{Ht-i553; '1'? 7" “' " ”'3';- [j fl._.u ’§_ 5 '. I, (F77. .-'...‘. viva-om v1.13" ‘2} write? . i.“ Entrees-Er *y J This is to certify that the dissertation entitled Simulation and Security Analysis Methods for Transients Due to Loss of Generation Contingencies presented by Mohsen Lotfalian has been accepted towards fulfillment of the requirements for Doctoral . Systems Science degree in GM MW MajOr professor Date September 29, 1982 MS U is an Affirmative Action/Equal Opportunity Institution 0‘ 12771 )V1ESI_J RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from “ your record. FINES will be charged if book is returned after the date stamped below. , "5; ”Wt-v '- M'ils; i"-f in ,t r t i “In. " '33? g» r.- 9""? 9”; —’w“-—v . '7 nffifi'“ . mi!“ SIMULATION AND SECURITY ANALYSIS METHODS FOR TRANSIENTS DUE TO LOSS OF GENERATION CONTINGENCIES By Mohsen Lotfalian A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1982 ABSTRACT SIMULATION AND SECURITY ANALYSIS METHODS FOR TRANSIENTS DUE TO LOSS OF GENERATION CONTINGENCIES By Mohsen Lotfalian The inertial and governor distribution of mismatch power due to loss of generation contingencies causes stability and security problems on boundaries and lines that are vulnerable to these power flows. Simulation methods based on load flow techniques are developed to allow direct assessment of stability and security problems associated with the inertial and governor power flows. A set of security measures for detecting the weak boundaries between generation groups which cause the stability and security problems in large networks is defined. The DC load flow methods for simulating the inertial and governor response of generating units to loss of gener- ation contingencies are compared to the midterm stability simulation of the same contingencies on a 49 bus test sys- tem, and the accuracy of the load flow methods is shown to be good. It is also shown that the inertial and governor response of generating units causes different power flows and thus different stability and security problems. The inertial and governor load flow methods permit simulation of loss of generation contingencies on immense Mohsen Lotfalian power system models that could not be handled using the Midterm Stability Program and present techniques. The security measures for inertial and governor power flows are shown to capture the strict synchronizing coherency (SSC) loss of controllability property for the inertial and governor state models that causes the vulnera- ble boundaries to inertial and governor power flows. These security measures are shown to be identical to the square of the r.m.s. coherency measure for the probabilistic modal disturbance loss of generation contingencies. A method is developed to identify and rank the vulner- able boundaries due to inertial and governor power flows from the weakest to the strongest based on the security measures and a commutative grouping algorithm. The method is applied to the 49 bus test system, and the weakest boundary to inertial and governor power flow due to loss of generation contingencies is identified. The identification of the weakest boundary is shown to be accurate, since it is shown that the loss of stability associated with iner- tial and governor power flows occurs across this boundary. ACKNOWLEDGMENTS I would like to express my deep appreciation to my major advisor, Dr. Schlueter, for his guidance, encourage- ment, and dedication that he gave me in academic matters, and most of all for his friendship and help in other aspects of my life. He will always be remembered as the best friend. I would also like to express my gratitude to the members of my guidance committee for their contribu- tion to my doctoral program. I would especially like to thank Dr. Park for all the help that he gave me in getting me started on research in his wind energy project, Dr. Yen and Dr. Khalil for the enjoyable experience of taking their classes, and Dr. Shanblatt for his friendship and guidance. TABLE OF CONTENTS LIST OF TABLES ....................................... vii LIST OF FIGURES ...................................... viii CHAPTER 1. INTRODUCTION ............................. 1 1.1. Review of Present Simulation Techniques ...... 6 1.1.1. Load Flows ........................... 7 1.1.2 Outage Distribution Factors (ODF) 9 1.1.3. Transfer Distribution Factor (TDF) 9 1.1.4. Decoupled Load Flow .................. 9 1.1.5. Transient Stability and Midterm Stability Programs ................... 10 1.1.6. Inertial Load Flow ................... 11 1.1.7. Long-Term Stability Programs ......... 13 1.2. Review of Present Planning Procedure ......... 13 1.3. Present Status of Security Measures .......... 16 1.4. Summary ...................................... 24 CHAPTER 2. LOAD FLOW METHODS FOR SIMULATING GENERA- TION RESPONSE TO LOSS OF GENERATION FOR TRANSMISSION PLANNING AND SECURITY ASSESSMENT ................... 26 2.1. Generation Response to Loss of Generation or Load Contingencies ........................... 28 2.1.1. Power Mismatch Distribution According to Synchronizing Power Coefficients .. 29 2.1.2. Inertial Distribution of Mismatch (PTOT) OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 30 iv 2 .2. 2.1.3. Governor Distribution of Mismatch p TOT oooooooooooooooooooooooooooooooooo 2.1.4. AGC/Operator Distribution of Mismatch PTOT .................................. Formulation of Inertial, Governor, AGC/ Operator Load Flow Problems .................. 2.2.1. Load Flow Formulation ................ CHAPTER 3. APPLICATION OF THE INERTIAL AND GOVERNOR LOAD FLOW FOR ASSESSMENT OF STABILITY AND SECURITY ON A 49 BUS TEST SYSTEM ............................ 3.1. 490 MW Loss of Generation in the External System ....................................... 3.2. 790 MW Loss of Generation at Bus Number 12 3.3. Simulation of 790 MW Loss of Generation at Bus 12 Using Conventional Load Flow Program 3.4. Line Outage Contingency of Line (40, 41) ..... CHAPTER 4. SECURITY MEASURES AND SECURITY ASSESSMENT 4.1. Security Measure Derivation and Justification 4.1.1. The Linear Power System Model ........ 4.1.2. Disturbance Model .................... 4.1.3. The r.m.s. Coherency Measure ......... 4.1.4. Inertial Security Measure and Its Relation to r.m.s. Coherency Measure.. 4.2. Transmission Boundary Vulnerability Justifica- tions Based on Loss of Controllability Condi- tions for the Classical Transient Stability Model and the Inertial Security Measure ...... 4.2.1. Observability and Controllability Conditions ........................... 4.2.3. SSC Loss of Controllability Condition 4.2.4. Grouping Algorithm ................... 4.2.5. Identifying Vulnerable Boundaries by the Security Measure and Grouping Procedure ............................ 32 36 37 39 50 52 64 74 75 79 81 82 86 89 92 100 101 105 4.3. Governor State Model, Coherency Measure for Governor Response, Governor Security Measure, SSC Property for Governor Load Flow .......... 117 4.3.1. Governor State Model ................. 117 4.3.2. r.m.s. Coherency Measure for Governor Load Flow ............................ 119 4.3.3. Governor Time Frame Security Measure.. 123 4.3.4. SSC Condition in Governor Load Flow... 130 CHAPTER 5. TESTING THE BOUNDARY IDENTIFICATION METHOD ON THE 49 BUS (EPRI) TEST SYSTEM ................... 138 CHAPTER 6. CONCLUSIONS AND FUTURE INVESTIGATION ..... 150 6.1. Overview of Thesis ........................... 150 6.2. Future Research .............................. 154 BIBLIOGRAPHY ......................................... 160 vi LIST OF TABLES Comparison of the Inertial Response Generator Angles Obtained from the Inertial Load Flow and the Midterm Stability Program for 490 MW Loss of Generation at Generator Bus 11 ................ 61 Comparison of the Governor Response Generator Angles Obtained from the Governor Load Flow and the Midterm Stability Program for 490 Mw Loss of Generation at Generator Bus 11 ................ 62 Comparison of the Inertial Response Generator Angles Obtained from the Inertial Load Flow and the Midterm Stability Program for 790 MW Loss of Generation at Generator Bus 12 ................ 65 Inertial Angles Across the Tie Lines Connecting the Internal and External System After 790 MW Loss of Generation ............................... 66 Governor Angles Across the Tie Lines Connecting the Internal and External System After 790 MW Loss of Generation ............................... 66 Angles Across the Transmission Lines After Outage of Line (40, 41) .......................... 76 Ranking Table of the Inertial Security Measures.. 140 Group Formation Table Based on Inertial Security Measure of Ranking Table ........................ 144 Group Formation Based on Governor Security Measure ......................................... 147 comment» 0 O I O O 00 ‘\I O‘ U“ 45 O I O O O .10. .11. .12. LIST OF FIGURES 49 Bus Test System ............................. Power Flow of Lines Connecting the Internal and External System After 490 MW Loss of Generation Frequency Deviation at Different Buses After 490 MW Loss of Generation .......................... Generator Angles After 490 Mw Loss of Generation Generator Angles After 490 Mw Loss of Generation Generator Angles After 490 MW Loss of Generation Generator Angles After 490 MW Loss of Generation Power Flow of Lines Connecting Internal and External System After 790 Loss of Generation.... Frequency Deviation at the Internal and External Buses After 790 Mw Loss of Generation .......... Generator Angles After 790 MW Loss of Generation Generator Angles After 790 Mw Loss of Generation Generator Angles After 790 Mw Loss of Generation viii 51 53 55 57 58 59 60 68 69 70 71 72 CHAPTER 1 INTRODUCTION The recent series of contracts for interregional power transfer for the purpose of either oil/gas displacement or economy will add significant additional stress to the trans- mission system that delivers power from Quebec, Ontario, and the Midwest to the Northeast United States. The multiple large transfers on an existing network will cause short lines to operate closer to their thermal overload limits, thus making them more vulnerable to contingencies caused by line overloads. The transfers will cause long lines to operate closer to their stability limits, which are approx- imately the same as or even less than thermal limits if the line is long enough. Thus, the transfers will make the system with long lines more vulnerable to stability prob- lems. Facility additions for coping with these transfers as well as the imposed transfers themselves may well inval- idate empiric predictions based on experience of vulnerable boundaries and lines and the set of critical contingencies. Present AD-DC load flow and transient stability simulation techniques and the associated planning methods discussed later in this chapter can, however, be used to determine the effects of transfers and facility additions on both 2 security and stability for a variety of operating condi- tions for an exhaustive set of line outage contingencies and for a set of fault contingencies. The security and stability problems associated with loss of generation contingencies on the large networks associated with interregional transfers cannot be: (1) adequately simulated using present techniques; (2) handled using present planning methods; (3) empirically predicted because of (a) these transfers, (b) facility additions, and (c) the lack of simula- tion techniques, planning methods, and experience with the kind of stability and security problems that occur for the loss of generation or load contingencies. The problem that is overlooked in present simulation techniques and planning methods is that a loss of genera- tion or load is at different time frames after the contin- gency distribution on all generation in the interconnection inertially (pr0portional to the inertia of each generator) or distribution on all generation under governor frequency regulations (proportional to the frequency response charac- teristic of each generator). Both distributions of the mis- match will cause inadvertant power flows throughout the network focusing to the point of the mismatch. Planners assume that the present planning methods will detect the security and stability problems associated with the flows, and that there are large margins between present thermal current loading and the thermal limits that can handle 3 these inadvertant flows without security or stability problems. The large multiple interregional transfers with the possible addition of long transmission lines may well eliminate the large thermal margins, cause stability mar- gins to decrease due to both the transfers and the addi- tions of long lines with inherently smaller stability mar- gins, and bring into question whether present simulation techniques and planning methods can uncover the stability and security problems associated with the distributions of loss of generation or load mismatch. The stability and security problems in the Northwest associated with the large interregional transfers from the Bonneville Power Administration (BPA) to California over long transmission lines for loss of generation contingencies [1] suggest that such problems actually exist with interregional transfers on long lines and may likely occur for the large multiple transfers being contracted in the Northeast for both oil displacement and economy. The study of these security and stability problems in the Northwest was simplified since the transfers were over one set of lines, and transfers were only in one direction at any time. The problems for the Northeast are much more complicated since the transfers are from north to east, north to west, and west to east simultaneously. Thus, the lack of simulation techniques for interconnection wide inertial and governor transient response to loss of gener- ation or load contingencies and the lack of security 4 assessment methods for detecting vulnerable boundaries and lines, and the critical loss of generation contingencies becomes an acute problem. There are two principal contributions in this thesis. The first is the development of load flow based simulation techniques for determining "snapshot" pictures of the sys- tem's transient response after a loss of generation contin- gency. The inertial load flow [2, 3] was developed after the 1965 Northeast blackout to simulate the inertial response of generation after a loss of generation contin- gency. This inertial load flow is utilized to capture the snapshot of the state of the system transient when the effects of the loss of generation (load) contingency have propagated throughout the system, and the deceleration (acceleration) of the system is identical and constant everywhere but before governors have had an opportunity to respond to arrest the deceleration (acceleration) of sys- tem inertia. A governor load flow is proposed and developed in this research to capture the snapshot of the system transient when the governor response to the drop in frequency is com- plete, but before automatic generation control or operator action can replace the lost generation or load in the utility that experienced the contingency. These inertial and governor load flows are applied to simulate a loss of generation contingency on a test system and are shown to be quite accurate in capturing system transient response as 5 compared to midterm transient stability simulations. The results suggest that the inertial and governor responses can be quite different and that each can cause stability problems that are not detected by normal first-swing transient stability simulations, load flow simulations, or line outage studies. The second major contribution of this thesis is the development of a security assessment methodology that can detect the boundary and line vulnerabilities and could be extended to determine the critical contingencies for the inertial and governor distribution of mismatch for loss of generation or load contingencies. This methodology will be based on a new set of security measures, theoretical understanding of the controllability and Observability of the classical transient stability model that causes these vulnerabilities, and methods for using these security meas- ures for detecting the boundaries and lines between genera- tor groups that cause these vulnerabilities for inertial or governor distribution of mismatch for loss of generation or load contingencies. These methods are obviously new in approach and are important in this application because planners have little experience with the security and sta- bility problems associated with inertial and governor dis- tribution of loss of generation mismatch and have no experience with such problems on the large network associ- ated with the transfers being contracted for oil displace- ment in the Northeast. 6 The remainder of this chapter is devoted to a review of presently available simulation techniques (section 1.1) and associated planning methods (section 1.2) with a dis- cussion of their limitations in simulating or pr0perly assessing security and stability for the inertial and governor distribution of loss of generation or load mis- match after such contingencies. A review of security meas- ures that have been proposed for security assessment is presented in section 1.3. A review of the research pre— sented in this thesis is then given in section 1.4. 1.1. Review of Present Simulation Techniques The present planning studies are performed in several steps with a variety of simulation techniques, such as AC and DC load flow programs for steady state analysis of the network and time domain dynamic simulation for analyzing the dynamic performance during the disturbance (transient stability program, midterm stability program, long-term stability program). To determine the performance of an interregional power transmission system, AC load flows are used to simulate large interregional and subregional power transfers under normal network conditions and single contingency condi- tions. A few important double contingencies are also analyzed. Linear techniques, such as DC load flow, outage distribution factors (ODF), and transfer distribution fac- tors (TDF), are used to identify critical facilities which 7 limit the interregional power transfers by the evaluation of an exhaustive set of first and second contingencies. Linear techniques are also used for contingency analysis procedure. Transient stability studies are performed to show the power system is capable of absorbing the first power swing and remaining stable upon the loss of any sin- gle transmission element, transformer, or generating unit. To clarify whether the present methods are able to assess the stability and security problems associated with inertial and governor distribution of loss of generation or load contingencies, a brief description of each technique is necessary. 1.1.1. Load Flows Load flow solutions provide bus voltage magnitudes and angles and real and reactive power flow on each element (line, transformer) in the transmission system for speci- fied generation and load injections. This information is needed to test the system's ability to transfer power from generation to loads and check for overload limitations in particular lines and voltage violation at buses. The physical characteristics of generation and load require that the load and generation injection to the load flow be represented in terms of active and reactive power rather than by bus current injections. Therefore, in a load flow study, the electrical condition at a bus is described by P, O, V, and e, which are active power 8 injection, reactive power injection, voltage magnitude, and voltage angle at each bus, respectively. For the generator buses, P and V are specified because these quantities are controllable. For load buses, one generally specifies the real power P and reactive power Q injections. At a bus called a swing bus (slack bus), V and e are specified. This bus is defined to account for losses in the transmission system which are not known before the load flow solution is obtained. The objective of load flow is to determine the two quantities at each bus that are not specified. For preliminary evaluation of planning and operating conditions, DC load flows [4] are used without reference to voltage conditions. With this network representation (power-angle relationship), it has become possible to carry out the thousands of load flows that are required for con- tingency analysis (security assessment) on large scale sys- tems. This method has been able to assess many overload- related system problems. DC load flow is used for the computation of outage distribution factors (ODF) and transfer distribution fac— tors (TDF). These distribution factors have been used to rapidly compute the change in real line flow with the change in bus power injection and the change in line flows for line outages. 9 1.1.2. Outage Distribution Factors (ODF) When a transmission line is opened, the power which the line was carrying is distributed to other lines according to the characteristic of the network. Thus, the ODF [5] represents the percentage of flow on line A trans- ferred on line B for loss of line A. The ODF's are used to determine contingency loading under all first and second contingency line outage conditions. 