Fast calculation of Thévenin equivalents for realtime steady state voltage stability estimation

Transcription

1 Graduate Theses and Dissertations Graduate College 2016 Fast calculation of Thévenin equivalents for realtime steady state voltage stability estimation Jorge Alberto Vélez Vásquez Iowa State University Follow this and additional works at: Part of the Electrical and Electronics Commons Recommended Citation Vélez Vásquez, Jorge Alberto, "Fast calculation of Thévenin equivalents for real-time steady state voltage stability estimation" (2016). Graduate Theses and Dissertations This Thesis is brought to you for free and open access by the Graduate College at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact

2 Fast calculation of Thévenin equivalents for real-time steady state voltage stability estimation by Jorge Alberto Vélez Vásquez A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Electrical Engineering Program of Study Committee: Venkataramana Ajjarapu, Major Professor Umesh Vaidya Siddhartha Khaitan Iowa State University Ames, Iowa 2016 Copyright c Jorge Alberto Vélez Vásquez, All rights reserved.

3 ii DEDICATION I would like to dedicate this thesis to my wife Catalina, my mother Bertha Inés and my brother Luis David without whose support, encouragement and loving guidance, I would not have been able to complete this work.

4 iii TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS ABSTRACT viii ix CHAPTER 1. INTRODUCTION CHAPTER 2. THE VOLTAGE STABILITY PROBLEM Introduction Voltage stability and voltage collapse Characteristics of the power systems affecting voltage stability Long-term voltage stability CHAPTER 3. REAL-TIME MONITORING FOR LONG-TERM VOLTAGE STABILITY Introduction Classification of approaches for long-term voltage instability detection Thévenin equivalents for log-term voltage stability analysis Literature review Voltage instability detection based on the concept of the single-port circuit Voltage instability detection based in the concept of coupled single-port circuit Voltage instability detection using linear index

5 iv 3.4 Determination of voltage stability margins based on the impedance matrix Calculation of Thévenin equivalents based on the impedance matrix Comparison of voltage stability margins using information of the admittance matrix and the impedance matrix CHAPTER 4. USING THE MATRIX INVERSION LEMMA AND KRON REDUCTIONS FOR FAST CALCULATION OF THÉVENIN EQUIV- ALENTS Introduction The matrix inversion lemma Admittance matrix modifications Updating the impedance matrix considering shunt elements Computational efforts for using matrix inversion lemma and Kron reductions. 40 CHAPTER 5. UPDATING THÉVENIN EQUIVALENTS IN REAL-TIME Introduction General description of the process Subprocess: Calculate VSMs Subprocess: Modify the Z bus with MIL and Kron reductions General comments CHAPTER 6. APPLICATIONS AND RESULTS CHAPTER 7. CONCLUSIONS CHAPTER 8. FUTURE WORK APPENDIX A. PROOF OF MATRIX INVERSION LEMMA APPENDIX B. BUS IMPEDANCE MATRIX MANIPULATION APPENDIX C. IEEE 118 BUS TEST SYSTEM DATA BIBLIOGRAPHY

6 v LIST OF TABLES Table 2.1 Some voltage stability incidents along the history (Taylor et al. (1994)) 13 Table 3.1 Comparison of parameters to calculate Thévenin equivalents Table 4.1 Complexity of operations using big-o notation (Wikipedia (2006)).. 41 Table 6.1 Errors in estimation of voltage stability margins based on Z bus Table C.1 IEEE 118-bus test system - Bus data Table C.2 IEEE 118-bus test system - Generator data Table C.3 IEEE 118-bus test system - Branch data

7 vi LIST OF FIGURES Figure 1.1 Classification of power system stability Figure 1.2 Time frame for voltage stability problems (Taylor et al. (1994)) Figure 2.1 Simple radial system feeding a load Z LD Figure 2.2 Normalized voltage, current and power in the system of Figure Figure 2.3 P-V curve for various power factors of the load Figure 3.1 Classification of long-term voltage instability detection according to Glavic and Cutsem (2011) Figure 3.2 Lumping of T- and Thévenin equivalents of a transmission corridor and surrounding environment (Zima et al. (2005)) Figure 3.3 Multi-port network system model (Wang et al. (2011)) Figure 3.4 Graphical representation of multi-port Thévenin equivalent (Wang et al. (2011)) Figure 3.5 Characteristics of the virtual loads. (a) Active power ratio P ratio (%). (b) Reactive power ratio Q ratio (%) (Wang et al. (2011)) Figure 3.6 Coupled voltages at different buses (each line for a different bus) (Wang et al. (2011)) Figure 3.7 Characteristics of the virtual impedance (Wang et al. (2011)) Figure 3.8 Voltage locus for E S = 1, power factor of -0.93, 1.0 and 0.93 (Matavalam and Ajjarapu (2014)) Figure 3.9 Linear index calculated using information of Y bus Figure 3.10 Graphical representation of sparsity for Y bus and Z bus matrices

8 vii Figure 3.11 Comparison between indices calculated using information of Y bus and Z bus Figure 4.1 Simple interconected power system Figure 4.2 Computation complexity of different big-o functions Figure 5.1 Flow chart for fast calculation of Thévenin equivalents for real-time steady state voltage stability estimation Figure 6.1 IEEE 9-bus test system Figure 6.2 Indices of voltage stability for IEEE 9-bus test system using information of Z bus Figure 6.3 Voltage Stability Indices under an n 1 contingency Figure 6.4 Linear index (Index 2) under an n 1 contingency Figure 6.5 Thévenin equivalents for bus #60 in IEEE 118-bus system Figure 6.6 Behavior of linear index (index 2) under different contingencies in IEEE 118-bus system Figure 6.7 Voltage stability margin comparison by updating Z bus Figure C.1 One-line diagram of IEEE 118-bus test system Figure C.2 Comparison of linear index using Y bus and Z bus in IEEE 118-bus test system

9 viii ACKNOWLEDGEMENTS I would like to thank Dr. Venkataramana Ajjarapu from the Electrical and Computer Engineering Department at Iowa State University for his guidance and help in all the time of research and writing of this thesis. I also like to thank the professors of my thesis committee: Dr. Umesh Vaidya and Dr. Siddhartha Khaitan for their valuable comments. My sincere thanks to Amarsagar Reddy Ramapuram Matavalam for his continuous help and support during my research. Last but not least, I want to thank my wife Catalina for her support, enthusiasm and help in the edition of this thesis.

