CT 28A - 7: Tr{C (M T)TTMfl M11—1} + 9, Tr{C]M _TTV T1TM T)2 +* 28A9 Tr{C [(M T)2 + (M T) (M T) 1M v MT CT 1 9. 4 11 12 21 —11— —11—1 + ZQAE-TrTC (M T) M0 MT (M T)TCT} (5-30) 9: —1 11 M11- 41 -—- —1 Similarly, the other cross product term may be found from the following relations A A T 6 A(2+3 u (2 “A 111-1 " h+2 1 T % =2 ( )AEMMMA (Maw 1 T’0 i+1 Substituting M,§,M,§, and EuTOT into equations (5-3l), after carrying out the multiplications, A J4 = Tr{§LT E{M(t)xT(t)}dt]§T} becomes 0 146 5 7 = §é_. 15A A MT J4 5: Tr{91M11M M1191} ' 7: Tr{91M 41— M11(M T)1191 7 15A . A T T ' 7: Tr{£](fll)flny_flngl} + 28A9 Tr{CM v MT [(M T)2+ +(M 1 M T 9: —1—11——11 11 —-—)12(_ _)213 £1} 9 9 28A ~ 2 T CT 70A A ‘ +—9: THEM” M11VM11 1+ -9—-,. Tr{9](M11MHy_ML(MT)Hc11 (5-321 Now, substituting equations (5-26), (5-28), (5-30), and (5-32) into equation (5-24), after cancelling the identical terms the following expression is obtained for the performance measure J. 9 A 9 = 70A 1C1 701 A . T '~ 1 1 J T "WHM I-M)1111-M11(MD1191}“ 9:T”§WMWHMMMM)E} - 70A9 Tr{C (M _) v MT (M T)TCT} - 70A911r1c (M A T)M 1 MT (M I)T cT 9: 11Mn— 41 — —1 T l——4F—H 11—fl (5-33) This expression shows that this quadratic error criterion is minimized (J = 0) if Mi_= (MI)1] (5-34) This result indicates that the optimal aggregation for short observation interval is the "inertial averaging" aggregation originally used by Podmore [25]. In this averaging method of aggregation the coherent generators are assumed to swing together as a single generator so that the network connecting these generators experiences no synchronizing power flows between generators although no coherent group ever is perfectly coherent as assumed. This assumption permits the effective shorting of 147 all generators high side transformer buses to a single equivalent bus through phase shifting transformers that will cause real power produced by each generator to flow from the single equivalent bus to all generator high side transformer buses. Thus, the generators of the coherent group are no longer connected to the high side transformer buses, but are connected in parallel at the single equivalent generator bus. The phase shifting transformers distribute the total generation of the generators of the coherent group at the single equivalent bus to the high side transformer buses in order to accurately preserve the injections into the rest of the network from each aggregated group of generators. The inertia of the equivalent generator at the single equivalent bus is the sum of all inertias of the generators of the coherent group that is aggregated. This method of network and generator aggregation perfectly pre- serves the network and flows between coherent groups and the initial acceleration due to faults or loss of generation contingencies. This inertial averaging (IA) method of aggregation does not preserve the steady state load flows, because the effects of the load flows and network within each coherent group is not utilized to modify the network impedances and flows outside the coherent group being aggregated. A basic feature of this IA aggregation is that the identity of the system outside the coherent group is retained. On the other hand, the optimal modal coherent (OMC) method of aggregating the network and generator models would preserve load flows by reflecting the network and flows within the coherent groups that are aggregated to single equivalent buses on the network and flows outside those coherent groups. General Electric is working on developing such a network and generator aggregation procedure. 148 Thus, it would appear that the IA aggregation is appropriate for short intervals, because it preserves the acceleration on each generator after the contingencies occur, while the OMC aggregation is appropriate for the steady state or load flow conditions (long interval). The OMC method of aggregation is also appropriate for mid-term transient and long term stability simulations as well as for developing reduced order models for analyzing and designing power system stabilizer and discrete supplementary controls, where load flow and eigenvalue preservation are important. In the next chapter the comparison of the behavior of the 39 bus New England System with the reduced order models obtained based on the IA aggregation and the OMC aggregation are given for the infinitesimulation interval. CHAPTER 6 COMPARISON OF THE "OPTIMAL MODAL COHERENT" AGGREGATION WITH THE "INERTIAL AVERAGING" AGGREGATION FOR THE INFINITE SIMULATION INTERVAL (T.I = 0°) To compare the behavior of the reduced order models produced based on the optimal modal coherent (OMC) aggregation and the inertial averaging (IA) aggregation, extensive testing of these two methods on the 39 bus New England System have been conducted. The one line diagram and the data for this system are given in Chapter 4. The principal objective of this chapter is to compare the eigenvalues, the coherency measures, and the simulation results of the full 39 bus New England System with the reduced order models