1.1.3. Transfer Distribution Factor (TDF) The transfer distribution factor [5] represents the percentage of the change in generation between two areas appearing on any specific line. That is, the transfer fac- tor is the ratio of the increase in loading on a facility (line, transformer) divided by the increase in power trans— fer (between two areas) that caused that increased loading. The TDF's can be used to calculate the power transfer between A and B that will increase the loading on a given line to its thermal limit. The TDF's can be applied in conjunction with ODF's to determine the line outage contin- gencies and transfers that would cause overloads on the system or a particular line. 1.1.4. Decoupled Load Flow To achieve a more accurate result than linearized load flow and also to be able to study voltage and reactive con- ditions, decoupled load flows [6] were developed. This 1O technique can rapidly calculate AC load flow quantites for specified contingencies. This method takes advantage of the weak coupling between (P-6) and (Q,V) components and solves separately the (P-6) and (Q,V) equations. The AC load flow or decoupled load flow outage dis- tribution factors and transfer distribution factors are used to simulate transmission limitations for a planned power transfer, selected outages, and generation participa- tion of different regions involved. 1.1.5. Transient Stability and Midterm Stability Programs In this method, the network solution is obtained in time step by time step computation of a steady state load flow solution and integration of machine differential equations. Transient stabiilty programs [7] generally utilize simplified representations of machines. This program is valid only for the study of first power swing after the dis- turbance. The Midterm Stability Program allows longer simu- lation intervals and detailed representation of synchronous machines, excitation systems, governor controls, and system loads. The Midterm Stability Program [8] can accurately simulate both the inertial and governor transient response to loss of generation or load contingencies since the gen- erator models include proper governor control models, and the program can simulate over intervals above a few seconds. The program cannot simulate governor response for large 11 systems because the number of generators that can have governor controls is very limited, and the cost of simulat- ing even small system models above twenty seconds becomes large. Both of these difficulties with the Midterm Stability Program in simulating inertial and especially governor response to loss of generation or load contingen- cies may be overcome in the future as this program under- goes further development. At present, it is not suitable for simulating inertial and governor response on the large data bases associated with the Northeast network involved in the oil displacement transfers. 1.1.6. Inertial Load Flow The inertial load flow determines a snapshot picture of the transient response to a loss of generation or load contingency at the time instant the effects of the contin- gency have propagated to every part of the interconnection, and the rate of change of frequency can be assumed identi- cal and constant everywhere. The load/generation mismatch at this time frame has not been compensated by any governor control action. The mismatch is distributed at this time frame to each generator by a participation factor that is the ratio of that generator's inertia over the inertia of all generators in the interconnection. The load flow solved for this inertial distribution of the mismatch is the inertial load flow. 12 Although inertial load flow programs exist, they are used only by a few utilities to check possible stability and security problems associated with inertial response to loss of generation or load contingencies. These inertial load flows are not utilized to determine security and stability problems for all first and second contingency loss of generation and line outages as done with DC load flow ODF and TDF methods. The utilities that utilize iner- tial load flows only utilize them as a last step check in the transmission planning process to determine if a spe- cific boundary would be vulnerable to inertial response induced stability or security problems for one or possibly two contingencies of concern. The inertial load flow simulations that have been per— formed may not accurately reflect true system responses if the large Multiregional Modeling Group (MMG) data bases were not utilized to produce dynamic equivalents that pre- serve total system inertia and if the inertia in these MMG data bases is not reduced from maximum generation (summer peak) conditions to actual system generation levels for some base case condition. The only known effort to utilize MMG data bases and reduce system inertia to apprOpriate levels was a post-mortem study of a 2,000 MW loss of gener- ation at the Nanticocke station in the Ontario hydro system [9]. 13 1.1.7. Long-Term Stability Programs A long-term stability program [10, 11, 12], which includes network solutions [10]. can simulate the network determined stability and security problems associated with inertial, governor, automatic generation control and economic dispatch distribution of mismatch for loss of gen- eration contingencies. Although these programs may be quite powerful, they have not achieved wide usage or indus- try acceptance. It is not known whether the latest version of this program can be applied to very large data bases and what the relative computational costs for the program are. It should be noted that inertial and governor response simulations for emergency conditions [13, 14], where tie lines have been lost and large frequency deviations are pos- sible, have recently been developed. These efforts are experimental, have not been utilized by any utility, and are not intended for use when tie lines are present and simula- tion is required for very large interconnections. 1.2. Review of Present Planning Procedure For a known operating condition, the planned power transfer is simulated between regions and subregions. This is possible when specific generation dispatches are used. Linear load flows are run for potentially limiting trans- mission facilities. Transfer distribution factors and outage distribution factors can be used to determine over- loads for any line outages and region-to-region transfers. 14 A simplified load flow program utilizing these factors sim- ulates load flow conditions for all first and second outage contingencies and for all anticipated area transfers. This program also computes the amount of power which can be transferred between two locations in the network under nor- mal conditions and for single and multiple outages. The AC load flows are run to verify the transfer limits and to determine if there are any potential voltage problems. The AC load flows are studied because the use of distribution factors assumes that the network is linear and voltage remains constant. Selective transient stability cases are run and capability of the power system to absorb the ini- initial power swing for different power transfers is checked. It is important to note that security and stability problems associated with inertial or governor response to loss of generation or load contingencies are not generally assessed by present planning procedures and methods. In these methods, stability is inferred if a comprehensive set of load flows with imposed transfers and all first and sec- ond contingency line outages do not exceed transfer limits set by overload considerations, if the network is capable of absorbing the first power swing for a selected set of fault contingencies, and if no voltage problems occur for another selected set of line outage contingencies. Inertial load flows are sometimes run to check whether particular boundaries or lines are vulnerable for a spe- cific contingency, but the results will not be accurate 15 unless the dynamic equivalent preserves total system inertia from a MMG data base and unless the system inertia from this data base is reduced to reflect actual operating conditions. Governor load flow could be run using inertial load flow programs if the participation factors utilized to dis- tribute the power mismatch for loss of generation or load contingencies are the ratio of frequency response charac- teristic of a particular generation over the frequency response characteristic of all generation in the entire interconnection. No utility has utilized the inertial load flow in this manner because the concept of a governor load flow does not appear in the literature and because many planners [1] assume governor response will be similar or identical to inertial response,which is not necessarily true. Lack of appropriate simulation techniques for inertial and governor transient response to loss of generation and load contingencies, lack of experience for empiric predic- tion of boundary and line vulnerabilities to inertial and governor response to such contingencies (due to lack of simulation techniques and the general lack of a need to investigate such vulnerabilities), and the large multiple interregional transfers with associated additions of long transfer lines to accommodate the transfers have together created the need for security assessment methods for iner- tial and governor response to loss of generation 16 contingencies. This need is even more apparent, consider- ing the large multiple transfers being contracted for oil displacement in the Northeast, the documented stability and security problems caused by inertial and governor response in the Northwest, and the additional large interregional transfer being contemplated in other regions of the country. Thus, there is a need to develop a contingency assessment methodology for identifying weak boundaries between generating groups in these large transmission net- works, the vulnerable transmission elements that make such boundaries vulnerable, and the loss of generation contin- gencies that cause stability and transfer limit violation problems across these vulnerabilities. This methodology will be consistent with present contingency assessment methods that utilize DC load flow and ODF and TDF pro- grams. 1.3. Present Status of Security Measures Security measures (security indices) are considered as part of operation and security assessment techniques. The indices are a starting point for more detailed analysis of limit violations (bus voltages, line and transformer thermal limitation, angle separation between generators). The purpose of developing security measures (indices) has been to express security of a system in terms of a set of numbers. These indices are supposed to measure different 17 abnormalities in the system. The security measures devel- oped have been limited for measuring overall relative security of the system for different operating conditions with different contingencies imposed. At present, there are a variety of security measures for line transformer outages and fault conditions. They are differentiated between steady state and transient meas- ures. In the steady state case, indices are defined for bus voltages and line and transformer MVA flow. The method used to generate these steady state secu- rity measures is the load flow programs described in sec- tion 1.1. For a given operating condition represented by a solved base case load flow, the line and transformer out- ages are simulated using AC load flow, or decoupled AC load flow for bus voltage index and line/transformer MVA flow index calculations. DC or any linearized load flow is also used for line/transformer MVA calculations. For bus voltage index calculations, the voltage magni- tude at each of a specified list of buses in the system is compared with the normal operating limits of the bus (V Vmax); then an index is generated. If the voltage is min’ between the limits (V V ), the index is 1. Otherwise, min’ max it assumes monotonically decreasing values from 1.0 to 0, based on linear, cubic, or quadratic functions. An overall index is calculated with consideration of a weighting fac- tor for each bus [15]. The line transformer MVA flow index is calculated the same way as the voltage index, except in 18 this case, the MVA flow is compared with the rating of line and then this index is calculated. In general form, the indices can be written as [15]: N N K i=1 i=1 K where: i ranges over all relevant components wiK is weighting factor f(.) is a function between 0.0 and 1.0 and the overall index for all contingencies is: 1fK/N H II II NZ K Hence, I is also between 0.0 and 1.0. A probability of occurrance aK can be assigned to each contingency; then the index T may be written as: These indices are useful in comparing the overall steady state security levels of different operating states, but they do not identify the specific potential boundary and lines which cause the problems. Transient security indices are also defined in a man- ner such that the index is zero if the system would lose synchronism and is 1 if the system remains stable [15]. (a) (b) 19 These measures are proposed as [16]: Maximum angle separation index 6K where: 6 = Max Max 9.. (t) K t i,j in where ein(t) is the angular separation between gener- ation bus i and j at time t for fault K, and where the value eMax varies for small or large systems. A con- tingency swing factor is defined as: UK = f(BK) where: 0K: 0 if GKZBMax 0K = 1 if 6K ~ 0 and for all faults, the overall transient index is defined as: K I = K/ 2 (1/ ) K 0" 1 Apparent impedance index. This index is calculated from the fact that loss of synchronism in a power sys- tem can be a separation of the network by circuit breaker operation when a line protection relay sees an out-of—step condition as an apparent line fault and the circuit breaker acts. Thus, to measure if loss of 20 synchronism would occur, it should be checked if any relays would trip. The apparent impedance seen by a relay at the ith terminal of line i,j is given by: 1 i z = _ lJ llJ Vi -Vj 13 or: V. 2*. = 1 13 Vi - VJ where 213 is line impedance, and Vi’vj are terminal voltages. * _ . Zij - Zij/Zij is normalized. Loss of synchronism is assumed if the normalized apparent impedance locus for any transmis- sion line crosses the line segment [0, 1]. A contingency swing factor is also defined for this case as: OK=g(dK) where: OK = 0 if dK = 0 oK=1ide-°° 21 where dK is the minimum distance to the line segment [0,1] from the normalized impedance locus. The transient security indices described in this sec- tion are valuable tools for operation and security assess- ment for fault studies, but they would not be able to assess the security and stability problems associated with inertial and governor load flow distributions. To develop a contingency assessment methodology, a set of security measures are defined in Chapter 4 of this thesis for inertial and governor load flow distributions. These security measures capture dynamic system structure that causes vulnerable boundaries between strongly bound genera- tor groups. The security measures developed in this research are related to but not identical to r.m.s. coher- ency measures developed in [ML 22] for producing dynamic equivalents for transient stability studies. It has been shown [17] that dynamic equivalents pro- duced by the r.m.s. coherency measure preserve both the eigenvalues and coherent properties of the unreduced system and that the r.m.s. coherency measure can be evaluated based on a probabilistic disturbance, which is computed by evaluating the coherency measure for a set of determinis- tic disturbances and summing the coherency measure for each disturbance. This coherency measure, evaluated for the probabilistic disturbance, has been shown to [17, 18]: 22 (1) produce dynamic equivalents where the eigenvalues with large imaginary values, which are shown to represent intermachine oscillation within the coherent groups, are eliminated; and, (2) detect strongly bound coherent groups that are charac- terized by a tree of "1'1 stiff interconnections between ni generators in each group i. These strongly bound groups were shown [18] to be one of five loss of controllability and Observability conditions that could be detected by the r.m.s. coherency measure for different types of disturbances. The concept of strongly bound groups (detected by the probabilistic disturbance) would indicate the boundaries of groups are composed of weak interconnections compared to the inertias of the generators in the groups they connect. It is shown in Chapter 4 that the security measures evaluated for a probabilistic loss of generation distur- bance (or alternately summed for an appropriate set of deterministic disturbances) has the coherency measure as a term and can detect the boundaries between weakly connected groups of generators. A ranking table of the security measures between generator pairs from smallest to largest is produced. Groups of generators are formed based on a commutative rule similar to that used for forming coherent groups for producing dynamic equivalents. As one proceeds down this ranking table, individual generators are included in groups and later groups are merged to form larger groups. 23 As groups are merged, the boundaries between groups should be continuously weaker. When this grouping was carried out on the 49 bus test system, the last group to be lumped into a single system group containing all generators was the group of machines in the external system. This boundary containing lines between the internal and external system was inferred to be the weakest set of lines and the most vulnerable for loss of generation contingencies. This proved to be true, since the loss of generation contingen- cies run only caused the loss of synchronism across this boundary or set of lines. It is hypothesized that the second to last group of generators to be aggregated would indicate the second weak- est and thus the second most vulnerable boundary, etc. Thus, a ranking of boundaries and hence the associated lines that connect strongly bound groups of generators was formed. Such a ranking and identification of weak bound- aries for loss of generation contingencies is extremely helpful in both security assessment and transmission plan- ning for identifying the critical element that bottles up generation, limits transfers, and causes security and stability problems for loss of generation or load mis- match distribution by inertial or governor load flows. The security measures developed for loss of generation contingencies have a similar form to those developed in [19] for security assessment applications on line outage contingencies. The security measures developed in this 24 research are much more flexible in that: (1) They can summarize the effects of a single contingency as in [19, 20] or any set of contingencies. (2) They can be written for a single transmission element, a set of vulnerable elements, or over-all system ele- ments as in [19]. This flexibility will greatly extend their usefulness in both transmission planning and security assessment applica- tions. 1.4. Summary The inertial and governor load flow are defined and discussed in Chapter 2. The DC inertial and governor load flow is shown to accurately capture the inertial and gover- nor generator transient response to loss of generation or load contingencies by comparing the angles determined by the inertial and governor load flows for specific contin- gencies with midterm transient stability simulations of these same contingencies. The results also indicate that the inertial and governor response to a contingency are quite different and that each cause different stability problems that cannot be detected based on a first-swing stability simulation, load flow simulation, or line outage study. The security measures for the security assessment pro- cedure are defined in Chapter 4. The controllability and Observability properties of the classical transient 25 stability model that cause coherent behavior and that lead to the boundary and line vulnerabilities for loss of gener- ation contingencies are discussed. The security measures, defined in Chapter 4, are then shown to detect these vul- nerabilities. Computational results on the 49 bus system using a security assessment procedure focused on the security meas- ures and their theoretical capability to detect the con- trollability property that causes vulnerability to loss of generation contingencies is presented in Chapter 5. These results show that boundaries between the generator groups that lose synchronism based on inertial or governor response can be determined. It is also shown that a par- ticular line in the boundary between the groups causes this vulnerability since the power flows from the entire inter- connection back to the point of mismatch focus on this one line that causes the instability. Conclusions are presented in Chapter 6. CHAPTER 2 LOAD FLOW METHODS FOR SIMULATING GENERATION RESPONSE TO LOSS OF GENERATION FOR TRANSMISSION PLANNING AND SECURITY ASSESSMENT Dynamic generation response to loss of generation or load contingencies causes different power flows for syn- chronizing coefficients, inertial and governor controls, and the automatic generation control/operator distribution of power mismatch. These distributions will stress spe- cific lines and boundaries in the transmission grid that may lead to stability and security problems. The fundamental hypothesis of this chapter is that each of the power distributions at different time frames is complete before the next distribution of power begins. This hypothesis will allow each of the distributions to be modeled as load flows. One can verify the accuracy and validity of these load flow models for each time frame by simulating the actual state of the system trajectories at different time frames using the EPRI Midterm Stability Program [8], which is the only package available at this time that can be accurate out to the governor response time frame. The objective of this chapter is to develop load flow models for each of the mismatch power distributions. The 26 27 load flow problem is stated given the real power injec- tions for all generator losses before the loss of genera- tion or load contingency occur and the magnitude of the mismatch caused by the loss of generation or load contin- gency. The real power generations are calculated for the inertial generation response, governor response, and the automatic generation control/operator action generation response to a loss of generation contingency. The load flows are then solved given the calculated real power injection at generation buses for a specific distribution of the loss of generation or load mismatch. The bus voltage angles and the real and reactive power flows for each distribution will be the result. This chapter is composed of two sections. The model for distribution of loss of generation or load mismatch based on synchronizing power coefficients, inertial response, governor response, and automatic generation con- trol (AGC)/operator action will be developed and discussed in section 2.1. The inertial, governor, and AGC/operator load flows for each of these three distributions of loss of generation or load mismatch will then be defined and dis- cussed in section 2.2. The comparison of the inertial and governor load flows with the simulation results provided by the EPRI Midterm Stability Program will be given in Chapter 3. 