10 ix ABSTRACT The voltage stability phenomenon has been a longtime major concern in the operation of power systems. This phenomenon appears when active and reactive power resources in a power system are not enough to supply the demand of the loads. Currently, with high penetration of renewable sources of energy like wind and solar, this problem is more severe because those resources represent a limited contribution of reactive power to respond to load increments. This situation becomes even worse if during heavy loaded conditions disconnections of electrical equipment (n k contingencies) occur. Those contingencies can be the result of programmed outages in the grid caused by maintenances and other unavailabilities or of nonprogrammed disconnections caused by the operation of protective relaying. Under these situations, the voltage stability of the system is compromised, and the risk of a voltage collapse is seriously increased. Therefore, numerous opportunities arise to develop and implement real-time tools to assist operators in taking proper actions toward an increment of the reliability of the network. One of the most studied tools to prevent voltage stability collapses is the estimation of voltage stability margins (VSMs) based on Thévenin equivalents of the grid at load buses. However, important aspects for a proper implementation of these tools in real systems are still under study. One of these aspects is to ensure the performance of real-time Thévenin equivalents of the network when it is subjected to equipment disconnections (n k contingencies). This study aims to present a novel approach to quickly update Thévenin equivalents of multinodal systems, which can be used to calculate long-term VSMs for real-time applications considering n k contingencies in the grid. To accomplish this purpose, a derivation of the Thévenin equivalent parameters of a multiport system is proposed. These parameters are expressed in terms of the elements of the impedance matrix of the network (Z bus ). A new methodology to quickly update the Z bus of the system when n k contingencies occur is explained using the matrix inversion lemma and Kron reductions. The speed of this approach

11 x is caused by the avoidance of inversion of matrices that is required when Thévenin equivalents are calculated using information of the admittance matrix of the network (Y bus ). Instead of this, only matrix multiplications between Z bus element and other vectors are necessary to obtain Thévenin equivalents. This approach proposes a solution for some of the current issues associated with the implementation of long-term VSMs in real-time systems. The structure of this work is organized as follows: Chapter 1 presents an introduction of this work; Chapter 2 describes the voltage stability problem and its characteristics; Chapter 3 presents a general review of the state of the art in using Thévenin equivalents for voltage stability analysis and explains in detail two recent developments to determine long-term voltage stability issues in real-time using those equivalents. Additionally this chapter will present will present a new method of determining Thévenin equivalents of the system based on the information of the Z bus. In Chapter 4, two effective tools of linear algebra to update matrices are presented as well as its application to determine Thévenin equivalents; Chapter 5 will describe the methodology proposed in this work to quickly update Thévenin equivalents and to calculate VSMs when the topology of the system changes due to n 1 contingencies. The proposed methodology is tested in the IEEE 9-bus and IEEE 118-bus systems and the results are presented and analyzed in Chapter 6. The conclusions of the work are presented in Chapter 7 and future work is proposed in Chapter 8.

12 1 CHAPTER 1. INTRODUCTION The continuous increase of loads in a power networks leads to the operation of this systems close to its limits. When those limits are related to the stability of the system (i.e., its ability to return to the initial conditions after having been subjected to a contingency), different categories of this phenomenon appears. These categories depend on the magnitudes affected and their duration. Stability of power systems can be classified as in Kundur et al. (2004), which is a widely accepted categorization of this phenomenon based on the variables of the system that are affected under a disturbance (Figure 1.1). This characterization allows an efficient way to separate efforts towards a practical solution to the industry needs. Based on this classification, this study will address some problems related to long-term voltage stability. Particularly, a thesis to solve the problem of quickly updating some voltage stability margins (VSMs) will be presented. This work will contribute to the efforts in implementing these margins in real-time systems. In general terms, the long-term voltage stability problem is characterized by a slow decrement of the voltage in a power system caused by the increase of its load followed by a rapid decline in the magnitude of the voltage when the system is close to instability. This problem becomes worse if during the load increase, disconnection of elements occurs. In this situation, the resulting topology of the system generates an increased reactive power transferred to the load in certain parts of the system, accelerating the conditions for a voltage collapse. The period when this type of phenomenon happens can vary between a few seconds to several minutes (Kundur et al. (1994)). Given this relatively long period of time, this phenomenon is called long-term voltage stability (marked in blue in Figure 1.1).

13 2 Figure 1.1 Classification of power system stability The long-term voltage stability analysis primarily aims to identify solutions that can be executed in the real-time operation of power systems. As shown in Figure 1.2, the time frame of long term voltage stability issues in power systems should allows system operators to take actions to mitigate this condition. Consequently, the speed of the calculations required to provide signals to system operators or to develop automatic schemes to avoid voltage collapse is a pressing need. In fact, the most recent methodologies to detect voltage instabilities are oriented to the use of centralized information of the network such as the information provided by SCADA/EMS and synchrophasor systems (Glavic and Cutsem (2011)) to timely determine this risky conditions. The analysis of voltage stability has been a research subject in recent years. Indeed, the use of synchronized measurements in power systems has demonstrated potential applications to address voltage stability issues. Hence, increasing efforts are exerted to incorporate synchrophasors as well as real-time information to improve the voltage stability of power systems, specifically by addressing long-term voltage stability issues.

14 3 Figure 1.2 Time frame for voltage stability problems (Taylor et al. (1994)) Within the current issues that should be tackled in terms of long-term voltage stability, the following are highlighted: Including the effects of disconnection of elements of the grid during large-scale voltage stability events Reaching reactive power limits of synchronous generators Including the actuation of devices that change the reactive power flow into the system, such as on-load tap-changers (OLTCs), static var compensators (SVCs), and flexible alternating current transmission system (FACTS) In particular, given that voltage stability problems are precipitated by single or multiple contingencies (Hartmann et al. (1996)), the detection of these disconnections and a fast update of VSMs are required for a proper application in real-time systems. This allows system operators take appropriate actions to mitigate the effects of those contingencies.