constructed based on the OMC aggregation and the IA aggregation at different levels of aggregation for the same disturbances used in Chapter 4 for the OMC aggregation. Since the results for the OMC aggregation are given in Chapter 4, only . the results for the IA aggregation will be given in this chapter. The 1 results given in this chapter are based on a linearized model of the E New England System. I The general test procedure used for the IA method of aggregation is as follows: (1) Apply a modal disturbance to the power system and generate a ranking table of the coherency measures between each pair of generators, ordered from the most coherent pair to the least coherent pair. 149 ISO (2) Identify the coherent groups by applying the commutative rule of aggregation (discussed in Chapter 4) to the above ranking table. In other words, generate a sequence of reduced order models at different levels of aggregation. (3) Aggregate the coherent groups at each level of aggregation based on the IA aggregation, i.e., for each level of aggregation compute the corresponding IA reduced order structure matrix NIIA_. This reduced structure matrix is the upper left (mxm) submatrix of matrix ET. given in equation (4-5b),i.e. MTIA = (£41)n (6-l) defined in equation (4-6). Where, m = Nu + Nc’ and N u the number of unaggregated generators. Nc the number of coherent groups. Once the reduced structure matrix is found, the linearized state models similar to equation (4-8) will be used for the simulation results, i.e. £=Ai+§£ we) where ._ T ._ T 5.- [x],x2,...,xm] , u_- [APM],APM2,...,APMm+]] 9 Elm F9 5.: _______ I _____ , §.= ---- I l _-(MTIA)E -QL"_ __EH]J and 151 NJ] is a matrix of the form (4-3a), where the equivalent generators are represented by the sum of the inertia constant of the generators of the coherent groups. The eigenvalues for the above linearized model may be found from equation similar to (4-9), i.e. 8.,31‘ = - 32%; Ji/oéwb]. , i = l,2,....,m (6-3) 1 l * A where Bi’Bi’ i = l,2,...,m, are the (2m) eigenvalues of A_ matrix and bi’ i = l,2,...,m, are the (m) eigenvalues of (-MTIA) matrix. In our analysis 0 is assumed to be (0 = 0.275). The RMS coherency measures between any pair of system generators may be found for different levels of aggregation using equation (2-23) once the following matrix is known at each level of ag- gregation. A you) = (2435)”): VT (241111 (6-4) T 1151 1-“111 where, the variance matrix, 54] is defined in Chapter 4 for different types of disturbances. (4) Apply' a l per unit (p.u.) step disturbance in the mechanical input power of one or more generators and compare the response of the full system with the response of the linear reduced models that correspond to successive levels of aggregation. The four steps outlined above clearly indicates that the general procedure for constructing IA dynamic equivalents is the same as the procedure for producing OMC dynamic equivalents with the only difference that the reduced structure matrix in this case is different than the 152 reduced structure matrix for OMC equivalents. Therefore, because of the similarity of the two procedures, the results and discussions given in Chapter 4 will not be repeated in this chapter. 6.l Global Equivalents for Step Input Disturbances: To identify the coherent groups required for producing global dynamic equivalents, as was mentioned in Chapter 4, the infinite interval RMS coherency measure must be evaluated for a modal disturbance of all generators. These coherent groups are identified in Table 4-5 of Chapter 4. Since the coherent groups are known for all levels of aggregation, the next step is to aggregate each coherent group at each level of aggregation into a single equivalent generator. The general procedure for aggregating coherent groups based on the inertial averaging (IA) aggregation was presented earlier in this chapter. To show how this aggregation method works consider the first level of aggregation where generators 6 and 7 are lumped into a single generator. Based on the foregoing discussion, the IA reduced structure matrix (MIIA) for this level of aggregation is the upper left (8x8) submatrix of the matrix ([41) given in Table 4-6. This reduced matrix is given in Table 6-l. Another way of computing this structure matrix for the first level of aggregation where generators 6 and 7 are combined into a single generator is as follows: (1) Compute the reduced synchronizing torque coefficient matrix, Il’ from the unreduced synchronizing torque coefficient matrix, 1, given in Table 4-4 by adding the row 7 of I_ to row 6 to get an intermediate matrix and then add column 7 of this intermediate l53 wwop.m¢ nmpm.mu vm_¢.mu Fmow.pu mmom.¢n oppm.mu —mmm.mu mmmm.mu m Ucm pomm.mu opwm.mm vumo.m- mmw_.Fu Nwmm.mn mmmo.mu ceme.mu omvm.m_u swom.ou Nm¢~.mu mmmm.m¢ Rump.mu omNO.NN- pwmp.wn mmcm.wu memo.mi mpom.Ps mnoo.mu umFP.¢i mmnm.m¢ mmom.mpi ommw.pu movm.Pu muvo.mi mmmw.mu mmpp.mn Nmmp.opi wmmm.mFi mmwm.mn onmm.vu mwmw.mu mmoo.mu vooo.mu mwwm.vn omnm.eu mmmn.~i vmem.mu memo.wm mmmm.FFn Nwmo.