28 2.1. Generation Response to Loss of Generation or [oad Contingencies. Following a loss of generation or load contingency, the power imbalance (PTOT) will first be distributed to the generators according to the synchronizing coefficient or electrical closeness to the disturbed bus. After a short period HL5 to 2 seconds), when the acceleration at every generator throughout the system becomes equal, the loss of generation mismatch will be redistributed to generators throughout the interconnection in proportion to the inertia of each generator as a percent of total inertia. Next, assuming the system is still stable after the inertia dis- tribution, the loss of generation or load power mismatch will be distributed according to governor control based on the roughly uniform frequency deviation throughout the interconnection. The power distribution at this point would be according to the governor frequency response character- istic of each generator. This third redistribution by gov- ernor action begins at approximately six seconds when the inertial distribution is complete. The governor distribu- tion requires twenty seconds to several minutes to complete depending on the magnitude of the mismatch and the response rate capability of all the generation in the interconnec- tion. Finally, if the system remains stable after governor action redistributed the power mismatch, the automatic gen- eration control or the operator of the utility experiencing the loss of generation will distribute this mismatch power in an attempt to reset frequency and interchange to 29 schedule. This action starts at 1 minute and can be com- plete in 10 minutes to several hours, depending on the size of the mismatch and the reserve available. 2.1.1. Power Mismatch Distribution According to Synchro- nizing Power Coefficients. Suppose that a loss of generation or load (PTOT) occurs at an arbitrary point in the network (bus k). This disturbance will cause a voltage angle change aek at bus k, and by this means, the power mismatch will be transferred to the various generators. Now if the synchronizing power coefficient of the equivalent lines connecting the dis- turbed bus k to various generators i are Psik’ the individ- ual power change for each generator will be: AP- (1) = P A8 1k Sik k and: P = z aP. (1) = A9 zP TOT 1 1k ki Sik 1.e. P (1) - PS” P (2-1) ik ' EPETT TOT i ik where: APik(1) = power changes for generator 1 for initial distri- butions of generator mismatch at bus k. P = 151vk Sik Xik COS elk 30 E. = voltage magnitude at internal generator bus i Vk = voltage magnitude at bus k eik = angle across equivalent lines connecting bus i to bus k. Xik = reactance of equivalent lines connecting bus 1 to bus k. The loss of load or generation power mismatch is immediately shared by the generators according to their synchronizing power coefficient with respect to the dis- turbed bus. The generators electrically close to the point of disturbance will pick up the greater share of the mis- match power regardless of their size. 2.1.2. Inertial Distribution of Mismatch (P101) After initial response Apik(1)’ every generator will be accelerated (retarded). This acceleration is: Apik ai = -—M—i—— (2‘2) The synchronizing forces pull all the generators toward a mean acceleration (retardation). At this point, the dis- turbance has propagated from the disturbed bus, and the acceleration at every generator throughout the system is approximately equal to the system mean acceleration (decel- eration). 31 where M1 is inertia constant of generator i. The propagation is like the ripple caused when a rock is thrown into a calm pond. The propagation first affects generation directly connected to disturbed bus k as indi- cated above. As these generators accelerate, they affect generators connected to them that in turn accelerate, affecting generators even further electrically from the dis- turbance. This propagation is already under way at .025 seconds and is complete at .5 seconds to several seconds depending on the size of the network. The synchronizing power flows in the entire intercon- nection will at this time keep the acceleration of each gen- erator close to the system mean acceleration. A transient can accompany this redistribution of mismatch power from generators directly connected to the disturbed bus to all generators in the interconnection by inertial distribution if the mismatch is very large. This transient is observed in synchronizing oscillation of bus angles around the values dictated by the inertial distribution of mismatch power. The mean acceleration at all generators distributes the mis- match PTOT at bus k to each generator in the interconnection by accelerating or retarding its inertia. The power drawn from (or sent to) generator i for a loss of generation (load) contingency for the inertial distribution of power mismatch at bus k is: 32 The power drawn from or sent to the generators in the interconnection flows over the network. These flows can cause security or stability problems if the flows exceed steady state stability or overload limits on the lines that carry these flows. 2.1.3. Governor Distribution of Mismatch (P101) After inertial distribution of mismatch power, each generator is controlled by its governor. A loss of genera- tion Head)contingency causes frequency deviation. Governor frequency regulation on each generator in the interconnec- tion work together to arrest the change in frequency. This frequency regulation begins approximately six seconds after the disturbance occurs. When the governor frequency regu- lation is complete, frequency is constant at a deviation Awo above nominal system frequency throughout the intercon- nection. The deviation of power at each generator from the basic analysis of governor response from Figure 2-1 is: CAPi Pik(3) = T “’1 Ami = Awo for all the generators then: CAPi PIk(3) = T A100 = BI Aldo . (2-5) l Am CAPi 4‘2 Tajyr— PGi ' ’V ‘———— R PGi min Figure 2-1. Linear Relation of Generation Change to Frequency Deviation by Governor. where: CAPi is megawatt capacity of generator i R is system regulation coefficient Bi is frequency response characteristic of generator 1 The total mismatch PTOT is the sum of all the power changes. Since the Aw. of all the generators are equal to Awo, one 1 obtains: i i 1 1 p TOT Awo - 281 p.u. Hz (2-6) 1 and from (2-5) and (2-6): BiPTOT ‘T e. i l APik(3) = 34 B1 is approximately equal to MW capacity of generator (CAPi) divided by system regulation coefficient (R). The other factors involved in governor response are generator and turbine damping and load dependence on frequency, but the dominant component is CAPi/R. This model of governor action that linearly relates generation change to frequency deviation is idealized. The governor deadband causes a generator to be insensitive to frequency deviation less than .036 Hz; if the frequency deviation sensed by a generator is greater than .036 Hz, the generator may still not respond to governor command if the unit is operating at a valve set point. The effects of governor deadband and valve set points have the effect of reducing the frequency response charac- teristics Bi of generators electrically distant from the point of mismatch (bus k). The effects of the loss of gen- eration on both the frequency and acceleration of genera- tors electrically close to bus k are larger than at generators distant to bus k, as the effects of disturbance ripple out and ultimately achieve an inertial distribution. The larger initial frequency deviations close to disturbed bus k will overcome governor deadband and valve set point nonlinearities, making the generator's actual change in generation at least BiAwo. The generators far from the disturbance do not feel the large initial frequency devia- tions but only the ripple of that disturbance, and thus the governor deadband and valve set point nonlinearities may 35 have the effect of reducing Bi for these generators, even though the same steady state frequency deviation is ulti- mately experienced. PJM [21] increased (R in per unit) in the external system from 5% to 16% to compensate for a lower measured frequency response for the external system than predicted when R (per unit) is 5% in both the external and internal system. The measured 8 for the external system were deter- mined from loss of generation tests in the PJM utility. The reduction of Bi for external system generation from those predicted by [21], when R is the same for internal and external systems, will not only increase the steady state frequency deviation after governor action is complete but will also reduce the percentage of the mismatch taken by the external system. The power flows caused by the governor mismatch dis- tribution will be different from the inertial distribution and cause quite different power flows back to the point of mismatch. The governor power distribution (Bi/zsi) is dif- ferent from the inertia distribution (Mi/2M1) since: (1) M1 is not proportional to CAPi for generators of differ- ent size and type. (2) The effective Bi is reduced on generation far from the disturbed bus k due to governor deadband and valve set pohfi;nonlinearities. 36 (3) Some types of units may not have governor regulations or sharply reduced regulation participation as on nuclear units. (4) Some utilities' automatic generation control dispatch is proportional to area control error as well as the integral of area control error. When AGC has propor- tional control, the effective Bi on generators under AGC is increased. If the governor distribution of power (Bi/821) mis- match is different than for inertial distribution, the power flows from the external system are channeled to the point of mismatch over different transmission lines. If these line flows exceed steady state stability or overload transfer limits, then stability or security problems result. The governor action is complete in 20 seconds to sev- eral minutes depending on the magnitude of the lost genera- tion or load and the response rate capability of all the generations in the internal and external system under governor regulation. 2.1.4. AGC/Operator Distribution of Mismatch P101 The final distribution of power imbalance is by the automatic generation control according to generator partic- ipation factors or operator action of the utility experi- encing the lost generation. The operator or the AGC distributes the mismatch power among generators under control 37 by means of participation factors 7i to reset the frequency. The redistribution in the utility for generator i is: (2-8) 2.2. Formulation of Inertial, Governor, AGC/Operator [bad Flow Problems. In section 2.1, the distribution of power mismatch over time due to loss of generation contingencies was dis- cussed. Synchronizing power coefficient distribution that happens at the instant of the disturbance causes signifi- cant rate of frequency change and power change. The study of this distribution is similar to transient stability studies for fault disturbances because a severe three-phase fault at the generator terminal causes an acceleration sim- ilar to loss of generation. The fault is more severe due to voltage changes that effectively weaken the transmission network connected to that generator. The transient stabil- ity studies for faults are widely done by the utilities, and the problems that may arise because of synchronizing power coefficient distribution would be less severe and are considered in these studies. Thus, there is no need to develop a load flow for this synchronizing distribution of mismatch. 38 The distribution of power mismatch based on Amnbperator participation factors for the utility experiencing the dis- turbance and the stability and security problems associated with this distribution are investigated by present load flow studies on all types of contingencies. These studies, dis- cussed in Chapter 1, establish the overload and transfer limit violation conditions for the utility of concern. Since the AGC/operator load flow is in common use as options in present load flow programs, it will be developed for com- pleteness but will not be investigated further. The inertial and governor distribution of loss of generation power mismatch can cause large power transfer to the region affected by the loss of generation. The power flows associated with these two distributions of mismatch power are quite different and place unique stress on the transmission grid that can lead to stability and security problems. Present planning techniques overlook the stability problems that may occur due to these generation responses to loss of generation or load contingencies, and the methods for assessing the stability problem are indirect, as indicated in Chapter 1. The objective of this section is to formulate inertial, governor, and AGC/operator load flow methods based on the description of section 2.1. This will permit direct assessment of the stability and security problems due to loss of generation or load contingencies. The load flow 39 methods will also be used for calculating inertial and governor security measures that will be defined in Chapter 4. 2.2.1. Load Flow Formulation Load flow solutions provide bus vultage angles and power flows in the transmission system for specified gener- ation and load conditions. To study the affect of generation response to loss of generation or load contingency for inertial, governor, and AGC/operator distribution, the real electrical power to be delivered into the network for each distribution must be specified, and load flow equations must then be solved for unknown, wanted variables. The load flow equations can be written as: PG.(K) - P0. = 2 P.. 1 1 jeAi 13 06. - QD. = 2 Q“ 1 1 jEAi 13 (2-9) __ 2 ' ' .. - Pij - ViYijSIn “ij + Viijij s1n (ai aj aij) _ 2 _, _ - - Qij .. vi(Yij cos aij % Bij) viVjYij cos (61 6j aij) where: A = set of buses connected to bus 1 PDi+JQDi complex power load at bus 1 40 a- = phase angle of voltage Vi PGi(K)+jQGi = complex power from generation at bus i = (90° - the impedance angle of line ij) Gij Bij = the total charging susceptance of the line i QGiMin 5 QGi 5 QGiMax for regulated buses 06. = reactive power generated by generator 1 1 These AC load flow equations can be solved given PDi + jQDi at load buses, lVil and 51 at the swing bus, and PGi(K) and IVil at generator buses. The variables calculated by solving the load flow equations are lvil and 6i at load buses, losses at swing bus, and QGi and 51 at generator buses. The load flow is calculated under the constraint: QG 5 QG. < QG 1mm 1 ‘ 1M“ at generator buses where voltage is held to specified levels. The inertial, governor, and AGC/operator load flows differ based on the generator injection PGi(K) for inertial (K = 1), governor (K = 2), and AGC/operator distributions described in section 2.1. Before deriving the formulas for PGi(K) for inertial, governor, and AGC/operator load flows, the equations similar to (2-9) for DC inertial, governor, and AGC/operator load flows are given. The DC load flow considers only the real power equa- tion and ignores the reactive power equations since reac- tive power and voltage are assumed to remain unchanged from 41 base case values for any contingency. Resistance is neglected in almost all cases due to the assumption for large X/R ratios. Thus, equation (2-9) becomes: PG.(K) - PD. = Z P-. 1 1 jeA. 13 l Pij = IViI lel Yij $10 (61 - oj - aij) and neglecting the line resistance: which leads to: Pij = |vi| |vj| bij siri (ai - aj) For linearization, assume 51 - aj is small so that: b-- (6- - a.) PGi(K) - P01: 2 ”1| |vj| U 1 J JEAi and define: Pi(K) = PGi(K) - PDi p.(K) = a. 2 H.) v. b.. - 2 v. v. 1).. 5. l ljeAi 1 'Jl lJ jEA l 1' ' Jl 1] J i or in matrix form: P(K) :99 (2-10) 42 where P(K) is the vector of real power injections at every bus for inertial (K=1), governor (K=2), and AGC/operator load flow (K=3), and g is the vector of voltage angles at every bus. The matrix Q is defined as: ll H- zlv,IIvJ-Ib,j 1 {J}-. = -'lvillvjlbij i 7 I For the base case, it is clear that: P°=gs° . 6°=1 For any loss of generation or load disturbance that produces injections A£(k), based on inertial, governor, or AGC/operator distribution, the change in angle A6 satisfies: 1) O + D 1, A K V II 2 (2° + A2) AP(K) = 1 A6 1 D 0" II I AB“) The DC load flow is of course less accurate than the AC load flow. The DC inertial, governor, and AGC/operator load flows may be accurate enough to determine vulnerabil- ities of elements to overload or stability problems. Even if this DC load flow may not be extremely accurate, it has the significant advantage of being able to be computed for all first and second loss of generation and line outage contingencies at reasonable cost and computer time. 43 The formulas for calculating the injections PGi(K) for all generators for inertial (K=1), governor (K=2), and AGC/operator (K=3) distributions of loss of generation or load mismatch are now presented. The value of {PGi(1)}?=1 for the inertial load flow assumes that the acceleration of every generator is con- stant dwi/dt = dwO/dt, and thus: dmo PGi(1) = PMi - Mi TH? (2-11a) where: N “”0 i§1(PMi - PGi(0)) = ‘ (2-11b) at N Z M. i=1 1 where: PGi(0) = real electrical power delivered to the network before loss of generation by generator 1 PMi = PGi(0) - APMi = mechanical power sent to the gen- erator after loss of generation APMi 1 r J PGi(1) = real electrical power delivered into the network PGi(0) 1 = j APMi = 0 . and j is the lost generator by generator 1 after loss of generation for inertial distribution of mismatch power The value {PGi(2)}?=1 for the governor load flow assumes that the governor action is complete and that fre- quency is constant everywhere m. 1 = m0 SO that: 44 PGi(2) = PMi - 81 mo (2-12a) _111 “’0' N 2 8- 1:11 PGi(2) = the real electrical power delivered into the network by generator 1 when governor frequency regulation is complete 81 = frequency response characteristic of generator 1 For the final distribution of power by the AGC/operator PGi(K) for (K=3) satisfies: PGi(3) = PMi - 71 PTOT (2-13a) where: in = 1 PTOT = f (PMi - PGi(O)) (2-13b) PGi(3) = real power delivered to the network by generator 1 after operator or AGC action is complete 1 = AGC/operator participation factor The load flow equations (2-9) or (2-10) are solved for bus voltage angles and power flow across the lines, given the values of PGi(K) from (2-11), (2-12), and (2-13) for the inertial, governor, and AGC/operator load flows, respec- tively. 