15 4 Based on two recent approaches for the analysis of voltage stability problems using Thévenin equivalents, this work will present a novel methodology for a fast calculation of parameters of the system that are required to obtain Thévenin equivalents when the system is subjected to contingencies or disconnections. These equivalents can be then used to estimate and update the VSMs of the system in real-time. As stated in Glavic and Cutsem (2011) and Kundur et al. (1994), the inclusion of equipment outages in the real-time calculations of VSMs is part of the recent challenges in this subject. Thus, the methodology proposed in this thesis presents a set of expressions to determine Thévenin equivalents based on the information of the impedance matrix of the system (Z bus ), and incorporates two mathematical tools of linear algebra to quickly update this matrix when disconnection of equipment occurs in real-time applications. The linear algebra tools are the matrix inversion lemma and the Kron reductions. The result of this thesis is organized as follows: Chapter 2 will present a detailed description of the phenomenon of voltage stability over its theoretical basis. This discussion will assist in understanding the issues currently addressed in this subject, particularly those related with long-term voltage stability. Chapter 3 will present a review of the state of the art in using Thévenin equivalents for voltage stability analysis and will explain the fundamentals of two approaches in determining voltage instabilities by using synchrophasor measurements. These approaches will be used to apply and validate the methodology proposed in this work. Furthermore, this chapter will present the derivations to obtain Thévenin equivalents from the Z bus of the system. Chapter 4 will present the mathematical background of the matrix inversion lemma (MIL) and Kron reductions, and the application of these tools to update the Z bus of a power system. Chapter 5 will describe the methodology to integrate the elements proposed in this work for a fast calculation of Thévenin equivalents for long-term voltage stability estimation. Chapter 6 will present examples of applications of this approach in IEEE test systems and the results obtained. Chapter 7 will present the conclusions of this work and Chapter 8 the future work proposed.

16 5 CHAPTER 2. THE VOLTAGE STABILITY PROBLEM 2.1 Introduction The most accepted definition of voltage stability describes this phenomenon as the ability of a power system to maintain acceptable voltages at all buses under normal conditions and after being subjected to a disturbance (Kundur et al. (1994)). Consequently, the system enters a voltage instability condition when because of a disturbance, a load increase, or a change in system conditions, a progressive and uncontrollable degradation of the voltage level occurs. The main factor that causes voltage instability is the inability of the power system to meet the active and reactive power demanded by the loads. This chapter will explain the voltage stability phenomenon and will describe the key factors that should be considered in voltage stability studies. 2.2 Voltage stability and voltage collapse This section will explain in detail the voltage stability phenomenon following the methodology proposed by Kundur et al. (1994). This author present the classical way to understand the voltage stability problem by analyzing the maximum power transferred from a source with voltage E S to a load of impedance Z LD (Figure 2.1). The current in the system in Figure 2.1 is obtained by the following expression: E S I = Z LN + Z LD where Z LN is the impedance of the transmission system. From Figure 2.1, assuming θ s = 0, the magnitude of the current in the circuit is given by the following expression:

17 6 Figure 2.1 Simple radial system feeding a load Z LD I = E S (ZLN cos θ + Z LD cos φ) 2 + (Z LN sin θ + Z LD sin φ) 2 = E S ZLN 2 (cos2 θ + 2 Z LD Z LN cos φ cos θ + Z2 LD cos Z 2 φ + sin 2 θ + 2 Z LN 2 LD Z LN sin φ sin θ + Z2 LD sin 2 φ) ZLN 2 = 1 F E S Z LN (2.1) where: F = 1 + ( ZLD Z LN ) Z LD Z LN cos (θ φ) (2.2) The expressions and normalized representations of voltage, current, and power in the system in terms of the ratio between the impedance of the system and the impedance of the load, can be obtained as follows: V R = Z LD I = 1 F Z LD Z LN E S (2.3) Then, normalizing with respect to E s, the voltage in the load is as follows: V R = 1 Z LD (2.4) E S F Z LN The maximum short circuit current in the system (I sc ) is given by the following expression: I sc = E S Z LN (2.5) Therefore, from Equation 2.1, the normalized current can be expressed as follows: I I sc = 1 F (2.6)

18 7 Figure 2.2 Normalized voltage, current and power in the system of Figure 2.1 Finally, the active power in the load, (P R ), is given by the follows: P R = V R I cos φ = Z LD F ( ES Z LN ) 2 cos φ = 1 Z LD ES 2 cos φ (2.7) F Z LN Z LN The maximum active power to the load (P RMAX ) can be defined as: P RMAX = E2 S Z LN (2.8) Hence, Equation 2.7 can be expressed as: P R = 1 F and the normalized power will be given by the following: Z LD Z LN P RMAX cos φ (2.9) P R P RMAX = 1 F Z LD Z LN cos φ (2.10) For a power factor of 0.95, tan θ = 10, and with a proper scale for P R /P RMAX, a graphical representation of Equations (2.4),(2.6) and (2.10) is shown in Figure 2.2. This figure, shows that the maximum power transferred to the load is reached when the impedance of the load equals the impedance of the source (i.e., when Z LN /Z LD = 1). At this point, the system voltage magnitude E s also equals the load voltage magnitude V R. For lowest ratios of Z LN /Z LD (i.e.,

19 8 when the load impedance is greater than the network impedance), the system responds by increasing the power delivered to the load. The opposite occurs when the impedance of the load is lower than the network impedance (i.e., when Z LN /Z LD > 1). In this case, the system responds by decreasing the power delivered to the load as a consequence of the decrement of the voltage at the load (Z LN = Z LD ), that dominates over the current increase. Therefore, the maximum transfer power point represents the limit of stable operation of the system. Operations beyond that point will represent a voltage collapse. The voltage stability phenomenon is generally analyzed by plotting the relationship between the voltage at the load and its active power. As presented in Bhaladhare et al. (2013), expressions for the normalized voltage into the load in function of its normalized active power can be derived and the so-called P-V curves can be determined. The procedure to obtain the normalized P-V curve is presented below. From Figure 2.1, the apparent power from the source to the load corresponds to the following expression: S SR = P SR + jq SR (2.11) where: P SR = E S 2 G E S V R G cos (θ S θ R ) + E S V R B sin (θ S θ R ) (2.12) Q SR = E S 2 B E S V R B cos (θ S θ R ) E S V R G sin (θ S θ R ) (2.13) with G and B, being the conductance and susceptance of the transmission network. Assuming G = 0, Equations 2.12 and 2.13 become in: P SR = E S V R B sin (θ S θ R ) (2.14) Q SR = E S 2 B E S V R B cos (θ S θ R ) (2.15) Then, the apparent power in the load can be written as: S R = P R + jq R = (P RS + jq RS ) (2.16)