¢i comm._u wmmp.mu mpmw.mu omwm.i mmmm.mu Fmpm.mn ommo.wm oumm.mn mmmm.mn umpo.mmu mpoo.mu wevm.pu ovum.oi wmow.¢u wmmv.¢n N¢N0.mm Ewpmxm ucmpmcm 3m: ms“ we cowpmmmgmm< o mcopmcmcmo wcp com chpmz mczuuzgum emcee umuauwm mcwmmew>< mepgmcH msh .Puo mpnmh 154 matrix to column 6 to find the reduced. (9XB) matrix Il' Note that the I_ and 14 matrices have one less column than row. The matrix, I, is actually square, but the last column is always deleted because it contains redundant information (found in the last row) about the reference generator. (2) Form the reduced inertia matrix, M4, from the unreduced inertia matrix, H, given in equation (4-3a) by putting the sum of the inertia of generators 6 and 7 in place of the inertia for generator 6 and eliminate the row 7 of matrix M_ to get the reduced (8x9) matrix NJ. (3) Compute the IA reduced structure matrix (M115) by multiplying the matrix, Eh, by the matrix, 14, i.e. MTIA = 9111] One can check that following the above procedure would in fact lead to the reduced structure matrix given in Table 6-l. Once the structure matrix (N115) for the reduced system is found, the eigenvalues, and the coherency measure between each pair of generators (or equivalent generators) can be found using equations (6-3) and (6—4), respectively. Table 6-2 shows the magnitude of the imaginary part of the system eigenvalues at each level of aggregation. Since all the eigenvalues are complex pairs and all have the same real part only the magnitude of the imaginary part is given in the eigenvalue tables. It was shown in Chapter 4, that the eigenvalue information given in Table 6-2 is A essentially contained in the RMS coherency measure matrix §X(w). 155 mpm.¢ Am.m.nxm.m.¢.m.N.Fv m mme.e mem.e Am.m.e.o.e.m.m._v A see.“ _Nm.m mem.e Aw.e.o.e.m.m._v e “mo.w Nam.“ weo.o Pee.e Ae.o.ev.flm.m.~.pv m mem.m owm.e moo.“ meo.o mme.e Ae.m.ev.fim.~v.fim._v e mom.m omN.m mew.“ ego.“ meo.e mme.e Ae.m.ev.flw.Pv m omm.m mpm.m om~.m _mm.e Ame.“ Pee.o wme.e Ae.mv.Am._v N mmm.m omm.m m_m.m omm.m _mm.e one.“ mmo.o eme.e Ae.mv _ _oa.m mmm.m m_m.a mpm.m mmm.m New.“ omo.e mmo.o mme.e eeoz o co_awmmgmm< we m_m>m4 Pym Low mszc>cwmwm Emumxw mo magma xgmcwmmefi mo mmuzpwcmuz cowpmmugmm< Fm>64 Acowuwmmcmm< mcwmmgm>< mepcmch weepmcmcmo :mh ppm mo mucmngzum_o Paco: cot mung mz—m>cmmwm .Nio m_nmh l56 Table 6—3 gives the RMS coherency measures between the reference generator 10 and each of the other generators (or equivalent generators) for different levels of aggregation. Since the analysis of the data in Tables 6-2, and 6-3 are almost a repeat of that done for Tables 4-7, and 4-8 given in Chapter 4, the descriptions of these tables will not be re- peated here. Instead, a brief comparison of each table with the cor- responding table in Chapter 4, will be discussed here. A careful comparison of the eigenvalues in Table 6-2 with the corresponding eigenvalues in Table 4-7 clearly shows the eigenvalues for each successively lower order equivalent are better preserved (closer to those of the unreduced system) in the case of OMC aggregation than the averaging aggregation for long simulation interval (T1 = w). Same con- clusion may not be drawn from the comparison of the RMS coherency mea- sures given in Table 4-8 and 6-3. These computational results confirm that the global equivalent produced based on OMC aggregation better preserve modal but not coherent structure of the full system than the IA aggregation for long simulation intervals. 6.2 Modal Disturbance of Generator l or 8: As was mentioned in Chapter 4, this type of parochial equivalents are appropriate for the study of a single contingency on a specific generator. Two generators selected in Chapter 4 for producing this type of equivalents were generators l and 8. To compare the behavior of the reduced models produced based on the OMC and IA aggregation these gen- erators will be used in this section. vmmoo.o waoo.o ommoo.o mmmmo.o mmooo.o mmooo.o mopmo.o mpwoo.o mwmoo.o mmmmo.o wmooo.o mmumo.o owpoo.o mmmmo.o cameo.o mpomo.o commo.o Nopmo.o mmpmo.o ommvo.o vmwvo.o mmmmo.o o~ooo.o mmooo.o memo.o mopmo.o mmmvo.o oomwo.o 157 mvmmo.o om~¢o.o moooo.o ommoo.o upmmo.o wmomo.c mpmeo.o ommvo.o mwmmo.o mmmwo.o mpmmo.o mmeo.o mmmoo.o Fmomo.o mnemo.o mmm¢o.o Pcovo.o III 82: 8:3 2:5 8:3 8:3 83; 8:2 8:3 8:: cowummmxmm< mo mpm>w4 Fpm Low op Loumcmcwu ucm Loumcmcmo comm :mmzpon mwgzmwmz xdcwemzoo Acowummmcmm< mcwmmgm>< _mwpgmch Am.w.n.m.m.v.m.m.