45 In this set of load flow techniques, the mismatch power is appropriately distributed among generators based on inertial, governor, and AGC/operator generation responses. The AGC/operator distribution is local within the utility experiencing the mismatch and thus places no stress on the transmission external to the utility. The inertial and governor distributions are over aU.generators in the interconnection back to the utility experiencing the mismatch. Severe stability and transfer limit violations may arise out of the power flows from inertia and governor generation reponse throughout the interconnection. These problems are even more severe when long distance power transfers are planned and when long transmission lines are utilized to carry these transfers, as discussed in Chapter 1. The long distance power transfers, as experienced in the BPA [1] system or planned from Canada and the Midwest to replace oil generation in the Northeast, utilize certain transmission corridors and bring loading much closer to their thermal overload limits. A loss of generation con- tingency in the area receiving the transfer causes large inadvertant transfers over the same corridors providing the large planned transfers. These inadvertant transfers are due to inertial or governor distribution of the mismatch from the loss of generation contingency and can be concen- trated on just one corridor. The planned transfers can be distributed among several corridors by establishing the 46 proper operating practice for each of the utilities in the interconnection where the transfers are planned. The inad- vertant transfers cannot be distributed among the various corridor options but are dictated solely by the inertial or governor responses of generation in the interconnection and thus can be concentrated in just one corridor. The combin- ation of the large planned transfers and the uncontrollable inadvertant transfers will be shown via simulation in the next chapter to cause thermal overload limit violations and steady state stability problems, even when the capacity of all possible corridors far exceeds the combined transfer requirements. Thermal overload can cause loss of equipment life and sagging of lines that can lead to fauH3 and other contin- gencies that ultimately can cause cascading outages and islanding. Stability problems may not be as likely or as severe a problem for systems with short lines carrying these long distance transfers because thermal overload limits, and thus normal operations, are far from the sta- bility limits. The results in the next section show that even in this case stability problems can occur. The use of long lines to carry these long distance transfers, such as for the BPA system and contemplated for some transfers to the Northeast from Canada, will possibly cause more severe stability problems because thermal overload limits on long lines do not restrict normal operation loads on these lines to be far below stability limits, and thus such lines 47 are heavily loaded. Therefore, the inadvertant transfers by inertial or governor distribution can more easily cause stability problems on long lines. Almost every major blackout and islanding problem can be associated with the interconnection wide power flows. An example of this type has been experienced in [1]. This problem is in great part associated with inertia or gov- ernor generation response. The load flow methods developed in this chapter allow direct assessment of stability and security problems due to generation response to loss of generation contingencies. The development of load flow models leads to a better understanding of the different power transfers associated with inertial and governor load flow time scales. The effects of these power flows on transmission gridswere also an objective. Lack of understanding of different power mismatch dis- tributions in present planning methods, in some cases, caused improper use of inertia load flow that was developed after the 1965 blackout when records of the power flow experienced did not agree with existing load flow analysis. The present planning methods have been considering the power flows due to generation response to loss of genera- tion almost equal for inertial and governor distribution of mismatch power. Conventional load flow is not able to address the stability and security problems associated with inertial load flow and governor load flow. The 48 present load flow techniques generally distribute the mis- match power to a large swing generator. This is not what will happen in a real power system, and the accuracy of this approximation to inertial or governor distribution will depend heavily on the representation of the system and the choice of the swing bus. The stability problems associated with generation response to loss of generation are related to system struc- tural weakness and generation response of synchronizing generators throughout the interconnection. The vulnerabil- ity of the lines and stability problems associated with them cannot be assessed by line outages or any fault studies. The recently developed Midterm Stability Package [8] could be used to assess the stability problems associated with dynamic generation response of generators, but this package cannot handle very large data bases required to analyze large interconnected networks associated with long distance power transfers where all the generators must have governor turbine representation. Even if it is upgraded, it would be very computationally expensive to run for the simulation time interval needed (> 20 seconds) to analyze governor action distribution. For assessing the stability problems associated with generator response to loss of generation or load contingen- cies, the load flow methods that were introduced in this chapter are superior and inexpensive for very large data 49 bases, compared with the solution time of several hundreds to thousands of differential equations that have to be solved in the Midterm Stability Package, especially for governor action response. The accuracy and validity of the inertial and governor load flow is shown in Chapter 3 by comparing these load flow results for specific contingencies with the Midterm Stability simulations of the same contingency. The results show that the DC inertial and DC governor load flows are reasonably accurate; that the inertial and governor load flows are quite different for the same contingency; that a separate stability problem can exist for each of these dis- tributions; and finally, that conventional load flow and line outage studies would not have detected these inertial and governor load flow stability problems. CHAPTER 3 APPLICATION OF THE INERTIAL AND GOVERNOR LOAD FLOW FOR ASSESSMENT OF STABILITY AND SECURITY ON A 49 BUS TEST SYSTEM To demonstrate the performance of the inertial and the governor load flow methods developed in Chapter 2, loss of generation contingencies were simulated using the inertial and governor load flow programs on the 49 bus Electric Power Research Institute (EPRI) test system. A schematic of this system is shown in Figure 3.1. The 49 bus test system was chosen since it was used and tested to validate the Midterm Stability Program, and the governor models for midterm stability studies were available for this system. Generator data, governor models, and base case load flow data used in this study are available through the EPRI Midterm Stability Package [8] and there- fore are not reproduced here. The inertial and the governor load flow results were compared with the Midterm Stability Program's results of the same contingencies to verify the inertial and the gov- ernor load flows. The following contingencies were simulated on the 49 bus test system to: 50 ...m mczmwu .Ewpmxm pmwh mam we m Jib JJMJ .QJ / / ewymxm Hmccwpcm z/fimccwpxm N. Empmxm 52 (a) establish accuracy of inertial and governor load flows against the midterm stability simulation results. (b) investigate possible existence of stability and secur- ity problems associated with inertial and governor power flows. (c) show the stability and security problems associated with inertial and governor load flow cannot be pre- dicted by present load flow methods and line outage studies. 3.1. 490 MW Loss of Generation in the External System The 49 bus test system was operating at the point given by the base case data with total generation of 24868.32 MW and maximum capacity of 28236 MW. At this operating point, the external system was importing power (1381 MW) through the three lines (43, 44), (40, 41), and (23, 24), with a combined steady state capacity of 3916 MW. A 490 MW loss of generation was simulated on the 49 bus system for 50 seconds using the Midterm Stability Program to see the effects of both the inertial and gover- nor response of generators to a loss of generation contin- gency on the system. The power flow on the three lines connecting the internal and external system is shown in Figure 3.2. The result shows that the internal system par- ticipates in supplying the mismatch after only 1 second through the reduction in frequency and thus loss of kinetic energy in internal system generation. The percentage of 53 oo.o_ Fl :o_pwcwcmw mo mmOg 32 omv quw< Empmxm floccmpxm new moccmpcm mew mcwpuwccoo mm:_4 do zofiu cmzoa .N.m wcsmwu mozoumm 2H mz_h ocum oeum ocus ocum ocum so“: oo.m oo.m so." oo.pu el Jill bl bl '3 3| 3 l4 6 E e a a E Nlmm ¢1m¢ :1O¢ +40 111111 P..-rar- _ _ _ 0 0 0 I 00'02 r 00'011 M013 ENIT 3-8 F 00'09 V 00'0”; SiiUMUOBW I OO'OOI ,0l* 00'021 54 the MW power transferred from the internal system via these lines (40, 41), (43, 44), (23, 24) is proportional to the ratio of the inertia of the internal system to the inertia internal) Msystem ' that the governor power flow is 24 MW larger than the iner- of the entire system (M The figure also shows tial power flow on line (40, 41) and 12.6 for line (43, 44). That is, for this disturbance, the response of internal generators by governor action is more than the internal response since a larger majority of the generation with governors is in the internal system and thus 12;::2;1 M . k2:::2;1 It is also shown that line (40, 41) picks up the larger share of the power flow, which is due to its location in the transmission system connecting the internal and external system generation. Figure 3.3 shows a plot of the frequency deviation at different buses in the system. It can be seen that the generators in the external system decelerate more rapidly since the loss of generation is in the external system. This difference in frequency sets up the changes in the angles that govern the system response during the inertial response time frame and are captured by the inertial load flow. The inertial time frame starts at approximately 1 second when the deceleration of all generators is equal. After 5 seconds, the change in frequency has been arrested, and frequency is constant. This indicates that the gover- nor regulation is complete for this loss of generation. .coflpwcmcwu mo mmOn 2: cos cmpm< mmmzm pcmcmmd_o pm :o_pm_>mo xocmacmcd mozoumm 2H 02:. .m.m ma=m_a 00. 00._0 00.5 00..0 00..m 00.3 00._m 00..N 00 00.0. 0 W 0 ‘1‘, ’1'..- II . 8f/‘3. TmU o E 0 Nu . . 3 IO “68 omw I44 . MU -bmw m. U 3 A . .0. 0 03 IA 3 .l .1...— e .|.. .1w wam [US 4 H .NN mam ”I./ + H “N mam 0% 0 u , Ne mam 4. u 2 mam 0 x u 3 25 -a 0 56 The DC inertial and governor load flows were run with 490 MW loss of generation contingency at generator 11. The deviation in generator angles due to inertial and gov- ernor response were added to generator angles at the operating point before loss of generation. The resulting angles were then compared with those of the Midterm Stability Program. The midterm stability results are illustrated by straight solid and dotted lines for inertial and governor responses respectively for some of the genera- tors in Figures 3.4 to 3.7. The lines reflect the average angles during the inertial time frame and the steady state angles after the governor response is completed. The final results are presented in Tables 3.1 and 3.2, which compare the midterm stability simulation results with those of the inertial load flow and the governor load flow respectively. Tables 3.1 and 3.2 show that: (1) The inertial and the governor angles are different and cause different power flows as they did in tie lines connecting the internal and external system. (2) The DC inertial and governor load flows can predict the inertial and the governor angles and thus capture the dynamics of the distribution of power mismatch with reasonable accuracy expected with a DC linear- ized model (15% error). The accuracy of the results is indicated by: (a) the agreement of angle differ- ences in both cases between the DC load flow and the Midterm Stability Program results, and (b) the changes 57 .couumcmcwo we mmog 3: ome cmuw< mm_mc< copccmcwo .e.m mc:m_u wozoumm zH 02—h oo.p. ocum oouo coup no“, ooum no“: ocum oouw loo._ oo.pu .23.». 53.0 o 2 Ju 0 0 .\\Lvh -mu 1 -lllnumml. . ill lllillyVhlflllll m HwflhHHIill¥ImMMHMMNMMMNHHWHNRHMHUNHU. \lJr \)( .\}/ rm :1 :1 rl Ifsllyrlxxlhqll . m .1111. . ll” M l. H 9 -mu 0 0 On cm 55 on 5.. com I +" 3P :3 1m x" E :8 .0 on me :3 0 I 00'021 OIUHBNEO $338030 NI BTBNU 8 58 .:o_uucmcmo mo mac; 2: owe cwpm< mw_u:< copocwcwu .m.m mesa“; mozouwm 20 mz_p 0N8 HDIUUENBO 00 .0." 00 ..m 00 ._0 00 ..~. 00 ..m 00 ..m 00 ..= 00 ..m 00 ..~ 00 .r_ 00 . 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Ida I YMN 00 03 9 H 3 53 Dev 0 0 v 61 00.0. 00.0_ 00.0 00 0.00 0_.0m 0.00 P0 00.00 00.00 00.00 0— 00.00 00.00 00.00 00 00.00 00.00 0.00 up 00.00 0.00 00.00 m. 00.00 0.00 _N.00 ~— 00.00 00.00 00.00 0. 00.00 00.00 00.00 0— 00.00 00.00 00.00 0, 00.00 0.00. _0.00 0 00.00, 0.0.? 0_.oo. 0 v0.00. 0.___ 00.00 v _0.00 00.00 0.00 m 00.00 00.00 00.00 F .. .. 0.0.0 000c< _o_pcmc0 000c< ~mfipcmc0 m_0:< .—— mam Lowwmeww H0 00wmewch $0 mmoq 3: 000 no» 5000000 000000000 500000: mcu 0:0 zofim 0000 ~m_pcm:0 0:0 5000 umc_mpno mw_0:< coumcmcmu mmcoamwm _m_pcw:0 may mo com_cmasou ._.0 m_nmp 62 00.0. 00.0. 00.0 00 00.00 0..00 0.00 .0 0.00 00.00 .0.00 0. 0.00 00.00 00.00 0. 00.00 00.00 0.00 0. 0.00 00.00 00.00 0. 00.00 00.00 .0.00 0. 00.00 00.00 00.00 0. 00.00 00..0 00.00 0. 0..00 00.00 00.00 0. 0.00. 0..00 .0.00 0 0..0.. 00.0.. 0..00. 0 0.0.. 00.0.. 00.00 0 00. 00.00 0.00 0 00.00. 0..00 00.00 . .0 0. 0.0.0.: m.0:< 00:00>00 0.0:< 00:00>00 000c< ... 0:0 000000000 00 0000000000 00 0000 2: 000 000 5000000 000000000 50000.: 0:0 0:0 30.0 0000 00:00>00 0:0 5000 00000000 mm.0c< 000000000 00000000 00:00>00 0:0 00 0000000500 .0.0 0.000 63 observed between simulation results in inertial and governor time frames are observed in the differences in the inertial and governor load flows. Table 3.2 shows that the governor time frame angles predicted by the DC governor load flow are larger than those obtained from the Midterm Stability Program. This overprediction of the angles is in part due to a decrease in load that is caused by the coupling between active power and voltage in the Midterm Stability Program which is omitted in the DC governor load flow. The reduction in load will reduce the mismatch that must be made up by the governor response of generators in the Midterm Stability Program, and thus reduces the angles at the internal gener- ator buses in the results of the Midterm Stability Program. A second reason why the angles at the internal genera- tor buses for the Midterm Stability Program are smaller than the DC governor load flow is that the generator with the lost generation is not dropped and thus participates in the governor response in the Midterm Stability Program but is dropped in the DC governor load flow. Thus, the remain- ing generators are required to pick up less generation in the Midterm Stability Program amihence have smaller angles. A third reason why the Midterm Stability results would have smaller angles is the reduction of transmission losses with the loss of generation. This effect is similar to the reduction of load with the loss of generation. The genera- tion changes observed in the Midterm Stability Program show 64 that the generating units picked up 42 Mw less than the total lost generation at the governor time frame due to the reasons given above which resulted in smaller internal gen- erator angles. The modification of Consumer Power Company's fast decoupled load flow to be able to compute power flows for inertial and governor response to synchronous generators has been proposed to EPRI, which would reduce the errors in both the inertial and governor load flows. 3.2. 790 Mw Loss of Generation at Bus Number 12 The second contingency simulated on the 49 bus system was loss of 790 MW generation at bus number 12 in the external system. This disturbance was again simulated by running the inertial and governor DC load flows and the Midterm Stability Program. The results given in Table 3.3 compare the inertial angles obtained from inertial load flow and the Midterm Stability Program. The overshoot was factored in for this case since the contingency was large. Table 3.3 shows that the inertial angles were predicted fairly accurately by the DC inertial load flow program (10%-15% error), with again good agreement in angle differ- ences. No angle across any line exceeded 90 degrees. That is, no loss of stability is predicted by inertial load flow. Table 3.4 shows the inertial angles across the tie lines connecting the internal and external system. The angle across the line (40, 41) is the largest angle 65 00.00 0.00 0.00 .0 00.00 0.00 00.00 0. 00.00 0.00 00.00 m. 00.00 0.00 .0.00 0. 00.00. 00.00. .0.00 0 00.00. 0.0.. .0.00 0 00.0.. 0.00. 0..00. 0 00.0.. 0.00. 00.00 0 00.00. 0.00. 00.00 . 20 00 0.0000 00. 0 00 5000 .2 up 00 “0.00.0002. 200000.000... 0.00.... .N— mam LOHwLwcmw Hm :O_#0Lm:mw $0 mmog 3: 000 000 5000000 000000000 500000: 000 000 3000 0000 00000000 000 5000 000.0000 00.00< 000000000 00000000 0000000. 000 00 0000000500 .m.m 0000. 66 Table 3.4. Inertial Angles Across the Tie Lines Connecting the Internal and External System After 790 Mw Loss of Generation. Line Inertial Angles Across Lines (41, 4D) 71.22 (44, 43) 32.21 (24, 23) 10.1 Table 3.5. Governor Angles Across the Tie Lines Connecting the Internal and External System After 790 MW Loss of Generation. Line Governor Angles Across Lines (41, 40) 148.2 (44, 43) 128.3 (24.23) 12.1 67 observed by inertial load flow, which shows this line was significantly stressed for inertial power flows due to 790 MW loss of generation in the external system. The governor load flow predicted a loss of stability across the tie lines connecting the internal and external system by showing very large angles across this boundary (angles >> 90°). This is shown in Table 3.5. This was further confirmed by the midterm stability results, where Figure 3.8 shows that the line (40, 41) exceeded its sta- bility limit at 5.35 seconds after the loss of generation and caused line (43, 44) and line (23, 24) to consecutively exceed their stability limits, leading to separation of the internal and external system. This can also be observed in Figure 3.9, where the frequency at a group of buses in the internal system oscil- lates with increasing magnitude against the frequency at the external buses, which leads to separation of two groups. The magnitude of this oscillation increases after 1 second, which is the inertial time frame indicating the -stability is due to governor generation response rather than inertial generation response. The separation of the internal and external system caused a very large swing in generator angles. These angles are shown in Figures 3.10 to 3.12. Figure 3.12 shows that after this swing, generators 16 and 17 lost syn- chronism. 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[83 O I Or :mw MUS d l NP me 0 0 C 0 0 73 associated with inertial and governor load flow can be very severe and lead to system collapse. The reasons Mu/the governor load flow led to the loss of stability of lines connecting the internal and external system are now given. The three lines (40, 41), (43, 44), and (23, 24) were importing power to the external system with ample addi— tional capacity. Lines (40, 41), (43, 44), and (23, 24) were importing 7.6 PU, 6.7 PU, and .11 PU with capacity of 11 PU, 20.16 PU, and 8 PU, respectively. The line (40, 41) was initially more heavily loaded than the others. The loss of 790 MW (7.9 PU) generation in the external system near the boundary caused the internal units to respond by 8internal Bsystem M . .833 > hnternal = .336). A significant percentage of system 8 this internal system response ( governor action much more than by inertia since ( internal ) system over the weakest line (40, 41), causing the loss of stabil- x 7.9 PU came ity over this line as can be seen when the power over the line reached 11 PU. Then the boundary lines exceed their stability limit one after another, causing separation of the external and internal system. The inertial response of the internal generators was not as severe as the governor response and did not cause stability limit violations over the boundary lines, but the inertial flow stressed these lines severely, which is an indication of vulnerability of this boundary to inertial power flow. A very large loss of generation, 20 p.u. at 74 generator 18 in the external system, showed a loss of sta- bility at .62 second across this boundary. The loss of stability occurred due to inertial response but would also have occurred due to governor response. It is clear that due to the differences in governor and inertial response loss of stability can occur due to only inertial, only gov- ernor, or both inertial and governor responses. When a loss of stability could occur due to both inertial or governor response, the actual loss of stability occurs in the iner- tial time frame since it comes first. 3.3. Simulation of 790 MW Loss of Generation at Bus 12 Using Conventional Load Flow Program The 790 Mw loss of generation at the external bus number 12 by the governor load flow and the midterm stabil- ity simulation was shown to cause loss of stability for governor distribution of this mismatch. This loss of gen- eration contingency was simulated on the system using a regular load flow program to study whether the regular load flow programs could predict an overload or a loss of stabil- ity across the external and internal boundary. In this load flow program, the loss of generation was made up on the large swing generator at bus number 40, which is also in the external system. The program does not report any violation of either line overload of bus voltage limits. This is expected since a shift of generation in the external system would 75 not significantly affect the boundary between the external and internal system because it does not require the inter- nal generators to respond to this generation shift. 