20 9 where: P R = P RS = E S V R B sin (θ R θ S ) = E S V R B sin (θ S θ R ) (2.17) Q R = Q RS = V R 2 B + E S V R B cos (θ R θ S ) = V R 2 B + E S V R B cos (θ S θ R ) (2.18) Defining θ RS = θ S θ R, Equations (2.17) and (2.18) become in: P R = E S V R B sin (θ SR ) (2.19) Q R = V R 2 B + E S V R B cos (θ SR ) (2.20) The apparent power in the load can also be expressed as follows: S R = V R I e jφ = V R I (cos φ + j sin φ) = P R (1 + j tan φ) = P R (1 + jβ), where β = tan φ (2.21) Then, the reactive power can be expressed in terms of P R, as: Q R = P R β. When the expressions for P R and Q R are equated to eliminate E S, the following equation is obtained: ( ( V R 2 ) 2 2PR β + B V R 2 + P ) R B 2 (1 + β2 ) = 0 (2.22) This is a quadratic equation in V R 2. When θ SR is eliminated and the second-order equation is solved, the following is obtained ( V R ) 2 = 1 βp R ± (1 P R (P R + 2β)) 2 (2.23) As shown in Equation (2.23), the voltage at the load location expressed in function of its active power has two solutions. The solution with the higher value is the stable one. Additionally, can be appreciated that the voltage of the load is influenced by its power, the reactance of the system, and the power factor of the load. A graph of Equation (2.23) corresponds to the P-V curve of the system in Figure 2.1. This curve is shown in Figure 2.3 for different power factors of the load.

21 10 Figure 2.3 P-V curve for various power factors of the load 2.3 Characteristics of the power systems affecting voltage stability The voltage stability problem has been thoroughly studied during the last three decades. Some characteristics of the power system have been found as key aspects in the study of this phenomenon. These characteristics are described below: Transmission system: The topology of the grid plays an important role in the establishment of the robustness of the system in terms of voltage stability. As explained in Section 2.2, the ratio between the impedance of the transmission system (Z LN ) and the impedance of the load (Z LD ) defines the proximity of the system to a voltage collapse. When this ratio is greater than 1, voltage instability occurs. Even under scenarios of no load increase, changes in the topology of the system that affects impedance the Z LD would lead to a voltage instability. In this sense, disconnection of equipment, such as transmission lines, power transformers, or shunt compensation devices, will create increases in

22 11 the system impedance (Z LN ) and will increase the risk of matching the impedance of the load (Z LD ) reaching the voltage collapse point. Generators: The capability of synchronous generators to feed the reactive power demanded by the loads also defines how far the system is from a voltage instability condition or from a voltage collapse. In fact, one of the most important aspects in voltage stability analysis is to consider the behavior of generators when they reach their reactive power limits. When synchronous generators reach these limits, they lose their responsiveness to increases in reactive power demanded by the loads, and voltage instabilities are more prone to occur. Loads: The type and the amount of load connected in the system are more critical for voltage stability analysis than for other forms of stability in power systems. Therefore, proper modeling of loads is a key factor to obtaining accurate results in voltage stability analysis. Particularly, long-term voltage stability studies use static models, that is, models where the load is function of the voltage and the frequency, which are the magnitudes that characterize this type of phenomenon. For other types of voltage stability analysis, such as the so-called short-term voltage stability studies, more detailed modeling of loads is required because the transient response of the system after a disturbance, is the subject of this kind of approaches (Morison et al. (2006)). Reactive compensator devices: When a voltage stability event occurs, reactive compensator devices attempt to recover the voltages. These devices are among others: automatic voltage regulators in synchronous machines, SVCs, automatic switch capacitors, and automatic transformer OLTCs. As the voltage is trying to be raised after a contingency, the loads start to recover their demand of active and reactive power, and when the amount of power demanded by the loads cannot be supplied because of the weakness of the system, a voltage collapse takes place (Morison et al. (2006)).

23 Long-term voltage stability Long-term voltage stability is a term used to categorize the analysis of the nonlinear dynamic behavior of power systems after certain contingencies, in periods of time long enough to capture the steady-state behavior of the network, i.e., after the interactions between devices that affect the flow of reactive power into the system, such as those mentioned in Section 2.3 (Kundur et al. (1994)) have took place. The typical periods where the long-term stability problems could happen are in the order of tens of seconds up to tens of minutes. Taylor et al. (1994) presents an interesting list of large-scale voltage stability incidents worldwide, as well as the time frame of those events (Table 2.1). This table shows that the majority of these incidents are categorized as long-term stability problems. Long-term voltage stability incidents are usually associated with the response of the system to large-scale disturbances and become particularly critical when they are problems of planning, design, or protection. Those problems have triggered major blackouts in the history that are characterized by a single trip of generation units or transmission lines, resulting in a redistribution of power flows that create deficits of the reactive power in certain regions of load. After the disconnection of generators or lines, the remaining generators should operate close to their limits of reactive power to assume this deficit. Consequently, in addition to the normal increase of active power flow due to the normal behavior of the loads, the reactive power flow into the grid also increases (Johansson (1998)). Then, as the generators reaching their limits of reactive power cannot help to maintain the voltages, high power flows, can trigger further disconnections of transmission lines, creating a cascade and ending in a voltage collapse.

24 13 Table 2.1 Some voltage stability incidents along the history (Taylor et al. (1994)) Date Location Time Frame Apr. 13, 1986 Winnipeg, Canada Nelson River HVDC link Short term, 1 sec Nov. 30, 1986 SE Brazil, Paraguay, Itaipu HVDC link Short term, 2 sec May 17, 1985 South Florida, USA Short term, 4 sec Dec. 27, 1983 Sweden Long term, 55 sec Dec. 30, 1982 Florida, USA Long term, 1-3 min Sep. 22, 1977 Jacksonville, Florida, USA Long term, few min Aug. 4, 1982 Belgium Long term, 4-5 min Nov. 10, 1976 Brittany, France Long term Jul. 23, 1987 Tokyo, Japan Long term, 20 min Dec. 19, 1978 France Long term, 26 min Aug. 22, 1970 Japan Long term, 30 min