~v w 8.m.e.e.e.m.m.: A afloéagi e taigmggi m Caiéééi e 3.0.3.8.: m 33.8.: N :5 _ mcoz o cowpmmuemm< Fm>m4 meoumcmcmw cop ~_m mo mucmngzpmwo _mcoz so; can: mesmmmz xocmgmnoo .mim mpnmp 158 6.2.1 Modal Disturbance of Generator l: The coherent groups for the modal disturbance of generator l were identified in Table 4-9 of Chapter 4. Tables 6-4, and 6-5 provide the eigenvalue and RMS coherency measure data for a modal disturbance of generator l, based on the IA method of aggregation of the coherent groups. The analysis of the Tables 6-4, and 6-5 are almost a repeat of that done for Tables 4-lO, and 4-ll discussed in Chapter 4. Comparison of Tables 6-4, and 4-lO clearly indicate that the eigenvalues for each successively lower order equivalent at different levels of aggregation are better preserved for the OMC aggregation than the IA aggregation. Comparison of Tables 6-5, and 4-ll obviously show that the RMS coherency measures are also better preserved if the coherent groups are aggregated based on the OMC aggregation. For instance, Table 4-ll indicates that the coherency measures between generators (1,8), and (l,lO) persist through levels 6 and 7 of aggregations respectively, but this is not true in the case of IA aggregation (Table 6-5). This means that the coherency measures between unaggregated generators and generator l, do not change in all levels of aggregation for the OMC aggregation, while this is not true in the case of IA aggregation. Therefore, OMC aggregation better preserves both the modal and coherent behavior of the unreduced system. These results have been confirmed by the simulation results for a l p.u. step disturbance applied to generator 1. Figures 6.ld and 6.le show the simulation response of the reduced order model versus the response of the full system, at levels 4 and 5 of aggregation, based on the IA aggregation. Note that the curves designated by D 159 opo.m mom.v Am.wxu.o.m.¢.m.mv emm.m em~.m emm.e Am.e.o.m.e.m.mv mmm.m e_m.m men.“ _Nm.e Am.e.o.ev.fim.m.mv mmw.m m_m.m mmw.e emm.e _Nm.e Ae.o.ev.Am.m.Nv mmm.m mmm.m Neo.m ewe.“ mmm.© eom.e Ae.©.ev.fim.mv mmm.m mox.m mmm.m Neo.m moN.e mmw.o mom.e Ae.ev.fim.mv mow.m mme.m mem.m mmN.m _me.e “mo.“ moo.m ome.e Ae.ev _om.m mmm.m o_m.m m_m.w mwm.m new.“ omo.e mme.o eme.e meez co_pmmmcmm< mo mpw>m4 cm>mm com mmzpm>cmmwm Emamxw mo mpcma xgmcwmmEH mo mmnapwcmwz cowpmmvcmm< Acowummmemm< mcwmmcm>< pmwpewch 00¢ Fm>m4 _ Lepmcmcmu mo mucmngzpmwo Pane: Low mums wz—m>:mmmm .eio mpamp I60 compo.o nmppo.o mumpo.o Noopo.o opmpo.o mum—0.0 poopo.o mxppo.o nmmpo.o _mmpo.o emp_o.o mmmoo.o mm—Fo.o mmmpo.o —wm~o.o «Nppo.o mmmoo.o mappo.o mmm_o.o unmpo.o Pwmpo.o ¢NFFo.o mmmoo.o Newpo.o mmppo.o mmmpo.o mum—o.o Pmmpo.o emppo.o mmmoo.o Pompo.o mepo.o cop—0.0 “mm—o.o mumpo.o —wm_o.o vm—Fo.o mmmoo.o mmFPo.o Nom_o.o mepo.o qmppo.o mmm_o.o mmmpo.o 8:: 8.: 8.: 8.: 8.: 8.: 8.: 8.: 8.: cowummwgmm< mo m—m>m4 cm>mm Lo: _ Loummema ccm Lopwgmcmw comm :mmzpmn mmgzmmmz xdcmemcou 8.N.N.o.m.e.m.N: N 8.N.m.m.:m.N: e 8.3.38.3: m 88.3.8.3: e 8.8.5.8.: N 8.3.8.: N 8.: P wcoz o :owummmgmm< Pm>m4 Acowpmmmgmm< mcwmmgm>< mepgmch _ Loumgmcmw :o mucmngzpmwo Paco: to: wane mgammmz aucwgmcou .muo w—nmh 161 Figure 6.l Simulation Response of the Reduced Order Models Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator l Based on the Inertial Averaging Aggregation of the Coherent Groups. Figure 6.l Aggregation Level System Generators (d) 4 l.(2.3.5).(4,6,7),8,9 (e) 5 l,(2,3,5),(4,6,7,9),8 Generator lO is the reference generator ""_’-"-_ u I'O') TIHE IN SECONDS ./4 -<- (DBOJBWONU 80108 6'N30 162 TIME IN SECONDS 1 Tfifi jfTT , PO) :/ -.( \. _ ': ) 5” f4'?/ m \ “\ /“ ,- "a .- ‘ 1‘ r.‘ -,// I ~ .3 \ ‘-‘.V ~10 I /" \V. I - "LD {\x \ .- i (t i r; “<- :1) : ' F" _. ) '0') \\ - \ - -.2 ‘ _fi 31 z b- ": ITfiII1fiffifi111 '" IDV'WN-"OTNI‘CDUDVWNF‘Q (DBOJBTONU 80108 B'NBO (OBDIBTONU 80108 I'NEO TIME IN SECONDS (d) 1" 4l- 3 _ ) 163 PO) 3 , r , :n l./‘ I- 0‘ :~ P’ "./" . F = I" I‘V' \ g 1 I I 4 TIHE IN SECONDS [Vlélllvrll Tfij 7111 Ififii (0 N .— D .-n N (OBOJBWONU 80103 8'N30 '2 -( \_ ,) /=/' gr" 3: /lJ (‘ 3) _/ 5" / ‘3 \\ >0 I _/‘ \\ J.) / t.’ \‘ '\=\ 3:. a" ’2‘] ./ "l K T)» '1 1 l l 1 i T 1 1 l l :T U3 V m N H D (OBOIHWONU 30108 I'NBO TIHE IN SECONDS (e) T64 represent the full New England System and the curves designated by <> represent the reduced order models. Comparison of these two plots with the corresponding plots given in Figures 4-2d, and 4-2e, obviously show that the reduced models produced based on the OMC aggregation do behave better than those produced based on the IA aggregation for long simu- lation intervals. 6.2.2 Modal Disturbance of Generator 8: The coherent groups for the modal disturbance of generator 8 were identified in Table 4-12 of Chapter 4. Tables 6-6, and 6-7 provide the eigenvalue and coherency measure information for a modal disturbance of generator 8. The analysis of the data in Tables 6-6, and 6-7 are almost a repeat of that done for Tables 4-l3, and 4-l4 discussed in Chapter 4. Compariosn of the eigenvalue and coherency measure data determined based on the IA aggregation (Tables 6-6, and 6-7) with the corresponding data obtained based on the OMC aggregation (Tables 4-l3, and 4-l4) clearly indicates that the reduced models constructed based on the OMC aggregation better preserve the eigenvalues and the coherency measures of the full system. The eigenvalue data of Tables 4-l3, and 6-6 indicate that the model accuracy should begin to deteriorate at about level 5, because there is no single eigenvalue pair that can be associated closely with the intermachine oscillation of generators l, and 9. This estimate is confirmed by the simulation results of a l p.u. step disturbance on generator 8. 