3.4. Line Outage Contingency of Line (40, 41) To determine whether the stability problem observed for governor load flow could be assessed by line outage studies, line (40, 41) was outaged using a DC load flow program. The result given in Table 3.6 indicates that the system is operating with no violations of the limits. The angles across the boundary lines (43, 44) and (23, 24) are not even close to the critical angle (90 degrees). This is due to ample capacity of the remaining lines connecting the external and internal system that can handle both the inadvertant flow due to outage of line (40, 41) plus the scheduled power flows. The results obtained from the simulation below ran on the 40 bus system and are now summarized. These results are extremely important because: (1) The inertial and governor load flows did predict the dynamic changes in power distribution in this system for loss of generation contingencies. This allows the direct assessment of system and boundary weaknesses to inertial and governor power flows with the methods introduced in Chapter 2. (2) The loss of stability caused by governor power flow after the first swing contradicts the generally held 76 Table 3.6. Angles Across the Transmission Lines After Outage of Line (40, 41). Line Angle Line Angle from Bus to Bus (Degree) from Bus to Bus (Degree) 30, 19 6.90 10, 34 8.80 15, 19 2.50 35, 46 .01 16, 19 2.3 14, 35 .20 22, 21 1.43 35, 48 .21 21, 46 .82 36, 46 .20 45, 22 3.33 49, 36 3.68 24, 23 5.84 11, 37 9.40 23, 43 14.04 12, 37 9.40 38, 24 .70 37, 43 1.19 49, 25 4.19 37, 40 28.35 48, 25 .1 45, 38 4.64 26, 39 7.51 39, 49 3.26 47, 26 9.41 41, 42 24.15 47, 27 32.35 41, 44 13.39 27, 28 1.48 6, 41 6.70 30, 29 2.36 7, 41 6.70 38, 29 1.80 8, 41 6.70 45, 30 4.07 42, 44 10.76 31, 44 6.79 42, 45 2.91 31, 47 1.99 44, 43 28.14 5, 31 8.50 45, 49 1.71 33, 32 3.29 4, 47 7.30 32, 34 1.88 49, 48 4.09 9, 32 2.91 17, 20 2.60 33, 47 1.21 13, 18 4.30 1, 33 5.40 18, 48 .11 2, 33 5.50 20, 40 1.2 3, 33 6.80 48, 28 9.55 34, 39 12.97 43, 40 27.16 (4) 77 understanding of stability that a system that survives the first swing is stable. Moreover, this stability problem is difficult to classify using present power system stability definitions. It would be classified as a transient stability problem due to the fact that it is caused by a large disturbance and requires a nonlinear rather than a linearized model for simula- tion or analysis. It also could be classified as a "steady state" stability problem because the loss of stability is dictated where line flows or angle dif- ferences exceed the steady state stability limit. These results suggest that some modification of power system stability definitions may be appropriate. The loss of stability caused by governor load flow was not observed in a regular load flow study of the loss of generation contingency where the loss of generation was made up on the large swing generator also lying in the external system. In the investigation of the 790 MW loss of generation, the loss of stability was caused by governor power flow distribution. The Midterm Stability Program and Long Term Stability Program are the only methods of analyzing the governor power flow distribution effects until now. However, these programs are very expensive to run for the governor time frame and are not able to handle very large data bases. 78 (5) The vulnerability of the boundary connecting the internal and external system could not be seen by a single contingency line outage on the boundary of line (40, 41). Thus, the stability problem associated with inertial and governor distribution of loss of generation cannot be assessed by line outage studies. The effects of inertial and governor load flow on very large networks was not investigated in this thesis due in part to the lack of data bases. This task has been pro- posed to EPRI, which also includes a method for modeling of governor frequency response characteristics of generators for large networks. CHAPTER 4 SECURITY MEASURES AND SECURITY ASSESSMENT The objective of this chapter is to propose and theoretically justify a security assessment methodology for inertial and governor generation response to loss of gener- ation contingencies. This security assessment methodology is unique in that: (1) (2) (3) It is a transient security assessment methodology based on the classical transient stability model. The vulnerabilities of the system for inertial (and governor) generation response to loss of generation contingenciesene associated with the strict synchro- nizing coherency loss of controllability condition on this classical transient stability model. The security measures are developed, can be shown to detect the loss of controllability condition, and thus detect the boundaries and lines between groups of generators that are vulnerable to a loss of stabil- ity due to the inertial or governor response to loss of generation contingencies. These intertial and governor security measures are probabilistic rather than deterministic, as are all previous security meas- ures [15]. These security measures can be evaluated 79 80 by summing the results of DC inertial and DC governor load flows for all single loss of generation contin- gencies. (4) The vulnerable boundaries between generator groups are determined by clustering generators into larger and larger groups with successively weaker boundaries. Security measures have not generally been defined over individual network elements and thus have not gener- ally permitted identification of vulnerable boundaries and lines. The justification that: (1) a specific loss of con- trollability conditions on the classical transient stabil- ity model causes vulnerability to loss of stability for inertial response to loss of generation contingencies, and (2) this loss of controllability condition is detected from the probabilistic security measures (defined from DC iner- tial and governor load flows) and the boundary identifica- tion procedure is given in this chapter. This justifica- tion is based on results derived in justifying the modal coherent technique for producing dynamic equivalents for transient stability studies. This chapter is divided into Uvee sections. The first section develops the linearized classical transient stabil- ity model, the probabilistic disturbance model, and the r.m.s. coherency measure. The inertial security measure is defined and related to the r.m.s. coherency by showing that the security measure, evaluated for all single loss of 81 generation contingencies, is identical to the r.m.s. coher- ency measure for the modal probabilistic disturbance when the base case condition is neglected. In the second sec- tion, the vulnerabilities of boundaries and lines to loss of stability for inertial generation response to loss of generation contingencies is related to the strict synchro- nyzing coherency loss of controllability condition and then shown to be detected by this security measure and a bound- ary identification procedure. In section three, the gover- nor response security measure is defined and shown to be computed by summing the results of the DC governor load flow for all single loss of generation contingencies. The governor security measure and boundary identification pro- cedure is shown to detect a loss of controllability condi- tion that causes the vulnerable boundaries and lines to the loss of stability for governor response to loss of genera- tion contingencies. 4.1. Security Measure Derivation and Justification The security assessment or contingency analysis method depends on the security measure defined in this section that is evaluated based on a DC inertial load flow simulation of loss of generation contingencies. The security measure is justified by showing that it is identical to the square of the ers. coherency measure evaluated for the linearized clas- sical transient stability model with a probabilistic dis- turbance model. The security measure is further justified 82 in the next section by showing that the strict synchroniz- ing loss of controllability conditions of this linearized classical transient stability model will cause the vulnera- ble boundaries in the transmission network and that these boundaries can be detected from this security measure using the boundary identification procedure. Thus, this section first develops the linearized classical transient stability model utilized to drive the r.m.s. coherency measure in this section and to define and discuss the loss of control- lability conditions for this model in the next section. The probabilistic disturbance model and the r.m.s. coher- ency measure are then defined. The security measure is then defined and shown to be evaluated as the summations of d.c. inertial load flow for all loss of generation contin- gencies and to be identical to the r.m.s. coherency measure for the probabilistic disturbance. 4.1.1. The Linear Power System Model To obtain a linearized model of power system, the synchronous machine is presented as: d Mi Hf Awi(t) = APMi(t) ' APGj(t) ' Di Ami (t) (4-1a) d at A6i(t) = Awi(t) , i 1,2,...,N (4-1b) where: Ami A6 83 is the subscript for generator i indicates that the variable is a small deviation about some specified (pre-calculated) steady-state operating point is the inertia constant of generator i in p.u. is the speed deviation of generator 1 is the rotor angle deviation of generator i (in radians) is the damping constant of generator 1 (in p.u.) is the change in mechanical input power at generator i in p.u. is the change in electrical output power at genera- tor i in p.u. The real power flow equations with real and reactive power decoupled are represented in polar form as: A A where: lo 11 lm 7Q APL. J A9. J p p G) 630/ a_a_ afig/ ag A_6 (4-2) afiL/ag aflL/ag A6 '— (P61,PGZ,...,PGN)T (PL1,PL2,...,PLK)T (61,62,...,6N)T (91,92....,9K)T is the number of load buses deviation in power injection at load bus j deviation in voltage angle at load bus j 84 The decoupling of real and reactive power is justified based on the strong dependence of real power on voltage angle and high X/R ratios (loss less network) for the transmission system. To solve the network equations (4-2), one of the angles must be specified as a reference bus since the power angle Jacobian matrix in the network equation (4-2) is sym- metric and singular. This allows a unique solution for Ag and Ag, given af§_and AEL. The generator angle of the Nth machine may by chosen as the reference angle; therefore, equations (4-1) and (4-2) may be written in the uniform machine N reference frame [30] as: d _ 1 1 . Hf Ami — M: (APMi - APGi) - MM (APMN - APGN) - koi (4-3a) i = 1,2,...,N-1 Di where: o = M: , i = 1,2,...,N. d A5 - A113 1 - 1 2 N-1 (4-3b) HT 1 - , s " a s 0: aPG aPQ/aé GEE/GE £9 aPL 6__L/6§ aP_L_/a§ A_0 (4’4) where: oi = oi - 6N , i = 1,2,...,N-1 6. = é. - é , ' = 1,2,...,K J J N J o = a - 0N , 1 = 1,2, ,N-1 A , _ ~ ~ T and. [61,62,...,6N_1] lo’ I ICD) I - [61,92,...,9 To express the model in state space form, equations (4-3) are written in vector form, and network equations (4-4) are used to eliminate Agg from the expression. APG in terms of A§ and afiL is: APG = TAo - LAPL (4-5) where: 1 = agg/aa - (agg/ag) tag/ary“ (aP_L/a§_) (4-6) is the synchronizing torque coefficient matrix and: L = -(a£§/a§) [aPL/aél‘1 (4-7) is referred to as the load reflection matrix. The resulting state model may be written as: is”) = A 1(t)+ B g(t) (4-8) where: A: EAL” 5. = 9 E = Ag VAPL Q l 9. 9. A = a E = (4'9) 1 13 l-1 I G H H: I: r. 86 and: (4-10) I3 u -1 N-1 M ‘M is a (N-1)N dimensional matrix; 5 is a 2N-2 dimensional state VECtOT‘ . 4.1.2. Disturbance Model The disturbance model presented in this subsection was developed by Schlueter [22]. The purpose of the distur- bance model is to allow modeling of deterministic as well as probabilistic system disturbances. The initial condition of linear model (4-8) is assumed random with: E{5(0)} = 9 115(0) EM} 1,03) since expected deviation from any operating state is zero, but the variance of such deviation is non zero. The ini- tial conditions are included not to reflect any specific type of disturbance but rather the effects on the state from some hypothetical disturbance whose statistic may be inferred from internal and external operating conditions. 87 The input composed of deviations in load power, aPL, arui the deviations in the mechanical input power, aPM, can be [1560 to model: i) loss of generation due to generator dropping ii) loss of load due to load shedding iii ) line switching These contingencies can be modeled by an input disturbance u(t) where: u for t 3 0 u(t) = (4-11) 0 for t < O that is, u(t) is a vector step function initiated at time t=0. Non-zero entries in u(t) will model loss of genera- tion, loss of load, and line switching type contingencies described above. The modeling of these three disturbances requires determination of u and possible modification of the network before determination of matrices A and B. The procedure for generator dropping and load shedding that is used in this study is discussed below: generator dropping - the transient reactance of the genera- tor dropped is omitted from the network, and the gen- erator output aPMi of generator dropped is set equal to the loss of generation. load shedding - the load deviation APLk for all buses k where load is shed should be set equal to the change in load caused by the load shedding operations. 88 To represent the random occurrance of generator drop- lbing, line switching, and load shedding, it is necessary to define: E1 92 E{[g(t) - m] [g(t) - mlT} = ‘1‘ - R (4-13) — —22 The matrices m1 and 311 describe the uncertainty in the location and magnitude of generation changes ABM. The matrices 32 and 322 describe the uncertainty in locations and magnitude of power injections on buses due to either load being shed or line being switched. It should be noted that ABM and AEL are assumed uncor- related because this model is to represent only one speci- fic type of contingency at a time. For the same reason, 3 is assumed uncorrelated with initial condition: E {5(0) ET} = 0 (4-14) The uncertain model of u can handle the case of specific deterministic disturbance by setting 5;Q and m=u for the particular disturbance. 89 4.1..3. The r.m.s. Coherency Measure The r.m.s. coherency, Ckz’ between generators k and 2 of a power system is defined as: T 1 Ckz = E} E{f0 [A6k(t) - A61t)]2 dt}]/2 (4-15) where E is expectation operator. This coherency measure first was used by Schlueter [22] to determine coherent groups ofgenerators which could be aggregated into a single generators uiform a reduced order power system model. It has been shown [23, 24, 25] the resulting equivalent by the r.m.s. coherency measure reflects the overall dynamics of the system better than other coherency measures [26]. The expectation operator E appears, because as shown in [17], the disturbance for detecting coherent groups that depend on the power system structure is probabilistic, and there is no single deter- ministic disturbance that adequately detects structural coherency. The results of [22-26] for the linear model of subsection 4.1.1 using a probabilistic step disturbance u(t) is now given. To facilitate the computation of r.m.s. coherency measure, the intermediate quantity, §x(T), is defined in terms of the state vector of the linear model as: T §X(T) = .1. f0 E{x(t) gm} dt (4-16) 90 which is a (2N-2)x(2N-2) symmetric matrix where x(t) is the state vector. Thus, the Ckz can be written as: T Ckz = [9kg 5 x(t) ek ]% (4-17) with ekz a 2N- 2 vector whose jth entry is defined by: (1 j . k - j = 2 } for k 1 N,.2= N 0 j i k 1 i = k _ 0 j g k } for k # N,2..11 (4-18) 1 j = 2 _ K0 j ¥ 2 } for k - N,.2¢ N For input function g(t), x(t) has the form: eAT 5(0) + ft e—1 g u dV (4-19) £(t) = 0 Substituting this expression for x(t) into (4-16) and tak- ing the expectation term by term and utilizing the assump- tion that 5(0) is zero leads to the expression: t eAV eAV s (T): 470 [foe dVB][R+mm ][f It avg]T dt (4-20) As shown in [17], §x(T) can be written in a closed form by letting T~o, that is: 91 sxm = [fig] [3 + mmTJ [MEN <4-21) for A and 8 given by (4-9), thus, (MT)'1 M (MT)'1 ML -1 ‘ __ _ ‘ A g = (4-22) 9 9 For step disturbance in mechanical input power, m and R, as defined by (4-12) and 4_13) become: m R 0 m = ‘1 . R = '11 — (4-23) 0 " 0 0 Substituting (4-22) and (4-23) into (4-21) leads to the expression: 5 ( ) [(Ml)’1flllfi11+m1m1TJHMD'1MJT 9] ‘x ' 9 9] Thus the coherency measure for any pair of generators is (4-24) defined by: §x(°) = [(fll)'1][311 + m1mT][(MT)“M]T (4-25) A disturbance which causes: m3” + -01) MT = I (4-26) shows that the r.m.s. coherency measure is a function of system structure only, and §x(o) can be written as: 50») = my") [(fll)'1]T (4-27) 92 The disturbance which satisfies (4-26) is: 2 2 2 D1 = g , 311 = DIAG (M1,M2,... :MN_13 0) (4-28) This disturbance is dependent on choice of reference gener- ators. It has been shown [17] a reference independent result can be obtained by allowing the covariance of the disturbance in ARM to be: 2 1 2 R N) _11 = DIAG (M ,M ,...,M (4-29) 2 The disburtance defined by (4-29) is called "the modal disturbance," and §x(o) produced by this disturbance is: [(_M_T_)'1J 5 [(MD'HT (n A 8 V II (4-30) where K is a constant matrix whose ij entry is defined as: (4-31) 0"! 7Q H-o H (_a I f—H -* N H H ‘IL ll {—1 (.1 4.1.4. Inertial Security Measure and Its Relation to r.m.s. Coherency Measure. The ability to transfer power from point A to point B is dictated largely by the angle across equivalent trans- mission lines between A and B. This angle can reflect the loading of the line and its closeness to the stability limit of the line. Thus, a set of these angles, if appro- priately compared and grouped, may be used to identify 93 weaknesses in the transmission system. Hence, in this sub- section, a security measure is defined based on these angles for inertial power flows. The inertial security measure for a single loss of generation contingency on generator i is defined as: s;(k,z) = [oM(k,i) - 5M(2,i)]2 (4-32) where oM(k,i) is the phase angle of the inertial load flow at bus k for loss of generation contingency, and: PMi = PGi(0) - APMi (4-33) where APMi is the lost mechanical power into generator 1. APMi is equal to PGi(0) for a tripped generator. The angle §M(k,i) is obtained by inserting (4-33) into (2-11) N which is in turn inserted in i=1’ (2-9) or (2-10) to obtain {5M(k,i)}E:T. to obtain {PGJ(1)} To account for the effects of all single loss of gen- eration contingencies, a contingency independent security measure (CISM) for inertial load flow is defined as the sum of the inertial security measures for a set of single contingencies: s§(k,2) (4-34) IIMZ SM(k,2) = i 1 where i is summed over all possible loss of generation con- tingencies or some subset. Since CISM considers the 94 effects of all contingencies, it is used to detect boundary vulnerabilities for inertial load flow in this research. It is shown in Chapter 2 that inertial distributions of mismatch power due to loss of generation is complete when the acceleration at all generators in the system is equal to the mean acceleration of the system. Knowing this fact, the linear relationship between the security measures and inertial load flow is now given: - 9 p=1,2,...,N and for the system loss of generation mismatches N j . J.;l31(PMJ. - PGj(0)) are. PMi - PG1 (1) 0 = p p , 1 = 1,2,...,N p *M p and: N i 2 (PM. - PG.(0)) j=1 J J 4 = N , i = 1,2,...,N 0 2 M. i=1 J N . M 2 (PM! - PG.(0)) PGi(1) - PMi - p 3“ J J p D N M 2 . i=1 3 (4-35) (4-36) (4-37) (4-38) 95 It is also known: PG;(1) = PGp(O) + ape;(1) , p = 1,2,...,N (4-39) i = _ i _ PMp PGp(O) APMp (4 40) and for loss of generation from (4-5), it is given: Ag§i(1) = 1 Ag; (4-41) where: A2; = [A6& ,A6; ,...,Mfi JT 1 2 N-1 are the inertial angle deviations at internal generator buses for loss of generation APM. Substituted for AE§i(1) from (4-41) into (4-39) and written in vector form: 3§i(1) = g§(0) + 1A9; (4-42) Inserting PM1 from (4-40) in vector form and P§k(1) from (4-42) into (4-38), the term AP§(0) is then eliminated from both sides of equation (4-38) to obtain in vector form: t 1 M1 M1 M1 , \ “6 "'6 "6 142.14= - (.1: ° ° 0 mm1 <4-43) "N MN L "6 "E J N where z Mj=MO. Multiplying both sides of equation (4-43) i=1 from the left by M, where: 11- 4‘ fl= 1 N 1 _1 ”11.1 “E and noting that: F 1 1— —1 1 _1 M1 "1, , M1 ”1’ “T “6 “B “6 M2 M2 “6' '"6 =9 1 _ 1 MN MN "1? ME] 115' ° ' '11; then (4-43) may be written as: 97 -M APMi = MT Abi ——— —— ——M Thus, the inertial angle changes for loss of generation i and generator buses are: i -1 i A911 = 4.1113 E ABM (4-44) where ARM] is a N dimensional column vector whose pth entry is defined by: {APMK} = , i,p = 1,2,...,N (4—45) 9 ap ' The inertial angles for loss of generation i for base case angle 90 are: _ i 9M ' 20 + ABM (4'46) Defining: i _ i i EM ' 3M 3M (4-47) the inertial line security measure may be expressed as: i T i SM(k’2) = [Ski EM Ekl] (4'48) Then the security for all possible loss of generation con- tingencies is given by: 98 N i T SM(k’2) = 3kg —M 3K2 (4‘49) where Ski is defined by (4-18), and: g; ggT (4-50) IIMZ N . N = 2 WI: 1 i 1 Using (4-44), MM can be written as: N . . ”M = z [(MINMJ [AEM‘ ABM”) [(M1)"M1T i=1 (4-51) N . + z 2 (MT)‘1M APMl a; + Naoog i=1 ‘ “’ ‘ ‘ ‘ Assuming 6020, the expression for MM becomes: -1 N T -1 T WM = [(MT) M] x [ABM ABM ] [(MT) M] (4-52) i=1 where: APM? 2 21PM2 N i 1T 2 [ 30 EM ] = . (4-53) i=1 . 