25 14 CHAPTER 3. REAL-TIME MONITORING FOR LONG-TERM VOLTAGE STABILITY 3.1 Introduction This chapter will present a general review of current trends in long-term voltage instability detection and its classification based on the type of measurements used for the calculation required (Section 3.2). Additionally, a summary of the state of the art in using Thévenin equivalents for voltage stability analysis is also presented in Section 3.3, emphasizing in three recent concepts developed to determine voltage instability by measurements at load locations. Finally, a new approach to calculate Thévenin equivalents by using the impedance matrix of the system instead of using the admittance matrix is presented in Section 3.4. The concepts presented in this chapter will be named as the concept of single-port circuit, the concept of multiport circuit, and the linear index. These concepts are categorized under long-term voltage stability approaches, which according to Glavic and Cutsem (2011), could be implemented by using wide-area measurements in real- time systems. 3.2 Classification of approaches for long-term voltage instability detection An interesting work to characterize recent approaches to determine VSMs is developed by Glavic and Cutsem (2011). This characterization is proposed based on the type of measurements available in the power system. A summary of the categorization proposed is presented in Figure 3.1. According to this figure, long-term voltage stability monitoring can be classified in two main categories:

26 15 Voltage stability detection using measurements gathered at one location Voltage stability detection based on wide-area measurements Figure 3.1 Classification of long-term voltage instability detection according to Glavic and Cutsem (2011) For both categories the use of Thévenin equivalents is a practical concept for the development of different approaches. In fact, the approaches based on the concept of the Thévenin equivalent matching impedance can be found within the first category. These methods can use either nonsynchronized or fast sampled measurements. The main characteristic of these methods is the assumption that the power system that is seen from a certain load location can be simplified as a voltage source in a series with an impedance. When the impedance of the load increases and equals the equivalent impedance of the system (Thévenin impedance), a voltage collapse will occur. The main disadvantage of this methodology is that, in addition to several assumptions that are difficult to have in real long-term voltage stability problems, prediction of VSMs is difficult due to the nonlinear behavior of the voltage under critical load increases. More details of these kind of approaches will be presented in Section 3.3.2

27 16 In the second category of voltage instability detection, both synchronized and nonsynchronized phasor measurements are used to detect the voltage status of the grid, thereby allowing the development of different approaches, such as monitoring reactive power reserves; designing decision trees for System Integrity Protection Schemes (SIPS); calculating multi-load Thévenin equivalents; and performing singular value decomposition, linear state estimation, and state reconstruction analysis. This category takes advantage of the potential applications of synchrophasor measurements of wide-area monitoring systems. One challenge of these sophisticated methods is that the calculations required for voltage stability estimation, should be performed fast enough to allow automatic or manual actions in the system to avoid voltage collapses. Nevertheless, all methods to detect VSMs have a major aim, which is to provide tools that can anticipate instabilities with enough time in advance to allow automatic or manual remedial actions to maintain the stability of the system under critical load increases and contingencies. Automatic actions primarily refer to the design of SIPS, whereas manual actions refer to the operative decisions that system operators should make in real-time to prevent voltage collapses. 3.3 Thévenin equivalents for log-term voltage stability analysis Thévenin equivalents has been used recently as the basis to develop different approaches to detect and predict voltage stability problems in power systems. These approaches to determine voltage instabilities includes both local-based and wide-area-based measurements methodologies. The advantages and disadvantages of every proposed method depends upon the applications intended Literature review The first approaches in using Thévenin equivalents for voltage stability, propose their calculation by using local measurements of voltage and currents at load busbars. As described by Vu et al. (1999) one way to determine long-term voltage stability issues is by defining Thévenin circles (circles with radius Z T hev ) and tracking the apparent impedance of the load. Theoretically, when apparent impedance hits Thévenin circle the system goes to a voltage collapse

28 17 condition. Thévenin equivalents of the system at every load location are calculated by using two or more measurements of its voltage and current (E and I) and by solving the following expression: E = V + Z T hev I (3.1) The main advantage of this approach is its easy implementation in devices such as protective relaying, as well as its independency of the communications infrastructure. Nevertheless, the solution of Equation (3.1) will be accurate only if the power system remains static between the two times where the measurements takes place. Therefore, if topological changes into the grid occurs, other methods should be used to determine the Thévenin equivalents. The limitations of the two-measurements technique to solve the Equation (3.1) that is required to obtain Thévenin equivalents of multiload systems, have been addressed from different perspectives. For example, Smon et al. (2006) propose the use of Tellegen s theorem and adjoint networks to obtain Thévenin equivalents in a single calculation instead of require two consecutive measurements of the currents and voltages of the system. Tellegen s theorem states the following relationship between the complex voltages and complex currents flowing across all network elements: U T I = 0 (3.2) This equation is valid, even though the voltages and currents are related with to two different networks of the same topology. Therefore, Thévenin equivalents can be obtained by solving Equation (3.2) in a one-step procedure. This approach makes the use of Thévenin equivalents very suitable for monitoring wide-area real-time voltage stability issues. A similar work has been developed by Julian et al. (2000) with the so-called Voltage Instability Predictor (VIP). In this approach, by measuring the distance between the apparent impedance of the load and the Thévenin equivalent impedance of the system, one can define the margin of the system before an unstable operation condition. These margins allow the prediction of voltage instabilities at load locations. The combination and centralization of this VIPs can be used to define automatic action schemes to avoid voltage collapses in both radial and meshed system configurations. One important advantage of this approach is that suggests

29 18 the use of Thévenin equivalents for local calculations and the centralization of this computation for its use in wide-area protection schemes. The classical approaches to determine Thévenin equivalents at any given load location by considering as static the rest of the loads of the system, has demonstrated important limitations in properly represent the independent behavior of its loads during the normal operation of power systems. To address this limitation, interesting approaches as the one presented by Li and Chen (2009) have been developed. In this work, the buses of the power system are divided into three categories: generator bus, load bus and tie bus (buses without generators and loads connected to it). As the current injections at tie buses is zero, a solution of the matrix equations of the voltages in the loads in terms of its currents can be obtained. In this process, Thévenin equivalents can be obtained from calculations over the admittance matrix of the system. Recent developments in the use of Thévenin equivalents to study voltage stability issues, proposes the collection of considerable amount of synchrophasor measurements in real time and the use of them to obtain Thévenin equivalents by solving voltage equations of the system through least square errors optimization methods (Parniani et al. (2006)). The main advantage of this kind of approaches is the minimization of efforts in system modeling to predict voltage stability collapses, making these developments very suitable for real-time applications. Higher accuracy of the simulations with respect to the data obtained from real disturbances, is presented by the authors of this approach as a field of research that is still under study. Thévenin equivalents can also be used as part of complex Wide-Area Monitoring and Control (WAMC) systems. As presented by Zima et al. (2005) WAMCs include within others, the development of application for frequency instability assessment, oscillations stability assessment, voltage stability assessment of transmission corridors and meshed networks and line temperature monitoring. Particularly, to implement voltage instability detection methods in transmission corridors, the approach mentioned uses Thévenin equivalents. These equivalents are obtained by calculating a T-equivalent of the corridor using the measurement of currents and voltages at both ends of the corridor. Then, by solving the T-equivalent system, the Thévenin equivalent can determined (Figure 3.2). The main advantage of this approach is that the equivalent network can computed from a single set of phasor measurements, and thus time