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At level 4, the reduced order model gives a very accurate representation of the behavior of the study system generators l, 8, and 9, as witnessed by Figure 6.2d. By contrast Figure 6.&eshows that the level 5 model is not a good repre- sentation of the system behavior. The explanation for why this occurs was discussed in Chapter 4. It was mentioned in Chapter 4 that by aggregating generators l and 9 the strongly bound coherent group (l,8) will be broken up and this is the source of the problem. The way to solve this problem is to avoid disturbing only part of the coherent generators which are strongly bound together. Comparison of Figures 6.2d and 4.3d clearly show that the OMC aggregation performs better than the IA aggregation at level 4 of aggregation for long simulation intervals. On the other hand, comparison of Figures 6.2e and 4.3e obviously indicate that the IA aggregation better performs for the short simulation interval than the OMC aggregation. In other words, it seems that the IA aggregation captures the amplitude of the first and second swing more accurately than the OMC aggregation. 6.3 Modal Disturbance of Generators 1 and 8: To produce a local dynamic equivalent for the New England System, the modal disturbance of generators l and 8 has been conducted. The reason for selecting generators l and 8 is that the generators l and 8 are strongly bound together. Thus, as was mentioned in the previous section, in order to find a good local equivalent, all the generators of the strongly bound group must be disturbed. A very good local Figure 6.2 Figure 6.2 (d) (e) 168 Simulation Response of the Reduced Order Models Versus the Response of the Full System for a One Per Unit Step Disturbance on Generator 8 Based on the Inertial Averaging Aggregation of the Coherent Groups. 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F r. __ 3 T— _. '1') .I 1'0) \-:\ i— ‘;\ b ('4’ —N ( ETN .; ) .- .‘> _- : "o-c : 'v-o 111fiIIIIIIII" Ti 11 I11fi1'2 N O (D (D <' N O O (D Q' N D I". (OBOIETONU 30103 9'N30 (DBDJETONU 30103 I'N30 TIME IN SECONDS (9) 178 Finally Figures 6.3f and 6.&ggive the simulation response of the reduced and unreduced systems, at level 4 of aggregation based on the IA and OMC aggregation for a l p.u. disturbance on generators l and 8. These simulation results again confirm that the OMC aggregation performs better than IA aggregation for long simulation intervals. 6.4 Comparison of IA Aggregation and OMC Aggregation for Impulse Input Disturbances when the Simulation Interval is Infinite (T1 = w) The purpose of this section is to compare the eigenvalue and coherency measure data determined based on the IA aggregation with the corresponding data obtained based on the OMC aggregation of the coherent groups identified based on the impulse coherency measure. For the sake of saving some space, only the results for the global case (disturbance on all generators) are presented in this section. The coherent groups are identified in Table 6-lO based on the RMS coherency measures for impulse using equation (2-23) with -IM 'T] + (n) M(R + m mT )M + '1( mmm mglmT (M_T) (6-5) _ I (m) ZEAXMT ) —-—ol —ol~ol i where the probabilistic disturbance which is being used in this section has the form 2 M2 M2 =9’R 2"“ ”9’ M10} _ - 2 _o] - DIag{M], M moi <6-6) Table 6-lO shows that the coherent groups indentified based on the impulse RMS coherency measures are different than the coherent groups produced based on the step RMS coherency measure (Table 4-5). This is becaUse of the fact that the analytical expressions for the RMS T79 Table 6-lO. Ranking Table of the Impulse RMS Coherency Measures for the Modal Disturbance of all lO Generators. Rank Generator Coherency Coherent Aggregation Order Pair Measure Groups Level 1. C(l, 8) = .l92947 (l,8) l 2. C(6, 7) = .l93214 (l,8),(6,7) 2 3. C(4, 7) = .20428l 4. C(4, 6) = .209986 (l,8),(4,6,7) 3 5. C(4, 8) = .2ll544 6. C(7, 8) = .213654 7. C(l, 4) = .2l399l 8. C(l, 7) = .2l6096 9. C(6, 8) = .221479 10. C(4, 5) = .223l24 ll. C(l, 6) = .224055 (l,4,6,7,8) 4 12. C(3, 8) = .227296 13. C(2, 8) = .227794 14. C(3, 4) = .228943 15. C(l, 3) = .229809 l6. C(2, 3) = .230038 (l,4,6,7,8),(2,3) 5 l7. C(l, 2) = .230512 l8. C(3, 7) = .230939 19. C(2, 4) = .231226 20. C(2, 7) = .233094 21. C(8, 9) = .235885 22. C(3, 6) = .238207 23. C(l, 9) = .240l74 24. C(2, 6) = .2406l5 (l,2,3,4,6,7,8) 6 25. C(5, 7) = .245784 26. C(4, 9) = .246003 27. C(7, 9) = .247945 28. C(5, 6) = .250947 29. C(5, 8) = .25lll4 30. C(l, 5) = .253989 3l. C(6, 9) = .254402 32. C(3, 9) = .263691 33. C(2, 9) = .264642 (l,2,3,4,6,7,8,9) 7 34. C(3, 5) = .266472 35. C(2, 5) = .268092 36. C(5, 9) = .28l299 (l,2,3,4,5,6,7,8,9) 8 37. C(8, 10) = .324067 38. C(l, l0) = .324654 39. C(2, lO) = .3384l8 40. C(3, l0) = .344713 4l. C(4, l0) = .35400l 42. C(7, l0) = .355602 43. C(6, 10) = .363920 44. C(9, 10) = .375939 45. C(5, lO) = .393312 I80 coherency measures for fault (impulse) and