2 APMN A The matrix MM is identical to the expression Sx for the r.m.s. coherency measures if: Ami ABET = R T (4_54) IIMZ which requires the statistic of disturbance to be repre- sented as a summation of N deterministic disturbances. Now, assume the loss of generation such that: APM (4-55) where C is a constant, and substitute for APMi from (4-55) into (4-53), then the expression (4-52) may be written as: ’2 2 7 1: M1 2 2 (2 M2 wM = [(M1)"MJ [(MT)"MJT 2 2 1: MN = c2 (M_1_)-‘ 19.1)” (4-56) 2 for i = j where {M}i. = (4-57) J 1 for i ,1: Equation (4-56) shows that the contingency independent line security measure depends only on the matrix [MIT1 (inertially weighted synchronizing torque coefficients) which determines the modal and coherent structure of the 100 power system. It is clear that the contingency independent security measure can be computed with the probabilistic disturbance (4-29). Using this disturbance, all the boundaries between generation groups for inertial distribu- tion of every loss of generation contingency are equally tested, and modal and coherent properties of the systems are captured by the security measure. A grouping and ranking algorithm, which is described in the next section, can identify the weakest boundary when it uses the security measures produced by this probabilistic disturbance. The next section also discusses the strict synchronizing loss of controllability condition, strict geometric coherency condition, and strict strong linear decoupling condition, and then shows the (CILSM) and boundary identification pro- cedure only detects the strict synchronizing loss of con- trollability conditions. 4.2. Transmission Boundary Vulnerability Justifica- tions Based on Loss of Controllability Condi- tions for the CTassical Transient Stability MOJeI and the Inertial Security Measure. The present literature has defined [27, 18] different conditions for which a group of generators behave as a sin- gle generator after specific disturbances. These conditions are based on controllability and obesrvability properties of the transient stability model (4-8)-(4-10) of the systems. The conditions, if satisfied, have been shown [18] to cause coherency [26] and modal [28] 101 analysis dynamic equivalents to be identical and also [18] to cause decoupling of fast and slow eigenvalues of the systems. These conditions are now given. 4.2.1. Observability and Controllability Conditions To discuss the controllability and Observability con- ditions requires a linearized model of the power system. The power system model used is a modified version of the linear model (4-8). This model is divided into an external and internal system and is expressed in second order form as: r .. — h‘ . ‘ " ‘ _ 1 ‘ 1- . ‘ Aém (‘41)11 : “E12 A9111 °lm: 9 A9111 I I ----- = -—-----fl——----—- ----- - ---1—----- ----- .. I I o A911-1 (”MT)21 : (“’22 A911-1 LO : °—In-1 A9w _. J ._ _ ._ J J _. J r— | - r- — M1:(flL)1 AB! (4-58a) ---.» ..... --- (4-58b) M '(ML) APL —2 ' — 2 — L. ' __ L... where: a , i = 1,2,...,n, are the internal generator angles of the external group. a , k = 1,2,...,m, are the internal generator angles of the m generators of the internal group. 102 1 , _ 1 "T : “T l 1 g _ 1 i; : ”T 1 I 1 I - : "E I I I I I I 1— | — 1: 4111312 “M: ---i--_- = ................. 1 ........................ I I 1 I 1 I I'M-— __9 ' M22_ : m+1 : i I I O 1 ‘ 1 I I E 1 _ 1 _ ' IM+n-1 MN_J (4-59) F— — — L11 L22 egg aE_L_ '1 _L.= -------- = --T — (4.60, as 69 £21 £22 ‘ ‘ where: 1m and ln-1 are identity matrices of dimensions m and n-1, respectively. LT: N x N-1 matrix of synchronizing power coefficients. _1==[M11M12] is a m x N and consists of the first m rows of M. (MEN = [911511 " fl12£12 fl11£12 1" 1412—522] is m X K and C°”' sists of the first rows of ML. 103 (ML)2==[M22521 M22522] is n-1 x K and consists of the last n-1 = N-m-1 rows of ML. At present, there are five conditions based on con- trollability and observability concepts. Any of these con- ditions has been shown [18] to represent modal and coherent properties, and these conditions, if satisfied, have pro- duced identical modal and coherent equivalents. Two of the conditions only hold for the linearized transient stability model (4—58) and are observability conditions. Thus, the three remaining conditions that are applicable to both linear and nonlinear models and are controllability condi- tions are presented here. (1) Strict synchornizing coherency (SSC) which requires that there exist (n-1) stiff equivalent lines con- necting internal generator buses and that these (n-1) lines form a tree for each n generator coherent group. These (n-1) equivalent lines are very stiff compared to the inertias in the group and cause a decoupling of the eigenvalues that describe oscillation of genera- tors against each other within coherent groups and those eigenvalues that describe the oscillation of one group against another. The eigenvalue association with oscillations within coherent groups would have large imaginary parts compared to the eigenvalues that describe group against group behavior. SSC causes (-Ml)é; ~ 0 in the limit as the interconnections between generators are progressively stiffened (2) (3) 104 compared to their inertias. This type of condition has been shown [18] to be detected by the r.m.s. coherency measure evaluated for the modal probabilis- tic disturbance (4-55, 4-29) or by singular perturba- tion methods described in [29]. Strict geometric coherency (SGC) which requires that the ratio of synchronizing torque coefficient of equivalent lines connecting the disturbed bus and an internal generator bus over the inertia of that gen- erator to be identical for all generators in the coherent group. This property causes uniform acceler- ation of the group over several seconds before gover- nor control takes over. SGC depends on the connection to the group and not the connections between members of the group as does SSC. SGC causes (-Ml)2fl= O. Strict strong linear decoupling (SSLD) which is a group formed by a combination of SSC and SGC which is different than either of these properties alone. A portion of the group satisfying SSLD will satisfy SSC, but the synchronizing torque coefficients of the lines connected to all members of the group must cause uni- form acceleration in the group. This property holds in the nonlinear model since SGC and SSC hold in the nonlinear models, but the condition was derived based on the linear model and thus its name. SSLD causes (mpg; @921» o. 105 In the next subsection, the SSC loss of controllabil- ity condition is discussed in more detail, and it is shown that SSC causes the submatrix (Ml)é; to go to zero in the limit. This results in a decoupling of the equations for the external group from the equations for the internal sys- tems and that this decoupling is detected in the contin- gency independent security measure (4-51). 4.2.3. SSC Loss of Controllability Condition The definition given below for strict synchronizing coherency is: A specified group of n-generators within a power system exhibits strict synchronizing coher- ency if there is at least one infinitely still connection joining each generator of the group (for a minimum total of n-1). [18] If the SSC condition holds for a group of n generators, this group can be replaced by a single equivalent genera- tor, and the response of the remainder of the system to a disturbance outside the group of n generators is preserved. The proof for this claim is now given. Consider a group of n generators within a power system and let yi be the admit— J tance between generators i and j of the group. Suppose yij"°’ then the voltage magnitude and phase angle must be the same at buses i and j or otherwise there will be an infinite flow of power between the two buses. Thus, gener- ators i, j can be replaced by a single composite generator of inertia Mi + M. as long as the power flow from the com- J bined generator to the rest of the system is preserved. 106 Next, let yi+j,k be the admittance between the com- posite generator and the generator k of the group, and let yi+j,k ~ o. By repeating the argument above, generator k and the composite of generators i and j can be replaced by a new composite of inertia Mi + Mj + Mk' Continuing from n-1 steps produces a single generator equivalent for the original group of n generators. This simply means that the internal behavior of the group of n generators is not con- trollable by disturbances inside or outside this group. The group formed by SSC conditions is thus called a strongly bound coherent group. Any generator connected to a strongly bound coherent group, but not part of it, has relatively weak connection with the group or otherwise it should be a member of the group. It is now shown for the model (4-21) that when there are n-1 stiff connections between n generators of external groups (SSC holds), the submatrix -(MT)£; ~ 0 in the limit as the interconnectnxm between generatorsane progressively stiffened. Suppose that the conditions for SSC are satisfied and that (-Ml)£; exists. The inverse of (~Ml)22 can be written: 107 C11 C12 C13 C1n c c . . . c -1 _ 1 21 22 ' 2n "fll)22 ‘ Det(-Ml)22 . (4'61) (:”1 cn2 . . . .1:nn where Cij Assume that n-1 of the interconnections that link all is the ijth cofactor of (~Ml)22. n generators of the external group are made infinitely stiff. In the linear model, the corresponding n-1 elements of (-Ml)22 become infinitely large. Now (-MT)22 is (n-1) x (n-1) so that Det (-MT)22 is the summation of (n-1) terms where each term is the product of (n-1) elements of (-MT)22 One of these terms is the product of all n-1 ele- ments that are being allowed to become infinitely large. Now each cofactor Cij is, in turn, the summation of (n-1)! terms where each term is the product of (n-2) elements of (-Ml)22. Thus, each cofactor Cij no more than n-2 of the elements that are becoming infin- can be the product of itely large. As a result, Det(-Ml)22 dominates every term in the summation of Ci i,j=1,2,...,n-1. Thus in the j! limit, all terms of (-Ml)§; tend to be zero. Now assume that (-MT)22 is finite and rewrite equa- tion (4-58b) as: 108 1 (-fll)§; Aén-1 = ‘(fl)25 ('fll)21A§m +A6 —n-1 -o(MT)2§1 A3 + (Mpg; [M2 A_P_M +(_1M)2 APL] n-1 Letting n-1 elements of (-MT)22 become infinitely large results in: A9n-1 = O for all t > 0 (4-62) This result in turn reduces equation (4-58a) to the form: A§m = -(MT)11 Agm - oA§m + [M1 ABM + (MB)1 ABM] (4-63) Thus, assuming zero initial conditions, the external groups of n generators behave, from the point of view of the remainder of the system, like a single equivalent genera- tor. The contingency independent security measure (4-56) will now be shown to depend on submatrix (Ml)2§1 of the strongly bound generators of the external system. This security measure detects the SSC loss of controllability of external systems and can be used to capture all the strongly bound groups in the power system model which have weak boundaries. Assume, as in (4-58), a power system with N = m+n generators, where n is the number of generators in the external system and m is the number of internal generators. Partition matrices (MI) and M. 109 (El)11 (51’12 [M_T] - (41’21 (51’22 K K E = —11 —12 521 522 where M is defined by (4-56). Assuming (M_T_)11 and (Ml)22 are nonsingular, the inverse of (MI) can be written as: T —1 -1 -f' 0 -(MT) (MT) P [Mll‘1 . — —— 11 —— 12— -1 -1 -1 -(fll)22(fll)21fl E J where: 2 = 0.1)“ - (012052021 3 = (MII22 ‘ (El)21(fll)11(fll)12 Then the security measure (4-56) MM can be written: 110 F -1 -1 -T‘ 7’ — Q -(fll)11(fll)1zp 511 K12 -1 -(M1)g;(M_1)21Q“ 3 : T T i7 0' -Q' (Ml)21(M__T_)§2 {‘(MDTMIHT 2” _ 4 L521 K22 -1 -T -1 -T -T -1 KM = 9 5119 ' 9 5123 (Ml)12(fll’11 ’[(fll)11(fll)12 1 T -T T - K22P (fll)1z(MT)11 3.152194J + (Ml)1I(MT)12P- M1 = 2“£119’T<fl>21T53 + 241422” +t1112 1 T -1 -T T -T -1 - - (055021281194 +t<fl>5l21i"5122‘T<fl>12 1 1 l 3 I K -T T -T —223 (51’12(fll)11 -T - -T - (51’11 + 3 5219- 3‘ 3 (Ml)22(fll)219-15119-T(fll)21(41’21 + “7152213-T §l212"£12£‘T - £"5212'T<41>£15£ The submatrix KMzz contains the information on the behavior of the n generators of the external group. In the linear model, either condition of strict synchronizing coherency (55C). [Ml]§;-*B; strict geometric coherency, 111 [MT]21 = 0; or strict strong linear decoupling, [Ml]§;[Ml]21-O will make: -1 _ -T —M ' _ 5223 -1 Applying the matrix identity on B, B can be written: 3" = (Mlig; + (M115;(M1)21 [(M1111 - (Mli12(M1)g; (Ml121i"(MT),2(MT)g; Thus: 3“ ~ .9 W ~ 0 _M22 _ if and only if [Milgg ~ 0 given 522 is positive definite so that ””22 only depends on the structure of the strongly bound external group of n generators for a disturbance either in the internal group or within the external group. This is a loss of controllability condition which is detected by the security measure. The next subsection introduces a boundary identifica- tion procedure based on a commutative grouping algorithm. The commutative rule requires that the groups formed have at least n-1 stiff connection in a n generator group. 4.2.4. Grouping Algorithm The grouping method is based on the cummutative rule [17]. This rule for forming a group requires that a group 112 is formed if all the generators are coherent with respect to each other; that is, if the group G1 is a group contain- ing generators A and B, then generator C is added to this group if and only if generator C is coherent with A, and C is coherent with B. This method has been used [17, 18] for clustering generators in coherent groups for producing dynamic equivalents of the system for transient stability studies. The values of the security measures are ranked from smallest to the largest forming a ranking table; then the groups are formed based on the following algorithm: (a) Form the first group (a pair) from the smallest security measure at rank 1, r=1. (b) Decide which of the following possibilities apply to generator k,2 at the rank r=r+1. (c) If r=Nx(N-1)/2, stop. N = number of generators (i) If neither k nor 2 has been previously identi- fied as belonging to a group, then this pair becomes a new group. (ii) If generator k(2) belongs to a group but generator 2(k) does not, then: (1) if 2(k) has been previously recognized as coherent with all members of the group to which k(2) belongs except for k(2), then add 2(k) to the group containing k(2). (2) if 2(k) has not been found previously to be coherent with all other members of the 113 group to which k(2) belongs, then recognize that k and 2 are coherent but do not add 2(k) to the coherent group containing k(2). Return to (b). (iii) If generators k and 2 belong to different groups, then: (1) if all possible generator pairs which can be selected from the members of the two groups except k and 2 have been previously recognized as being coherent, then merge the two groups to form a single group containing all members of the separate group. (2) if at least one pair of generators which can be selected from the two groups other than k and 2 has not yet been recognized as a coher- ent pair but do not merge the groups. Return to (b). The algorithm continues the procedure to the bottom of the ranking table, and when every generator pair is checked, it terminates. As one proceeds down the ranking table, individual generators are included 'to groups and later groups are merged to form larger groups. As groups are merged, the boundaries between groups should be contin- uously weaker since a coherency measure between generator pairs indicates stiff connection of the generators compared to the inertia of the generators, and the security measures 114 are ranked from the smallest to the largest in this ranking table. Thus, the boundaries may be ranked from the weakest to strongest based on the reverse order of the group forma- tion; that is, the last two groups to be lumped into a sin- gle group have the weakest boundary between them, and the second weakest belongs to the second to last group aggre- gated, etc. The commutative grouping rules assure that all the generators in a group formed have stiff connection by requiring a generator to join the group if and only if it is coherent with all the generators in the group, and groups are merged if and only if all the generators in group one are coherent with every generator in group two. This ensures at least n-1 stiff connections between generators in an n generator group. Now to sum up the procedure for identifying vulnerable boundaries, the following steps are given: (1) Compute the security measure SM(k,2) for all generator bus pairs. (2) Rank the security measures from smallest to largest and form a ranking table. (3) Form groups by the commutative grouping rules and set a group formation table. (4) Rank the boundaries from the weakest to the strongest based on the reverse order in the group formation table. 115 4.2.5. Identifying Vulnerable Boundaries by the Security Measure and Grouping Procedure. It was discussed in Chapter 2 that inertial load flow effects all the generators in the interconnection and therefore is a global phenomenon. The strict synchronizing coherency (SSC) indicates the vulnerable boundaries in the inertial load flow model independent of the location of a contingency and thus identifies vulnerable boundaries in the model in a global manner. The contingency independent security measure defined in (4-20) and the boundary identification procedure described in subsection 4.2.3 can detect vulnerable bound- aries for inertial load flow because of the following reasons: i) SM(k,2) (4-34) is defined from s;(k,2) (4-32), which calculates the effect of inertial load flow for a single loss of generation on the boundaries and lines. ii) Inertial distribution of a single loss of generation contingency significantly effects the boundary close to the contingency since the inertial flows would all flow through this boundary to cope with the lost gen- eration. The summation of security measures for all possible losses of generation contingencies assures that every boundary is equally tested for inertial flow of all loss of generation contingencies; thus, the weakest among all boundaries is identified. iii) iv) v) vi) 116 The security measures are related to the r.m.s. coher- ency measure. It was shown that the CISM is identical to the r.m.s. coherency measure for the probabilistic modal disturbance. This confirms that the security measure can capture the coherent properties of the system which causes groups of generators to swing together and behave like a single generator. The SCC loss of controllability property that forms strongly bound coherent groups is detected by the security measure SM(k,2) summed over all loss of gen- eration contingencies. Thus, the boundaries between the strongly bound groups which have n-1 stiff inter— connections in an n generator group are composed of relatively weak interconnections for the inertia of the group they connect, or otherwise the generators they connect would be in the same strongly bound group. These security measures would have large values for generator pairs that belong to the differ- ent groups but small values for generator pairs in the same strongly bound group. The commutative grouping rules require at least n-1 stiff interconnection in an n generator group. Hence, the group formed by this method is a strongly bound coherent group. The fast eigenvalues (high frequency) which represent the stiff interconnection between generators [29] and slow eigenvalues (low frequency) which represent less 117 stiff connections between generators are a property similar to SSC [18] that is detected by the security measure and grouping algorithm. The fast eigenvalues are associated with intermachine oscillations within strongly bound coherent groups, and the slow eigen- values are associated with group against group oscillations. 4.3. Governor State Model, Coherency Measure for Governor Response, Governor Security Measure, SSC Property fOr Governor Load Flow. This section develops a state model for governor response, a coherency measure to capture the behavior of coherent groups of generators, a governor security measure based on governor load flow results, and then discusses the SSC condition for governor load flow detected by both the coherency measure and governor security measure. 4.3.1. Governor State Model To represent the state model of the system during the governor response, the algebraic equation for the power flows among the generator and load buses are the same as equation (4-2). The linear differential equations for the generators are: On) _ 1 1 A i "Tfi—(APMi - APGi) - EN (APMN - APGN) (4-64) i = 1,2,...,N-1 118 where: A61 is the rotor angel deviation of generator 1 Bi is the frequency response characteristic of genera- tor i APMi is the change in mechanical input power at generator 1 APG. is the change in electrical output power at genera- tor i The ABB in (4-5) is expressed in terms of AB and ABM as: APG = lAé - BABB (4-65) where l and B are defined by equations (4-6) and (4-7), respectively. A state model may be derived by writing the equations in (4-64) in vector form and (4-65) to eliminate BB from the expression. The state model follows as: IO") A = [- B1] AB + [BJEAEM + BABB] (4-66) 119 where: _;L. - JLE 81 BN .1. _ .1. 