30 19 delay issues associated with multiple measurements are not present. Nevertheless, the independent behavior of the loads of the equivalent system is not considered during the analysis of voltage stability margins at certain load location, which is a limitation of this methodology. Figure 3.2 Lumping of T- and Thévenin equivalents of a transmission corridor and surrounding environment (Zima et al. (2005)) In general, recent approaches in long-term voltage in stability detection have proposed its application as part of other advanced systems of measuring and monitoring of power networks. In consequence, the development of tools and methodologies to perform calculations in realtime, are required to successfully accomplish the implementation of these approaches in real systems. In this sense, the following subsections presents the details of three concepts used for long-term voltage in stability detection in real-time systems. They are the single-port circuit, the coupled single-port circuit and the linear index. These concepts are based on the use of Thévenin equivalents and can use information of the real-time operation of the system. Additionally, those concepts will be used to illustrate the contributions of this thesis for the implementation of real-time tools for monitoring long-term voltage stability Voltage instability detection based on the concept of the single-port circuit The theoretical approach to determine voltage stability problems in power systems based on the concept of impedance matching between the Thévenin equivalent of the system and load impedance, has important restrictions when it is applied to multiload systems (Glavic and Cutsem (2011)). The main limitation of this methodology is the assumption that power system does not change during load increase. This condition restricts its implementation in large power systems because that assumption is not realistic in this kind of systems. In fact, as described previously, loads are dynamic and nonlinear, and this behavior is enhanced during long-term voltage stability events.

31 20 As explained in Chapter 2, the maximum power that can be delivered from a generator to a load in a system such as that shown in Figure 2.1, is obtained when the impedance of the load Z LD equals the impedance of the system Z LN. In the concept of the single-port circuit, Z LN is the Thévenin equivalent of the system obtained by considering a static behavior of all its loads except the load under analysis. This method has been recommended for local or distributed load shedding schemes placed in power systems to avoid voltage collapses (Glavic and Cutsem (2011)). The assumptions of this approach are as follows: 1. The monitored bus is considered separated from the rest of the system. 2. Apart of the load under study, the rest of the system is treated as a Thévenin equivalent circuit. 3. Loads are considered linear. 4. The load shedding schemes proposed under this approach should be active when other corrective actions based on the control of reactive power are exhausted. 5. Generators do not reach their reactive power limits. In summary, the concept of single-port circuit is simple and relatively easy to implement. This method does not depend on communications infrastructure or high data processing capabilities because only local measurements are required. The main disadvantage of this method is that the inaccuracies associated with the assumption that the system does not change during load increases, could lead to actions that might be insufficient in preserving its stability or could lead to actions more severe than required. The importance of this method is that constitutes the basis for more advanced approaches to determine Thévenin equivalents in multi-load systems as will be explained in following subsections Voltage instability detection based in the concept of coupled single-port circuit This concept considers all loads in the system as constant-power type loads. One interesting approach based on this concept is presented by Wang et al. (2011). Following this approach,

32 21 in the present work all loads and generators of the grid are kept out of a network, which has an equivalent impedance Z L as shown in Figure 3.3. Figure 3.3 Multi-port network system model (Wang et al. (2011)) In matrix form, system in Figure 3.3, can be described as follows: I L Y LL Y LZ Y LG V L 0 = Y ZL Y ZZ Y ZG V Z I G Y GL Y GZ Y GG V G (3.3) The currents in this system can be obtained by solving Equation (3.3) as follows: I L = Y LL V L + Y LZ V Z + Y LG V G (3.4) 0 = Y ZL V L + Y ZZ V Z + Y ZG V G (3.5) I G = Y GL V L + Y GZ V Z + Y GG V G (3.6) To obtain an expression of V L in terms of V G and I L, from Equation (3.4), Y LL V L is given by the following expression: Y LL V L = Y LZ V Z Y LG V G I L (3.7) In addition, from Equation (3.5), V Z can be written as follows: V Z = Y 1 ZZ ( Y ZLV L Y ZG V G ) (3.8) If Equation (3.8) is integrated into Equation (3.7), the following expression is obtained: Y LL V L = Y LZ Y 1 ZZ Y ZLV L + Y LZ Y 1 ZZ Y ZGV G Y LG V G I L (3.9) Equation (3.9) can be reorganized as follows: (Y LL Y LZ Y 1 ZZ Y ZL)V L = (Y LZ Y 1 ZZ Y ZG Y LG )V G I L (3.10)

33 22 If we call Z L = (Y LL Y LZ Y 1 ZZ Y ZL) 1, Equation (3.10) can be written as follows: V L = Z L (Y LZ Y 1 ZZ Y ZG Y LG )V G Z L I L (3.11) Finally, if we define K = Z L (Y LZ Y 1 ZZ Y ZG Y LG ), V L in Equation (3.11) can be written as follows: V L = KV G Z L I L, (3.12) Furthermore, for every bus j in the system, the voltage V Lj can be expressed as follows: V Lj = E eqj Z eqj I Lj E coupledj (3.13) where E eqj = n G i=1 K ji V Gi (3.14) Z eqj = Z Ljj (3.15) E coupledj = n L i j;i=1 Z Lji I Li (3.16) where n G is the number of generators and n L the number of loads into the system. Therefore, Equation (3.13) can be written as: V Lj = V Lj = V Lj = n G i=1 n G i=1 n G i=1 K ji V Gi K ji V Gi K ji V Gi n L i=1 ( nl i=1 ( nl i=1 Z Lji I Li Z Lji I Li I Lj ) I Lj Z Lji S L i S L j V L j V L i ) I Lj (3.17) And from Equation (3.17), one can define the Thévenin equivalent impedance of the system (Z T hj ) as follows: Z T hj = n L i=1 Z Lji S L i S L j V L j V L i (3.18) Can be noted that Thévenin equivalents described by equations Equations (3.14) and (3.18), can be affected by the topology of the network (represented by factors K ji and Z Lji ) and by the voltages at generation buses and by the voltages and currents at load location. That is