step are different. Thus, the coherent groups that are detected by these two measures for the modal disturbance of all generators could be expected to be different. Another reason why the fault and step disturbances produce different strongly bound groups are that the impulse disturbance is a wide band disturbance, while the step disturbance is a narrow band disturbance and thus the modes and coherent structure are excited differently by these two disturbances. Tables 6-ll and 6-l3 exhibit the eigenvalue and RMS coherency measure data for the IA aggregation and Tables 6-l2 and 6-l4 provide the eigenvalue and RMS coherency measure data for the OMC aggregation of the coherent groups. Comparison of Tables 6-ll and 6-l2 clearly indicate that the eigenvalues are better preserved for the OMC aggregation than the IA aggregation. It is also obvious from comparing Tables 6-l3 and 6-l4 that the RMS coherency measures are also better preserved if the coherent groups are aggregated based on the OMC aggregation. In fact, Table 6—l4 shows that for the OMC aggregation the RMS coherency measure between each unaggregated generator and the reference generator 10 persists through seven levels of aggregation, while this is not true in the case of IA aggregation (Table 6-l3). Thus, in summary the conclusion of this chapter is that the IA aggregation performs better for short simulation intervals where capturing the initial acceleration is important (transient stability). 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CHAPTER 7 CONCLUSIONS AND FUTURE INVESTIGATIONS This research was initiated primarily to derive optimal ag- gregation methods based on the types of disturbance (impulse, pulse, or step), the time interval over which the reduced model must be accurate, and the specific application it will be used for. These optimal ag- gregations were derived'based on the linearized model of the power system which was divided into two parts called the "study system", where the disturbances occur and whose detailed model is of interest, and the "external system" whosedetailed behavior is of no interest but whose effect on the study system must be accounted for. The main problem was to find a reduced order model (dynamic equivalent) of the external system thatfaithfully and accurately preserves the interaction between the internal and external systems. The optimality was derived using a parameter optimization for the reduced model, where the performance index J, which was a function of the differences between each pair of generator bus angles in the study system of the aggregated and unaggregated systems, was minimized. The major results of this thesis are now summarized on a chapter by chapter basis and the related topics for future research are proposed at the end of this chapter. 7.1 Overview of Thesis In the first chapter, several methods of producing dynamic T86 187 equivalents which exist at present were discussed; namely,modal analysis, coherency based methods, identification methods, and singular pertur- bation techniques. Some of the advantages and disadvantages of these different techniques have been discussed. Different structural conditions (SSC, SGC, SSLD, PC, WLD) that cause the external group of generators to behave as a single generator were defined. It was mentioned that the RMS coherency measure can detect the SSC, SGC, and SSLD coherent groups if different porbabilistic disturbances are utilized. It was also mentioned that the RMS coherency measure is an appropriate measure for producing dynamic equivalents that retain both frequency domain (modal) and time domain (coherent) dynamic structures. This RMS co- herency measure is thus utilized in this thesis for deriving "optimal- modal-coherent" aggregation methods that maintain modal and coherent dynamic properties. Three distinct types of dynamic equivalents were defined in Chapter 1; namely, a disturbance dependent equivalent (parochial), a partial disturbance independent equivalent (local), and a total disturbance independent equivalent (global). It was mentioned that these three types of equivalents have completely different applications and exploit different coherent structural pr0perties (SGC, SSC, SSLD) and can be produced by evaluating the RMS coherency measure for different pro- babilistic or deterministic disturbances. Chapter 1 is closed with a statement of the thesis objective which was to determine different optimal aggregation techniques for different types of disturbances, different simulation intervals, and different applications they will be used for. 188 In the second chapter, the linearized power system model, the generalized disturbance model, and the generalized RMS coherency mea- sures used for identifying coherent groups for specific type of distur- bances(step, pulse, impulse) were defined. It was shown that the expected value of the RMS coherency measures, evaluated over an infinite simulation interval, are algebraically related to the parameters of the power system state model and the