32 BN E = . . (4-57) __1_ - .3. BN-1 8N g _l ~ . ~ T A_6_ = [Ao ,A62,...,A6N_1] _ T ABM - [APM1,APM2,...,APMN] T ABM = [APL1,APL2,...,APLK] 4.3.2. r.m.s. Coherency Measure for Governor Load Flow The purpose of this subsection is to show that an r.m.s. coherency measure can be obtained for the linearized model of (4-66). This measure could be used to determine coherent groups of generators during the governor response for producing a dynamic equivalent of a power system. For the linear model (4-66), the r.m.s. coherency measure Ck between generator k,2 is defined as: , 2 Ck2 [} E{f [A6k(t) - A6£(t)]2 dt}]% 1:“:- EIf [(A6k(t) - A6N(t)) - (A52(t) _ AéNItDJZdtIJVZ O—IO—l 120 T 1 [4 E{f0[A5k(t) - A5£(t)]2dt 1k [312 §x (T) ek£]% (4-68) where: 1 T - -T §x(t) = T f E {Ag(t) A9 (tll dt (4-69) 0 and 9k2 is defined in (4-18). Let us assume disturbances in mechanical input power ABM, where: E {ABM} = m1 (4-70) and: E {(ABM — M1)(ABM - m1)T} = 511 (4-71) For the input ABM, A§(t) has the form: 29(t) = eE’EIJtAg(O) + IE e['£l]v BAEM dv (4-72) Substituting this expression for A6(t) in (4-69) and carry- ing out the expectation operation term by term and utiliz- ing the assumption Ag(0) is zero leads to the expression: §X(T) = 1.13 US$41“ dVEJ [311 " MI] (4-73) [f8 eE‘EIJV deJT dt 121 given: t f e[‘El]V dv = [-311" [eE‘EIJt - 13 (4-74) 0 and defining: 8 (4-75) expression (4-73) becomes: [__T_]-1f;[e[-fl]tfie[-B_T_]Tt _ eE-BIJtE T U1 A —-I V n -1|- (4-76) - pe[‘—l] t + 3] dt i-gli‘” Letting T ~ w, the first three terms in the integral (4-76) vanish, that is: §x<~1 =[(_l)'1§] [311 + M1M1TIL(_I)'1§]T (4—77) =[(§1)'1]B (1211+ 1111111119T [(_1>“]T A disturbance: 5(311 + 3131) ET = l (4'78) causes the r.m.s. coherency measure: §x(~) = [(91)“J[(g1)“JT (4-79) to depend solely on system structure and is determined by governor frequency response characteristics and 122 synchronizing torque coefficients. Moreover, the coherent groups identified for aggregation using the coherency measure evaluated from (4-79) are determined by line stiff- ness weighted by the governor frequency response character- istic of generators at the end of each equivalent line. The disturbance which satisfies (4-78) has the statistics: E {ABM} = M1: 0 (4-80) T _ 2 2 2 E {AP—MAE! } - DIAG (81:823-0098N_1 a 0) Another disturbance with: _ _ 2 2 2 M1 — 0 and B11 - DIAG «0,82,...,BN) (4-81) results in: EAE1EI + 311)ET = 5 (4-82) and: S _ -1 -1 T 3 -11‘“) — [(21) mug) ] (4-8) where: } 2 i j {K .. ‘3 11111 This disburbance is over all the generators of the system and is reference independent. 123 4.3.3. Governor Time Frame Security Measure The difference between distribution of power mismatch for inertial response and governor response of generating units requires the investigation of the significant effect and difference of governor load flow in contrast to inertial load flow on the transmission network and assessment of security and stability problems associated with this gover- nor power flow. Like the inertial security measures, the governor security measures are defined based on angles across the transmission lines at the governor time frame discussed in Chapter 2. Governor response security measure for a single loss of generation is defined as: i _ . _ . z _ 58(k,2) - [68(k11) 68(211)] (4 84) whereob(k,i) is the phase angle of governor load flow at bus k for a loss of generation contingency: PMi = PGi(O) - APMi (4-85) where PMi is the lost mechanical power into the generator i. The angle 58(k,i) for loss of generation i is obtained by inserting (4-85) into (2-12) to obtain {PGi(2)}?=1 which is in turn inserted into (2-9) or (2-10) to obtain 68(k,i). Similar to inertial contingency independent security measure, the governor response contingency independent 124 security measure is defined as: 55(K‘) = f s;(k,2) (4-86) where i is summed over all possible loss of generation contingencies. Now the linear relationship between the security meas- ures and governor load flows is derived, and the governor security measure is then shown to be identical to the square of the governor coherency measure when the base case load flow condition is neglected. After inertial distribution of mismatch power due to loss of generation contingency, governor frequency regula- tion begins to arrest the change in frequency. When the governor regulation is complete, frequency is constant throughout the interconnection, and the rate of change in frequency is arrested. It can be written: “’i = ”o where: PMi - PG (2) w = P P (4-87) P a p i . - . 9 g (PMJ PGJ( )) ”o = (4-88) zsj .1 PMi PGi ? PMI — PG. 0 the”: __.B-__B= .1( J J( )) (4-89) B B 28. p p jJ 125 The above expression can be written as: . . e . 1 _ 1 _ __E_ 1 _ . _ PGp(2) - PMp 28. 2(PMJ PGJ(0)) (4 90) j J J where: PG;(2) = the electrical power delivered to the network when governor frequency regulation is complete. 1 _ _ i - PMp — PGp(0) APMp (4 91) is mechanical power input at generator p after loss of generation APMg. The electrical power delivered to the network after governor load flow response is related to the power delivered to the network before the loss of generation by the following equation: PGi(2) = P (0) + Again) (4-92) and from (4-5): A3902) = 1A2; (4-93) where: Aoi [Aoi A61 A51 ]T —B - M,1’ M,2’° " M,N-1 126 are the derivation bus phase angles for governor load flow N P=1' Write the equation (4-90) in matrix form and substi- due to loss of generation {APMé} tute for BBk(2) from (4-92) and (4-93) and represent: PMi - §M§(0) .- A5341 then the equation (4-90) becomes: 51.51. 51. Bo Bo Bo . .5212 ‘12. 39(0)1IA51;3§(0)-APMF+ Bo Bo Bo ABM? 3252 “_N Bo Bo Bo /' B1 B1 . . B1 '\ . B2 B2 B2 . TAol - - J1 - ;L > APMl (4-94) --a 30 . . -- . a J 127 Define: .E = 2 ' (4-95) _ .1 and multiply both sides of (4-94) by E3 2142’ = 24221 where: — 1 1_' F — —- --—— B B .1. . = B 1 1 BN-I - 73-111 8" ° ' B" L _. _ J Therefore, the vector Ag; has the form: 1 _ -1 i Ag - 43] _s 45! (4-96) where ABMi is defined by equation (4-44). 128 8 coefficient matrix acts the same way as the coeffi- cient matrix M in inertial load flow. To obtain the security measures, define: (4-97) where £8 = EOTAEL is a vector of phase angles at buses for governor load flow, and 30 is an operating point before loss of generation. Hence: (4-98) 58(k’2) = [ek2 —Be —k2] where ek2 is defined by equation (4-18), and for all possi- ble loss of generation contingencies: _ T 53(k") ‘ Skn 53 9K2 (4'99) where: N . N . . M8 = 21 w; = .21 3; SET (4-100) 1: 1- Using (4-96) and substituting for 28 into (4-100) leads to: 11 _ 2 [(fll'1BIEABMiAPMiTJL(£1)'1ng B - i=1 (4-101) i T T 2[(§_) 3] APM 90 + ”9090 + IIMZ - B, M may be written as: Assuming 30 — 8 129 APMiAPMiTJHglflgiT <4-102) w = [(gl)'1§][ 1 -B "NZ 1 which has the same form as §x(w) for the r.m.s. coherency measure for governor load flow. N .E PMi BM1T is the summation of a single loss of gen- eration contingency and can be written as: r _ APM? APM N i iT 2 AP AP = . (4-103) 2 APMN ... .._1 This disturbance can be presented by a probabilistic dis- turbance with the covariance equal to (4-103). With a disturbance proportional to Bi for each generator, (4-102) can now be expressed as: 8 w = [(8 )"gi 2 [(3114ng (4-104) 2 8N _. _1 Rearrangimgtheiiand BT and carrying out the multiplication: 130 ”a = L(_B_1)’1]§[(§1)'1JT (4-105) which is identical to the Sx(o) for the r.m.s. coherency measure given by equation (4-83). Thus, the security meas- ure evaluated when 30:2 is the square of the r.m.s. coherency measure. The security measure with the disturbance described above depends on synchronizing torque coefficients weighted by the governor frequency characteristic of the generators. This disturbance equally tests all the boundaries between generation groups for the governor distribution of power mismatch due to every single loss of generation contingency since Bi is proportional to the capacity of generator i. The governor security measure (4-99) with the boundary identification procedure detects the cumulative effects of governor load flow on the boundaries, and the weak bound- aries for governor load flow loss of generation mismatch can be identified. The justification of boundary detection by the secu- rity measures based on an SSC loss of controllability property for governor laod flow is given in the next sub— section. 4.3.4. SSC Condition in Governor Load Flow. The SSC loss of controllability condition, which was discussed for the classical transient stability model, is applicable for the linear model (4-66). It was shown that 131 governor coherency measure for a probabilistic disturbance depends only on matrix (91)-1 which reflects the structure of the system when the generators of the power system respond based on their governor frequency characteristics. The definition of SSC condition for governor response of generators is now given: If there is at least one stiff connection joining each generator compared to the governor frequency characteristic of generator in an n generator group, then this group of n generators exhibits SSC. If SSC condition is satisfied for n generators of external systems, there results a decoupling of the equa- tion of the external group from the remaining internal system due to the fact that submatrix (21);;1 which repre- sents the structure of the external group, goes to zero in the limit as the interconnection between n generators ofthe external group are progressively stiffened compared to their frequency response characteristic. This is now shown. The linear model of (4-66) is divided into two groups as the internal system and the external system. Thus, the power system model (4-66) may now be written as: A21 ('fll)11 (‘21)12 A21 A32 ('fiI)21 (‘fl)22 A§2 (4'1063) £11 £12 ABM + L APL (4-106b) 11 1 Q ABM2+ BZZAPL2 922 132 where A§1 = [A61,A62,...,A6m]T are the generator angles of A the internal group, and A§1 = [A5 ,A6 are m+1’A5m+2’°'° m+n-1] the generator angles of the external group. The generator N in the external group is chosen as the reference genera- tor and: JL : _ _L 51 I 3N ’- — 1 8 B 1 ' 1 —11 —12 —— ' -—— = 82 I 8N I I 9— 522 ° ; " '_ , 1 I _L: 8m: .............. .4..................... I 1 : Bn+1 I . . I I o I i 1 _ _L 1 anv1 8N where: N = m+n _ _ aB_ aPL -1 E- - [£11 , L22] ' “1a— [711‘] T [APL1 , A302] T [ABM1 . ABMZJ D D 'U '0 I3 If" 11 It is assumed (~Bl)§; exists and there are (n-1) stiff interconnections between n generators of the external system. Then (-Bl)é; ~ 0 in the limit for the governor load flow if (n-1) equivalent lines between n 133 generators are progressively stiffened compared to the gov- ernor frequency characteristic of generators. The proof is as simple as before for inertial load flow given in section 4.2.3. The proof argues that the cofactors of matrix (-Bl)22 are small, compared to the Det (-§I)22 which causes the ('El)§; to go to zero. Equation (4-106b) can be written as: (21);; A332 = (215521011521 + Aéz + ('BT)22222[A3N2 + L22 APLz] If SSC is satisfied for the external groups, then: (-eT)§; ~ 0 and A§2 = O for all t > 0. Thus, assuming zero initial conditions, the external group of n generators behaves, from the point of view of the rest of the system, as a single generator. Hence, this group can be replaced by a ficticious generator with capac- ity of the n generators it represents. The SSC condition is captured by the contingency independent governor security measure (4-105). That is, if (21);; goes to zero, then the security measure (4-105) M822 also goes to zero. The proof is similar to the inertial load flow security measure given in 4.2.3 where MB and M are partitioned and written as: 134 ”w w " ’1 K " = ‘311 ‘312 and ‘1‘ ‘12 w M K “321 ‘322 ‘2‘ _1 - 17: II |7< Then equation (4-105) can be written as: r — r —-1 —- — — --T Np11 3252 (El)i1 (El)uz .Erl 512 (El)r1(§l)uz w w ' (an (an K K (3T) (an :321 43224. _— 21 —22J :21 ‘22.._—21 _22] in a manner similar to subsection 4.2.3. M822 can be written as: _ -1 -1 -T T -T -1 -T 5822 ' (21’22(£l)219 511Q (21’21(£l)22 + 3 5223 -1 -1 -T -1 -T T -T ' (21’22(§l)219 5123 ' 3 K219 (21’21(El)22 where: 9 = (El)11 ' (El)12(£l)22(£l)21 3 (£1122 - (g1)21(BT)}}(§I)12 Applying the matrix identity: -1 _ -1 -1 -1 -1 -1 B - (flIzz + (_B_T)22(_Bl)21[(_8_1)11”(fl)12(§l)22(§l)21] (_B_T1zHfl)22 The SSC for the n strongly bound generators of external systems causes (gl)5; ~ B. This leads to: 1 - -T E822 = E 5223 135 and: 2“-9 Thus, w ~ 0 ‘822 “ The submatrix M522 contains the information on the behavior of the n generator of the external group at governor time frame. Hence, the loss of controllability conditions of the strongly bound group of external system is detected by the governor security measure. It could be shown that if K22 is positive definite, only (BT)£; * 0 could cause * O. EB22 Governor load flow is a global phenomenon in the inter- connected power system in a manner similar to inertial load flow and also depends on the synchronizing power coeffici- ent matrix I, which represents the degree of stiffness of the lines connecting the generators of the system. Instead of generator inertias in inertial load flow, governor frequency response characteristics of generators dictate the distribution of power mismatch in the governor load flow, and its effects are represented by the matrix B. In a more general form, ([11)’1 represents the effects of governor load flow on the power system. The governor security measure (4-99) was shown to be a function of (Bl)'1, and for a strongly connected n 136 generator group of the external system, a function of (BT)§;. This submatrix reflects the structure of the strongly bound group of n generators for governor load flow and will make the security measure reflect strongly bound groups and the relative strengths or weaknesses of the boundaries that surround them. The governor coherency measure for a probabilistic disturbance (4-81) was shown to be solely dependent on (BT)'1. Thus, this coherency measure captures the SSC loss of controllability condition, which causes a group of gen- erators to swing together. The governor security measure is related to the governor coherency measure, and it was shown for the probabilistic disturbance that this security measure is identical to the square of the coherency measure when the base case voltage angles are ignored. Hence, the security measure also captures the SSC property for the governor load flow. The governor security measure summed over all the security measures for single loss of generation distur- bances tests all the boundaries equally. Each boundary is affected by the loss of generation contingencies close to this boundary, and the effect of every loss of generation contingency on all boundaries is captured by the summation of the effects (which is presented by the governor coher- ency measure for a probabilistic disturbance of form (4-81)). Every Bi is proportional to the capacity of gen- . . ~ aPi, erators by the relationship of Bi - —§——, thus, the 137 disturbance is an appropriate disturbance which propor- tionately affects all the boundaries in the system accord- ing to the capacity of generators it connects. The commutative grouping rules require the SSC prop- erty to hold before forming a group. Hence, only strongly bound groups which have "1'1 stiff interconnections com- pared to their governor frequency characteristic in an ni generator group are formed. The boundaries between these groups are vulnerable to governor load flow and are possi- ble candidates for causing stability and security problems. Thus, the security measure (4-99) and boundary identi- fication procedure can detect and identify the weak bound- aries. The procedure for ranking the boundaries is identical to the one described for inertial load flow. In the next section, the boundary identification pro- cedure is carried on the 49 bus test system, and the weak- est boundary for inertial and governor load flow is identified. This boundary is shown to be the boundary between the external and the internal system where each group is a strongly bound group of generators. CHAPTER 5 TESTING THE BOUNDARY IDENTIFICATION METHOD ON THE 49 BUS (EPRI) TEST SYSTEM In this chapter, the method developed in Chapter 4 for identifying and ranking vulnerable boundaries to inertial and governor load flow is applied to the 40 bus test system. First, the inertial DC load flow and governor DC load flow are run for contingencies: i Mi 1:3 {ABM }. = (5-1) J 0 1111' to produce inertial angle changes A3; for i = 1,2,...,22 and: CAP. . B. = —-R—1 l = J {Aml }j = 1 (5’2) 0 1 f j to produce Ag; for i = 1,2,...,22 where 22 is the number of generators in this system. CAPi and M1 are megawatt capacity and inertia of the generator i with the regulation coefficient R. The angles are then used to compute the SM(k,2) and SB(k,2) from equations (4-34) and (4-86), respectively. 138 139 This procedure for computing the inertial and governor security measures is equal to applying the probabilistic modal disturbance to the system for both inertial and gov- ernor load flows and computing the r.m.s. coherency meas- ures. These security measures are ranked for each pair of generators from the smallest to the largest for both the inertial response and the governor response load flows to form ranking tables of the security measures. The group formation tables, which specify strongly bound groups of generators for inertial and governor load flows, are deter- mined by applying the commutative rule to these ranking tables. These groups are merged to form larger groups at each level of group formation. Moving down to the bottom of the ranking table results in a single large system group containing all the generators. The result of this group formation is tabulated in a group formation table. The ranking of the boundaries from the weakest to the strongest is based on the reverse order of the group formation table. That is, the weakest boundary is between the last two groups to be combined together to form a single system group containing all the generators. The above procedure was carried out for the inertial angles obtained from inertial load flow results to the dis- turbance (5-1). Table 5.1 is the ranking table for the SM(k,2) security measure for generator bus pairs. In this table, pairs are specified by generator numbers. 