34 23 reason why under topological changes in the grid, these equivalents change and therefore their update are required for accurate calculations of VSMs. Equation (3.13) can be used to determine VSMs at any load j. This expression considers a Thévenin equivalent of the network (E eqj and Z eqj ) and the effect of the rest of the system loads that is represented by E coupledj. Figure 3.4 Graphical representation of multi-port Thévenin equivalent (Wang et al. (2011)) Now, the issue is how to model the coupling term (E coupledj ) to properly reflect the behavior of the rest of the system loads into the Thévenin equivalent (Figure 3.4). The options to solve these issues are as follows: 1. Modeling E coupled as an extra power load. In this case, the behavior of the Thévenin equivalents under load increase are nonlinear. In fact, as shown in Figure 3.5 the active and reactive powers on this load are non-linear along the load increase. Therefore, this model is not appropriate for the type of computations required to determine VSMs. 2. Modeling E coupled as an extra voltage source: The same flaws as the previous approach (Figure 3.6). 3. Modeling E coupled as an extra impedance: The extra impedance is called a virtual impedance Z vj (Figure 3.7). In this case, the behavior of Thévenin equivalents is totally linear and suitable for VSM calculations. Therefore, as proposed by Wang et al. (2011) if E coupledj is considered as an extra impedance, the maximum power to the load is given by the following expression: S max = E2 eq [ Z eq (imag(z eq ) sin δ + real(z eq ) cos δ)] 2[imag(Z eq ) cos δ real(z eq ) sin δ] 2 (3.19)

35 24 Figure 3.5 Characteristics of the virtual loads. (a) Active power ratio P ratio (%). (b) Reactive power ratio Q ratio (%) (Wang et al. (2011)) Figure 3.6 Coupled voltages at different buses (each line for a different bus) (Wang et al. (2011)) Finally, as proposed by Wang et al. (2011) a VSM based on the concept of single-port circuit, can be calculated as follows: margin n = S max,n S Ln S Ln (3.20) Index 1 = min(margin 1, margin 2,..., margin n ) (3.21) P max1 = P (1 + Index 1) (3.22) VSM1 = P max1 P (3.23) where P is the actual active power of the system.

36 25 Figure 3.7 Characteristics of the virtual impedance (Wang et al. (2011)) Voltage instability detection using linear index This approach is introduced by Matavalam and Ajjarapu (2015) and will be explained in this section and then will be used to illustrate the methodology proposed in Chapter 5. The linear index is an estimation of the VSM of a power system. This estimator has a linear behavior under load increase. The main advantage of this index is the capability of predict VSMs because of its linear behavior. Additionally, this index has the advantage of responding properly under different directions of load increase. The linear index uses a combination of the coupled single-port concept described in Section and the fact that under load increase, the voltage phasor at the load in an equivalent system such as that shown in Figure 2.1, lies into a locus defined by a circle (Matavalam and Ajjarapu (2014)). Following the approach in Matavalam and Ajjarapu (2015), active and reactive power into the system in Figure 2.1, if the resistance on the transmission line is neglected, can be expressed as in Equation (2.17) and Equation (2.18). Assuming θ S = 0, to eliminate θ R from equations Equations (2.17) and (2.18), PR 2 and Q2 R are as follows: P 2 R = E2 S V 2 R sin2 θ R X 2 (3.24) Q 2 R = E2 S V 2 R cos2 θ R 2E S V 3 R cos θ R + V 4 R X 2 (3.25)

37 26 Then, by adding PR 2 and Q2 R, the following equation is obtained: P 2 RX 2 + Q 2 RX 2 = E 2 SV 2 R sin 2 θ R + E 2 SV 2 R cos 2 θ R 2E S V R cos θ R V 2 R + V 4 R (3.26) From Equation (2.18): Q R X + VR 2 = E s V R cos θ R (3.27) So, Equation (3.27) can be integrated into Equation (3.26). By doing so, the following expression is obtained: (P 2 R + Q 2 R)X 2 = E 2 SV 2 R(sin 2 θ R + cos 2 θ R ) 2(Q R X + V 2 R)V 2 R + V 4 R (3.28) Considering that sin 2 θ R + cos 2 θ R = 1, Equation (3.28) can be written as: (P 2 R + Q 2 R)X 2 = E 2 SV 2 R 2Q R XV 2 R 2V 4 R + V 4 R 0 = (P 2 R + Q 2 R)x 2 + (2Q R X E 2 S)V 2 R + V 4 R (3.29) Assuming constant power factor of loads during load increase and decomposing the voltage phasor of the loads in its real and imaginary parts, the following expressions are obtained: V Rr = V cos θ R, V Ri = V R sin θ R (3.30) V 2 R = V 2 R r + V 2 R i (3.31) Now, as the ratio of the powers at the load can be expressed as follows: P R Q R = E S V Ri E S V Rr (V 2 R r + V 2 R i ) (3.32) The following equation is obtained: ( ) VR 2 r E S V Rr + VR 2 QR i E S V Ri = 0 (3.33) P R Finally, if tan θ = Q R /P R and considering that (1 + tan 2θ) = sec 2θ = (1/pf 2 ), the equation becomes: ( V Rr E ) 2 ( S + V Ri E ) 2 ( S 2 2 tan θ ES = 2 which is a locus corresponding to a circle with radius ) 1 2 (3.34) cos θ ( E S 2 cos θ and center in ES 2, E S tan θ 2 ). In Matavalam and Ajjarapu (2014) this locus is illustrated for different power factors as shown in (Figure 3.8).