statistics of the system disturbances. An efficient algorithm for identifying the coherent groups, based on the RMS coherency measure for different types of disturbance was also defined. Finally, the inertial averaging and singular perturbation techinques by which coherent groups can be aggregated to form an equivalent were discussed in more detail than Chapter 1. Chapter 2 was closed with the statement that although there are different aggregation techniques there is no theory at present on which aggregation is best for different contingency type and time interval combinations. This was the objective of this research. In Chapter 3, it was shown that a singular perturbation like aggregation method called optimal modal coherent (or 0MC)aggregation produces equivalents that are Optimal (produces no error) for either any step or impulse disturbance restricted to the "study system" if the simulation interval is long (T1 = 00). This OMC aggregation allows aggregation for strict geometric coherency (SGC) and strict strong linear decoupling (SSLD) coherent structural conditions and can be also used for impulse input disturbances. Singular perturbation (or SP) aggregation can not be applied under either of these conditions. SP aggregation is only applicable for the system which satisfies strict 189 synchronizing coherency (SSC) structural condition. Although OMC ag- gregation is not SP aggregation for the disturbances restricted to the "study system" and for faults, the actual method of aggregating the network and the generators model is identical with the procedures that one would use for the SP aggregation. The equivalents produced based on this OMC aggregation are optimal for long term simulations, for use in design of generator controls (such as excitation system and power system stabilizer), and for planning applications. A computationally efficient algorithm, extendable to the large scale systems, was developed in Chapter 4 for aggregating each coherent group of generators, identified using the RMS coherency measure for a particular type of disturbance, into a single generator based on the OMC aggregation. The performance of the dynamic equivalents, derived based on the infinite interval RMS coherency measure and OMC aggregation, were judged based on their ability to reproduce the coherency measures (co- herent behavior), the eigenvalues (modal behavior), and the simulation response observed with the unreduced model at different levels of ag- gregation. Simulation results have been conducted on the 39 bus New England System for producing three different types of dynamic equivalents; namely, parochial, local, and global equivalents. From the simulation results in Chapter 4, it was concluded that the equivalents produced based on the OMC aggregation closely approximate the eigenvalues, the coherency measures, and the simulation response of the unreduced system, as long as strongly bound coherent groups have not been broken up. In other words, these equivalents preserve both modal and coherent dynamic structure of the unreduced system. The strongly bound coherent groups 190 are the coherent groups identified based on the modal disturbance of all generators. These groups divide the overall power system into areas, called principal groups (or tightly interconnected groups), that react in consort to disturbances anywhere in the system. Thus, if the main concern is how a disturbance propagates among the principal groups, then the modal disturbance of all generators can provide the proper model. This type of global dynamic equivalent can be used for the study of inter-areapower transfer limit in transmission planning and islanding applications. The results obtained in Chapter 4, also suggested that the RMS coherency measure in conjunction with the OMC aggregation seem to provide selective eigenvalue retention, based on the location of the partial modal disturbance of generators, as long as all generators of a tightly interconnected group are disturbed. That is,the eigenvalue retention seemed to be based on the frequencies most excited by all the disturbances located in a certain region of the power system. This is why these local type of equivalents can be used for the design of excitation system and power system stabilizer where all eigenvalues, fast or slow, that are excited and must be appropriately damped by the excitation system for disturbances in a local region must be preserved in the local equivalent. This type of local equivalent can be also used for discrete supplementary control. Finally, the computational results for single contingencies also indicated that the OMC aggregation better preserves the modal and coherent structure of the unreduced system (than the IA aggregation) for long simulation intervals. These parochial type of equivalents can be used for mid-term and long term stability programs when the unreduced 191 system is very large. They can be also used for on line transient stability for security assessment, where a low order accurate model for a specific