140 Table 5.1. Ranking Table of the Inertial Security Measures. Generator Security Generator Security Rank Pair Measure Rank Pair Measure 1. S (18, 22) .8628 46. S ( 1, 10) 2.4341 2. S ( 7, 8) 1.3336 47. S ( 9, 13) 2.4500 3. S ( 6, 7) 1.3330 48. S ( 2, 7) 2.4549 4. S ( 6, 8) 1.3335 49. S ( 1, 7) 2.4550 5. S ( 1, 2) 1.4813 50. S ( 2, 8) 2.4551 6. S (13, 14) 1.5508 51. S ( 2, 6) 2.4551 7. S ( 2, 17) 1.6118 52. S ( 1, 8) 2.4552 8. S ( 1, 17) 1.6118 53. S ( 1, 6) 2.4552 9. S ( 2, 3) 1.6844 54. S ( 9, 14) 2.4725 10. S ( 1, 3) 1.6844 55. S ( 3, 13) 2.5106 11. S ( 9, 17) 1.8093 56. S (13, 15) 2.5176 12. S ( 2, 4) 1.8276 57. S ( 3, 14) 2.5340 13. S ( 1, 4) 1.8276 58. S (14, 15) 2.5499 14. S ( 9, 10) 1.8581 59. S ( 5, 9) 2.5634 15. S ( 3, 17) 1.8649 60. S (10, 15) 2.5674 16. S (15, 17) 1.9084 61. S ( 3, 10) 2.5834 17. S ( 3, 4) 2.0115 62. S ( 4, 15) 2.6283 18. S ( 2, 5) 2.0188 63. S ( 3, 7) 2.6964 19. S ( 1, 5) 2.0189 64. S ( 3, 8) 2.6965 20. S ( 2, 9) 2.0189 65. S ( 3, 6) 2.6965 21. S ( 1, 9) 2.0190 66. S ( 5, 15) 2.7408 22. S (13, 17) 2.0541 67. S ( 7, 13) 2.7681 23. S (14, 17) 2.0680 68. S ( 8, 13) 2.7683 24. S ( 7, 17) 2.1032 69. S ( 6, 13) 2.7683 25. S ( 6, 17) 2.1034 70. S ( 7, 14) 2.7708 26. S ( 8, 17) 2.1034 71. S ( 8, 14) 2.7710 27. S ( 4, 17) 2.1473 72. S ( 6, 14) 2.7710 28. S (11, 12) 2.1843 73. S ( 4, 13) 2.7778 29. S (10, 17) 2.1855 74. S (10, 13) 2.7809 30. S (12, 21) 2.1879 75. S ( 4, 10) 2.7958 31. S (11, 21) 2.1879 76. S ( 4, 14) 2.7977 32. S ( 3, 9) 2,1936 77. S (10, 14) 2.8007 33. S ( 5, 17) 2.2111 78. S ( 7, 9) 2.8276 34. S ( 3, 5) 2.2115 79. S ( 8, 9) 1.8277 35. S ( 2, 15) 2.2211 80. S ( 6, 9) 2.8277 36. S ( 1, 15) 2.2211 81. S ( 5, 13) 2.8307 37. S ( 9, 15) 2.2642 82. S ( 5, 14) 2.8480 38. S ( 2, 13) 2.3362 83. S ( 5, 7) 2.8673 39. S ( 1, 13) 2.3362 84. S ( 5, 8) 2.8675 40. S'( 4, 5) 2.3497 85. S ( 5, 6) 2.8675 41. S ( 2, 14) 2.3592 86. S ( 7, 15) 2.9267 42. S ( 1, 14) 2.3592 87. S ( 8, 15) 2.9268 43. S ( 3, 15) 2.3844 88. S ( 6, 15) 2.9268 44. S ( 4, 9) 2.4241 89. S ( 5, 10) 2.9415 45. S ( 2, 10) 2.4341 90. S ( 4, 7) 2.9983 141 Table 5.1. (Continued) Generator Security Generator Security Rank Pair Measure Rank Pair Measure 91. S ( 4, 8) 2.9985 136. S (10, 16) 4.3331 92. S ( 4, 6) 2.9985 137. S (13, 16) 4.3578 93. S (17, 19) 3.0306 138. S (14, 16) 4.3812 94. S (18, 21) 3.0791 139. S (19, 20) 4.4045 95. S ( 2, 19) 3.1949 140. S ( 4, 16) 4.4540 96. S ( 1, 19) 3.1949 141. S ( 7, 20) 4.5121 97. S (21, 22) 3.1992 142. S ( 8, 20) 4.5121 98. S (15, 19) 3.2187 143. S ( 6, 20) 4.5121 99. S ( 7, 10) 3.2305 144. S ( 5, 16) 4.5749 100. S ( 8, 10) 3.2306 145. S (16, 19) 4.6789 101. S ( 6, 10) 3.2306 146. S ( 7, 12) 4.7210 102. S ( 9, 19) 3.2606 147. S ( 7, 11) 4.7210 103. S (12, 18) 3.2743 148. S ( 8, 12) 4.7215 104. S (11, 18) 3.2743 149. S ( 8, 11) 4.7215 105. S ( 3, 19) 3.2971 150. S ( 6, 12) 4.7215 106. S (12, 22) 3.3723 151. S ( 6, 11) 4.7215 107. S (11, 22) 3.3723 152. S ( 7, 16) 4.9300 108. S ( 4, 19) 3.4422 153. S ( 8, 16) 4.9300 109. S (10, 19) 3.4991 154. S ( 6, 16) 4.9300 110. S (15, 16) 3.5161 155. S (16, 20) 4.9912 111. S (13, 19) 3.5464 156. S (17, 21) 5.1597 112. S (14, 19) 3.5671 157. S ( 2, 21) 5.4864 113. S ( 5, 19) 3.6248 158. S ( 1, 21) 5.4864 114. S (17, 20) 3.6883 159. S (14, 21) 5.5907 115. S (15, 10) 3.7847 160. S (13, 21) 5.6051 116. S ( 9, 20) 3.8221 161. S (12, 17) 5.6643 117. S ( 2, 10) 3.8594 162. S (11, 17) 5.6643 118. S ( 1, 20) 3.8594 163. S ( 3, 21) 5.6775 119. S ( 3, 20) 3.9384 164. S ( 5, 21) 5.7219 120. S (10, 20) 4.0074 165. S ( 9, 21) 5.8295 121. S (13, 20) 4.0160 166. S (15, 21) 5.8881 122. S (14, 20) 4.0376 167. S ( 4, 21) 5.9293 123. S (16, 17) 4.0699 168. S ( 2, 12) 5.9810 124. S ( 7, 19) 4.0898 169. S ( 2, 11) 5.9810 125. S ( 8, 19) 4.0899 170. S ( 1, 12) 5.9811 126. S ( 6, 19) 4.0899 171. S ( 1, 11) 5.9811 127. S ( 4, 20) 4.1060 172. S (12, 14) 6.0673 128. S ( 9, 16) 4.1631 173. S (11, 14) 6.0673 129. S ( 7, 21) 4.1782 174. S (12, 13) 6.0810 130. S ( 8, 21) 4.1787 175. S (11, 13) 6.0810 131. S ( 6, 21) 4.1787 176. S (10, 21) 6.1334 132. S ( 5, 20) 4.2168 177. S ( 3, 12) 6.1640 133. S ( 2, 16) 4.2284 178. S ( 3, 11) 6.1640 134. S ( 1, 16) 4.2284 179. S ( 5, 12) 6.2114 135. S ( 3, 16) 4.2924 180. S ( 5, 11) 6.2114 142 Table 5.1. (Continued) Generator Security Generator Security Rank Pair Measure Rank Pair Measure 181. S ( 5, 11) 6.2114 206. S (14, 18) 7.9041 182. S ( 9, 12) 6.3104 207. S (13, 18) 7.9249 183. S ( 9, 11) 6.3104 208. S (12, 16) 7.9329 184. S (12, 15) 6.3622 209. S (11, 16) 7.9329 185. S (11, 15) 6.3622 210. S ( 5, 18) 8.0092 186. S ( 4, 12) 6.4150 211. S ( 3, 18) 8.0195 187. S ( 4, 11) 6.4150 212. S ( 2, 22) 8.0263 188. S ( 7, 18) 6.4472 213. S ( 1, 22) 8.0263 189. S ( 6, 18) 6.4477 214. S (14, 22) 8.0869 190. S ( 8, 18) 6.4477 215. S (13, 22) 8.1088 191. S (10, 11) 6.6059 216. S ( 9, 18) 8.1735 192. S ( 7, 22) 6.6182 217. S ( 5, 22) 8.1824 193. S ( 6, 22) 6.6187 218. S ( 3, 22) 8.2014 194. S ( 8, 22) 6.6187 219. S (15, 18) 8.2278 195. S (19, 21) 6.8692 220. S ( 4, 18) 8.2314 196. S (20, 21) 7.1077 221. S ( 9, 22) 8.3569 197. S (12, 19) 7.3194 222. S ( 4, 22) 8.4064 198. S (11, 19) 7.3194 223. S (15, 22) 8.4128 199. S (16, 21) 7.5230 224. S (10, 18) 8.4310 200. S (17, 18) 7.5278 225. S (10, 22) 8.6094 201. S (12, 20) 7.5280 226. S (18, 19) 9.1063 202. S (11, 20) 7.5280 227. S (19, 22) 9.2806 203. S (17, 22) 7.7109 228. S (18, 20) 9.3111 204. S ( 2, 18) 7.8439 229. S (20, 22) 9.4917 205. S ( 1, 18) 7.8439 230. S (16, 18) 9.7169 231. S (16, 22) 9.8983 143 The commutative rule was applied to this ranking table. The group containing generators 18 and 22 (18, 22) is the first group of strongly bound generators. At rank 2 of the ranking table,group (7, 8) is formed as the second group. At rank 3, generators 6 and 7 do not form a group since 7 is already a member of group (7, 8),and generator 6 cannot join this group because at this rank generator 6 has not been shown to have a tight connection to 8 with respect to its inertia. At rank 4, generator 6 is combined with the group (7, 8), forming a new group of (7, 8, 6) since generator 6 is now tightly bound with both generators 7 and 8. Moving down to the bottom of the ranking table results in larger and larger groups which later are merged and finally form a single generator. This is shown in group formation Table 5.2, which also shows the ranks of ranking table at which groups are merged or a new pair group is formed. Level 20 of Table 5.2 shows the last two groups. Group No. 1 identifies the external system generator, and group No. 2 contains all the generators in the inanmal sys- tem. Hence, for this system, the boundary which connects the external and the internal system is the weakest bound- ary to inertial power flow due to loss of generation con- tingencies. This was shown in Chapter 3 to be true, since this boundary was shown to have the largest angle across the lines forming the boundary for several loss of genera- tion contingencies. Table 5.2. 144 Group Formation Table Based on Inertial Security Measure of Ranking Table. Level Rank Group 1O 11 14 15 27 28 31 DON-P mthN—b (flabUON-fi 01-5de mwa—P wa-i me-P DON—b N—* N-9 —- (ML 22) (13, 22) (7. 8) (18, 22) (7. 8. 6) (18, 22) (7. 8. 6) (1. 2) (18, 22) (7. 8. 6) (1. 2) (13, 14) (18, 22) (7. 8. 6) (1, 2,17) (13, 14) (ML 22) (7: 89 6) (1. 2.17) (13,14) (9. 10) (18, 22) (7. 8. 6) (1, 2, 17, 3) (13, 14) (9. 10) (18, 22) (7. 8. 6) (1, 2, 17, 3) (13, 14) (9. 10) (18, 22) (79 89 6) (1. 2. 17.3.4) (13, 14) (9. 10) (11, 12) (ML 22) (7. 8. 6) (1. 2.173 3.1” 145 Table 5.2. (Continued) Level Rank Group 11 (cont) 31 (13, 14) (9,10) (11, 12, 21) (18, 22) (7. 8. 6) (1, 2, 17, 3, 4, 5) (13, 14) (9. 10) (11, 12, 21) (18, 22) (7.8. 6) (1, 2, 17, 3, 4, 5) (13, 14, 15) (9. 10) (11, 12, 21) (18, 22) (7. 8. 6) (1, 2, 17, 3, 4, 5) (13, 14, 15, 9, 10) (11, 12, 21) (18, 22) (7. 8. 6) (1, 2, 17, 3, 4, 5, 13, 14, 15,9, 10) (18,22) (7,£L 6,1, 2,17,1L 4,!L 13,14, 15,9, MD (TL 12,21) (18, 22, 11, 12, 21) (7,EL 6,1, 2,17,2L 4,!L 13,14, 15,9, MD (ML 22,11, ML 21) 2 (7,€L 6,1, 2,173 3,43 5,13, ML 15,9, ML 19) 19 143 1 (ML 22,11, ML 21) 2 (7,£L 6,1, 2,17,2L 4,5L 13,14, 15,9, ML 19,20) 20 155 1 (18, 22, 11, 12, 21) 2 (79 8: 6: 1: 29 17s 3: 49 59 139 14a 15, 9, 10, 19, 20, 16) 21 231 1 (ML 22,‘H, 12,2H, 7,£L 6,1, 2, 17,3,1L 5,13,14, ML 9,10,19, 20, 16) 12 4O 13 58 14 77 15 89 16 101 17 107 hJ-h 0‘ RJ—* UJRD—b C‘h-Q)RD—h CTU1¢DQJRJ—I OWU1¢IUJhJ-b OWUT¢D 18 126 d 146 In a manner similar to that for the inertial security measure, the governor security measure SB (k,2) was com- puted for disturbance given in equation (5-2) using the DC governor load flow. The ranking table for the governor security measure was formed. The commutative rule was applied to the governor security measure ranking table from the top to the bottom. The final result is presented in Table 5.3. This table contains the strongly bound groups of generators at different levels of group formation for governor response of each generating unit. Groups formed based on the governor security measure shown in Table 5.3 are somewhat different from those given in Table 5.2. However, the last two groups in Tables 5.2 and 5.3 are identical. This indicates that the weakest boundary for governor power flow in this system is also between the internal and external system. This is also true since the loss of stability across this boundary was shown to occur for governor power flow due to loss of gen- eration contingencies in the external system. The results of Chapter 3 showed that the inertial and the governor security measures with commutative grouping procedure accurately detected the weakest boundary for inertial and governor power flow in this system. The investigation of the second and third weakest boundaries was not performed in this and is a subject of future research. 147 Table 5.3. Group Formation Based on Governor Security Measure. Level Rank Group (18,22) (18,22) (15, 16) (18,22) (15,16) (13, 14) (18,22) (15, 16) (13,14) (17,20) (ML 22) (15,16) (13, 14) (17,20) (7. 8) (18, 22) (15, 15) (13, 14) (17, 20) (7.8. 6) (18, 22) (15, 16) (13, 14) (17, 20) (7.8. 6) (2. 3) (18, 22) (15, 16) (13, 14) (17, 20) (7. 8. 6) (2. 3. 1) (18, 22) (15, 16) (13, 14) (17, 20) (7. 8. 6) (2. 3. 1. 4) (18,22) (15, 16) (13, 14) 10 25 (AN-9 mm4>wm—- mthN—P mthN—P macaw—e (Tl-DOOR)"m bWN—P (JON—P I\)-'b —- Table 5.3. (Continued) 148 Level Rank Group 1O (cont) 11 12 13 14 15 16 17 25 28 38 58 6O 82 91 103 G>\J RJ-i GD\JUW hJ-* \JUT RJ-b \JOWUTRJ—b \JO\UIQJR)-i \JOWUWQJRJ-D \JOWUIQ)RJ—I \JOIUW¢D (HR 20) (7. 8. 6) (2. 3. 1. 4) (11,12) (18, 22) (15, 15, 17, (13, 14) (7. 8. 6) (2’ 3) 1’ 4) (11, 12) (ML 22) (15, 16, 17, (13,14) (7. 8. 6) (29 39 19 4a (11, 12) (ML 22) (15, 16, 17, (13, 14) (7. 8. 6) (29 39 1: 4, (11,12, 21) (ML 22) (15, 16, 17, (7. 8. 6) (29 39 1: 49 CH, 12, 21) (18, 22) (15, 16, 17, 4, 5) (7. 8. 6) (11, 12, 21) (ML 22) (ML 16,17, 4, 5) (7s 89 6) (11,125 21) (9. 10) (ML 22) (15, 16, 17, 20) 20) 20) 5) 20, 13, 14) 5) 20, 13, 14, 2, 3, 11 20, 13, 14, 2, 3. 11 20, 13, 14, 1, 2, 11 4. 5. 7. 8. 6) (11, 12, 21) (9. 10) 149 Table 5.3. (Continued) Level Rank Group 18 131 (ML 22,11, ML 21) (15, 16, 17, 20, 13, 14, 2, 3, 1, 4 EL 7,EL 6) (3, 10) (ML 22,11, ML 21) (15, 16, 17, 20, 13, 14, 2, 3, 1, 4,!L 7,£L 6) (9, 10, 19) (ML 22,11, ML 21) (15, 16, 17, 20, 13, 14, 2, 3, 1, 4,5L 7,£L 6,SL 10,19) 21 231 1 (18, 22, 11, 12, 21, 15, 16, 17, 20, 13,14,2L 3,1, 4,5,]L 8,€L 9) 19 140 20 146 N-P oo N-b (D N—h CHAPTER 6 CONCLUSIONS AND FUTURE INVESTIGATION 6.1. Overview of Thesis A review of present planning methods was presented, and the lack of detailed planning procedures for the possi- ble stability and security violation caused by inertial and governor response to loss of generation contingencies was pointed out. The presently available simulation methods for gover- nor and inertial response to loss of generation contingen- cies were also reviewed. The lack of simulation techniques that can handle the large data base required to simulate the inertial and governor response on large interconnected systems was pointed out. The need to develop simulation methods and contingency assessment methods for these iner- tial and governor transient responses to loss of generation contingencies was noted. In the second chapter, the dynamic distribution of power mismatch to loss of generation contingencies was dis- cussed in detail, and it was shown that the generating units respond to the mismatch: first, proportional to syn- chronizing power coefficient of equivalent lines connecting the generation to the disturbed bus; second, proportional 150 151 to their inertia; third, by governor action proportional to governor frequency response characteristics; and finally, by automatic generation control participation factors or the operator. Chapter 3 also developed a simulation method for governor load flow to complement the inertial load flow. These governor and inertial load flows allow direct assess— ment of stability and security problems due to generation response to loss of generation contingencies on large net- works that is presently not possible. The inertial and gov- ernor power flows were indicated to be different and cause different stability problems. The reasons given are: (a) Mi is not proportional to CAPi for generators of dif- ferent size and type. (b) The effective Bi is reduced on generation far from the disturbed bus due to governor deadband and value set point nonlinearities. (c) Some types of units may not have governor regulation or sharply reduced regulation participation. (d) Some utilities' automatic generation controls dis- patch in proportion to area control error as well as the integral of area control error. When AGC has proportional control, the effective Bi on generators under AGC is increased. Chapter 3 demonstrated the performance of the DC iner- tial and governor load flow on a 49 bus test system. The results of DC inertial and governor load flow for a 490 MW loss of generation at the external system, when compared 152 with the Midterm Stability Program results for this contin- gency, showed that the inertial and governor power flow angle changes were captured reasonably accurately by the DC load flow methods. The results also indicated that the power flow for inertial response of generating units is different from the governor power flows. The 790 MW loss of generation at the external system caused a line stability limit violation over the boundary between the external and internal system since the governor response of generating units in the internal system was much greater than the inertial response. This further jus- tified the difference between inertial and governor power flows and their uniqueness in causing different stability and security problems. This 790 MW loss of generation, when simulated on the 49 bus test system using regular load flow, did not cause any stability limit violation. The line outage study of line (40, 41), which exceeded its stability limit due to governor power flow, also did not indicate any problem across the internal and external system. These two simulation runs showed that the stability and security problem associated with inertial and governor power flow cannot be assessed by present load flow and line outage studies. The use of the Midterm Stability Program for assessing inertial and governor power flow stability prob- lems is prohibited because of its cost and inability to handle large data bases. Thus, the inertial and governor 153 load flow method is a significant contribution to power system planning and security assessment studies. In Chapter 4, a set of security measures was proposed for the inertial and governor time frame. The inertial security measure was shown to be identical to the square of the r.m.s. coherency measure evaluated for the linearized classical transient stability model for a probabilistic dis- turbance when the base case was ignored. This security measure was further justified by showing that the strict synchronizing loss of controllability condition of this linearized classical stability model, which causes the vul- nerable boundaries in the transmission network, is detected by this security measure. It was argued that the inertial security and stability problem depends on SSC loss of controllability condition since the groups formed by this property have weak bound- aries. The cummutative grouping algorithm was chosen to guarantee that the groups formed are strongly bound groups. Thus, the inertial security measure and commutative group- ing algorithm were used to identify the vulnerable bound- aries to inertial power flows. In Chapter 4, the governor security measure was also defined. It was shown that the governor power flow security problem depends on the state model for this time frame and SSC condition for this model. It was also argued that the governor security measure and the commutative 154 governor power flows due to loss of generation contingen- cies. In both the inertial and governor boundary identifica- tion methods, the weakest boundaries lie between the last two groups of strongly bound groups to be combined together to form a single group. The results of testing the vulnerable boundary identification method on the 49 bus EPRI system were summarized in Chapter 5 and indicated that the method worked very well since the weakest boundary identified was the boundary which caused stability limit violation shown in Chapter 3. Knowing weak boundaries gives planners and operators the insight needed for transfer limit studies. The bound- ary vulnerability ranking can assist operation planners in adjusting unit commitment, economic dispatch, and line maintenance schedules to minimize or eliminate the most significant boundary and line vulnerabilities. It can also assist expansion transmission planners to assess security of various alternate expansion configurations and their effect on present network vulnerabilities and new vulnera- bilities produced by each expansion alternative. 6.2. Future Research The results obtained for inertial and governor response of generators to loss of generation contingencies in Chapter 3 are based on a linearized DC load flow. A useful investigation would be the comparison of the 155 performance of inertial and governor decoupled load flow with the midterm stability simulation in an effort to obtain more accurate results than achieved with the DC load flow used in this research. The Consumer Power Com— pany's decoupled load flow could be a prime candidate for this study. This program, at present, can handle 3,000 buses, 5,500 lines, 750 generators, and 750 transformers and can be made, with pr0per input data, to compute iner- tial and governor load flow. This program is almost the size needed to handle the large data base required for analyzing the power transfers to the Northeast United States. Thus, this program is an excellent candidate for performing the inertial and governor load flows for large interconnected networks. Another useful investigation would be the development of a better governor model for simulating loss of genera- tion contingencies on large interconnected networks where the average frequency deviation would be below governor deadband (.036 hz). The loss of generation must be above 1.2% of total system capacity to exceed governor deadband. If the loss of generation were not above governor deadband for all generators, no generators in the system could respond to this generation loss. The transient after the loss of generation affects generators electrically close much more than those further away and will cause these gen- erators that are electrically close to exceed governor deadband. An algorithm for determining which generators 156 participate in governor generation response in large inter- connected systems is thus another subject of future research. The line power flow measurement for the recent Ontario Hydro Nanticocke 2000 MW loss of generation is available. Thus, the post mortem on this Nanticocke 2000 MW loss of generation is a valuable investigation using the inertial and governor load flow. An investigation of this actual contingency with the inertial and governor load flow can: (1) determine the principal transmission network for those regions that participate in governor regulation and can lead to proper modeling of governor load flow for large networks; (2) give an insight on how to determine a proper base case load flow and an equivalent for inertial load flow. This can be based on knowledge of generation availa- ble, generation loading, transfers, and network con- figurations. A new security analysis procedure that utilizes the information developed by identifying the weak base case boundaries in the system could be developed. This secur- ity analysis procedure involves: (1) determining the groups of generators that are sepa- rated by the weakest transmission boundaries; (2) determining the network elements that comprise these base case transmission boundaries; (3) 157 identifying network elements that are vulnerable to stability or security violations for all loss of gen- eration contingencies, all line outage contingencies, and all loss of generation/line outage combination contingencies. The security or stability vulnerabil- ity of a network element could be assessed based on an r.m.s. average over all contingencies as well as the enumeration and ranking of specific contingencies that caused overload or stability constraint viola- tions for this element. Vulnerable elements that lie in weak boundaries and that do not lie in the weak base case boundary could be enumerated. The assess- ment of network element security by an r.m.s. average over all contingencies of a particular type deter- mines elements and ranking that are continually vul- nerable,and the enumeration of contingencies that cause overload induxnes elements that are vulnerable for a very specific contingency or contingencies; a contingency security analysis procedure that could determine the effects of a specific loss of genera— tion, line outage, or loss of generation/line outage combination contingency on stability or security by r.m.s. average of the effects on all vulnerable net- work elements for that contingency type as well as enumerating and ranking the overloaded elements for a specific contingency. The r.m.s. average over vulner- able network elements pick out contingencies that have 158 a rather pronounced widespread effect on network security and the enumeration of overloads indicates the effects of contingency on a specific element or elements. The loss of generation line outage or loss of generation/line outage combination contingencies that affect and do not affect the base case boundary network could be identified. The identification of vulnerable network elements based on a security measure based average over all contin- gencies of a particular type and the identification of whether these elements belong to the weak base case bound- aries will determine whether these weak base case bound- aries dictate the security problems a utility experiences as the transmission network is weakened by line outages or is stressed by line outage/loss of generation combinations. This information would be helpful to operation and expan- sion planners. The security measure based average overall contingencies of a particular type will indicate how vul- nerable a network element is on the average, which is not possible to assess if only overload or stability limit vio- lations for specific contingencies are enumerated. Using sums of the security measures for vulnerable network ele- ments as a system security measure would allow planners or operators to assess specific operation or expansion alter- natives and even perform such an evaluation using optimiza- tion procedures. Thus, the investigation of network 159 element security measures and procedures for operation and expansion planning is a tOpic for further research. The identification of contingencies that cause loss of security or stability based on a security measure average over all network elements or all weak base case boundary network elements for a specific contingency indicates the effect of that contingency on network security better than just enumerating overloads or stability limit violations for that contingency alone without the contingency security measure evaluation. 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