contingency is desired. In Chapter 5, it was shown that the network and generator ag- gregation used in the EPRI (Electric Power Research Institute) dynamic equivalents package, called "Inertial Averaging" aggregation, where inertia and power injections for the coherent generators to be aggregated are summed, is optimal for any disturbance restricted to the "study system" if the simulation interval is small (T1 = A). In this inertial averaging (or IA) method of aggregation the coherent generators are assumed to swing together as a single generator so that the network connecting these generators experiences no synchronizing power flows between generators although no coherent group is ever perfectly coherent as assumed. This assumption permits the effective shorting of all generators high side transformer buses to a single equivalent bus through phase shifting transformers that will cause real power produced by each generator to flow from the single equivalent bus to all generator high side transformer buses. Thus, the generator of the coherent group are no longer connected to the high side transformer buses, but are connected in parallel at the single equivalent generator bus. The phase shifting transformers distribute the total generation of the generators of the coherent group at the single equivalent bus to the high side transformer buses in order to accurately preserve the injections into the rest of the network from each aggregated group of generators. The inertia of the equivalent generator at the single equivalent bus is the sum of all inertias of the generators of the coherent group that is aggregated. 192 This IA method of network and generator aggregation perfectly preserves the network and flows between coherent groups and the initial acceleration due to faults or loss of generation contingencies. This IA method of aggregation does not preserve the steady state load flows, because the effects of the load flows and network within each coherent group is not utilized to modify the network impedances and flows outside the coherent group being aggregated. On the other hand, the OMC method of aggregating the network and generator models would preserve load flows by reflecting the network and flows within the coherent groups that are aggregated to single equivalent buses on the network and flows outside those coherent groups. Finally,in Chapter 6 the comparison of the behavior of the full 39 bus New England System with the reduced order models obtained based on the IA aggregation and the OMC aggregation are given for the infinite simulation interval. The results in Chapter 6 confirmed that the reduced order models obtained based on the OMC aggregation better preserves the coherency measure, the eigenvalues, and the simulation results of the full New England System (comparedto the IA aggregation) when the simulation interval is infinite (T1 = 0o). The results in Chapter 6 also indicated that the IA aggregation is appropriate for short intervals, because it preserves the initial acceleration on each generator after the contingencies occur, while the OMC aggregation is appropriate for the steady state or load flow condition (long intervals). The OMC method of aggregation is also appropriate for mid-term transient and long term stability simulation as well as for developing reduced order models for analyzing and de- signing power system stabilizer and discrete supplementary controls, 193 where load flows and eigenvalue preservation are important. 7.2 Topics for Future Research Based on the analysis in Chapter 3, it was concluded that the quadratic error criterion may not be appropriate as the performance index in deriving optimal modal coherent aggregation for impulse type disturbances based on the RMS coherency measure for impulse. One could propose other types of coherency measures (not necessiarly the RMS coherency measure for impulse) or other types of performance indices for deriving optimal aggregation. For example one may want to derive an optimal equivalent which optimally preserves only the modal behavior (eigenvalues) of the system which may be called optimal modal aggregation. Another interesting item for future research is to try to develop a methodology for producing nonlinear optimal modal coherent aggregations. This is not an easy task and needs a lot of research. General Electric is already working on producing such equivalents. One other area of future investigation is to develop identification methods that could produce approximately the OMC and IA aggregations. In other words, try to produce recursive algorithm that could continually update the dynamic equivalents for on line applications; such as dynamic security assessment for dynamic stability or transient stability, or possibly automatic generation control. In Chapter 6, the behavior of the reduced order models obtained based on the IA and OMC aggregations, for the impulse disturbances, were only compared for the global type of equivalents. 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