38 27 Figure 3.8 Voltage locus for E S = 1, power factor of -0.93, 1.0 and 0.93 (Matavalam and Ajjarapu (2014)) In the mentioned reference, it is demonstrated that the critical angle of loadability of the system, δ cr, is presented when angle OAB = BOA in Figure 3.8. If X is the center of the circle, there can be demonstrated that OXB = 90 (θ + φ) = 2δ cr so: δ cr = 45 (θ + φ) 2 (3.35) Additionally, it is demonstrated that the critical voltage at load location is given by the following: V cr = E S 2 cos δ cr (3.36) Equation (3.36) is an expression to calculate the minimum voltage of the system before a collapse (critical voltage). This expression is derived under the assumption that there are no limits of reactive power in generators and power factor of the loads are constant during its increase. To apply this concept to a multi-load system, Equation (3.36) can be written for every load bus j as follows: where, V crj = E eq j 2 cos δ crj (3.37) δ crj = 45 (θ ( ) ( ) j + φ j ), θ j = tan 1 QRj, φ j = tan 1 Re(ZT hj ) 2 P Rj Im(Z T hj ) (3.38) Note that E eqj in Equation (3.37) and Z T hj in Equation (3.38), are given by Equations (3.14) and (3.18) respectively.

39 28 The linear index proposed by Matavalam and Ajjarapu (2015), is given by the follows: M j = 1 V R j E eqj V Rj V crj 2 (3.39) Then, the VSM using the linear index can be expressed as follows: Index 2 = min(m 1, M 2,..., M j ) (3.40) P max2 = P 1 Index 2 (3.41) VSM2 = P max2 P (3.42) where P is the actual active power of the system. To illustrate the behavior of this index, it was implemented in the base case of the IEEE 9-bus test system. This index has a range between 0% to 100%, where 0% represents the point of voltage instability. Therefore, for this particular example, can be appreciated that bus #7 is the one that could lead the system to a voltage collapse (Figure 3.9). In summary, the approach proposed by Matavalam and Ajjarapu (2015) to develop the linear index, uses information of the admittance matrix of the system to determine the Thévenin equivalents of the system. Then, by using a geometrical approach, a linear index of the voltage stability of the system can be developed. 3.4 Determination of voltage stability margins based on the impedance matrix Despite the important advantages of the coupled single-port circuit concept in terms of the consideration of different phenomena of the grid into one single calculation, determining VSMs in the methods presented in Sections and requires the inversion of matrices to obtain the parameters of Thévenin equivalents (matrices K and Z L in Equation (3.12)). This condition can represent a risk for practical applications due to the computational efforts required for this type of tasks in large power system. This aspect becomes more relevant during log-term voltage stability events where the degradations of the grid caused by equipment disconnections are usually present.

40 29 Figure 3.9 Linear index calculated using information of Y bus This chapter will present the matrix derivations and calculations required to obtain Thévenin equivalents from the Z bus of the system. Additionally, a novel way to apply the concept of coupled single-port circuit (or multi-port equivalent) using the information of the impedance matrix of the system (Z bus ) instead of the information of the admittance matrix Y bus, will be presented Calculation of Thévenin equivalents based on the impedance matrix The system described in Equation (3.3) can also be written as follows: V G Z GG Z GZ Z GL I G V Z = Y 1 bus I = Z busi = Z ZG Z ZZ Z ZL 0 Z LG Z LZ Z LL I L V L (3.43) where Y 1 bus = Z bus and represents the impedance matrix of the system. An expression for V L in terms of V G and I L that is equivalent to the one obtained in Equation (3.12) can be derived as follows:

41 30 From (3.43), the matrices for generators and load voltages V G and V L, are given by the following: From Equation (3.44), I G can be expressed as follows: V G = Z GG I G Z GL I L (3.44) V L = Z LG I G Z LL I L (3.45) I G = Z 1 GG (V G + Z GL I L ) (3.46) When I G in Equation (3.46) is replaced into Equation (3.45), the following expression is obtained: V L = Z LG Z 1 GG (V G + Z GL I L ) Z LL I L = Z LG Z 1 GG V G + Z LG Z 1 GG Z GLI L Z LL I L = Z LG Z 1 GG V G (Z LL Z LG Z 1 GG Z GL)I L (3.47) A new expressions for K and Z L, equivalent to those obtained for Equation (3.12) are given by the following: K = Z LG Z 1 GG (3.48) Z L = (Z LL Z LG Z 1 GG Z GL) (3.49) With this new K and Z L, the VSMs presented in Sections and can also be obtained. Table 3.1 shows the comparison of the parameters K and Z L for the initial approach based on Y bus and those obtained with the approach of this thesis, which is based on the information of the Z bus. Table 3.1 Comparison of parameters to calculate Thévenin equivalents Parameter Y bus Z bus Z L K (Y LL Y LZ Y 1 ZZ Y ZL) 1 Z L (Y LZ Y 1 ZZ Y ZG Y LG ) Z LL Z LG Z 1 GG Z GL Z LG Z 1 GG As will be presented in Chapter 4, the main advantage of obtaining the parameters to calculate Thévenin equivalents from the Z bus, is the possibility of applying fast matrix calculations

42 31 Figure 3.10 Graphical representation of sparsity for Y bus and Z bus matrices to update this information when the topology of the system change. This feature make this approach convenient for real-time applications. It must be noticed that using Z bus instead of Y bus to determine Thévenin equivalents, implies losing the sparsity of this last matrix. As shown in Figure 3.10, the sparsity of the Y bus matrix of IEEE 118-bus test system is bigger than the sparsity of the Z bus of the same system. In fact, can be appreciated that Y bus matrix has 13,448 zero entries while Z bus matrix has no zeros. Nevertheless, the presence of several amount of zeros in Y bus matrices (sparsity) is an important advantage to perform iterative calculations such as those required to resolve power flows. However, as no iterative methods are required to conduct the voltage stability estimations proposed in this work, there is no added value of having a sparse matrix. Therefore, the use of Z bus to determine Thévenin equivalents is proposed because as will be presented in Chapter 4, the update of this matrix when the system suffer topological changes is straightforward and the determination of its inverse can be easily performed. Additionally, having the Z bus to determine Thévenin equivalents, will represent significant benefits in terms of computer time savings and therefore will make this approach very suitable for real-time applications.

43 Comparison of voltage stability margins using information of the admittance matrix and the impedance matrix To illustrate the validity of this approach, the IEEE 9-bus test system will be used to determine the multi-port Thévenin equivalent of the system and to compute the linear index as described in Section The results are presented in Figure 3.11 where the linear index is calculated by using the information of both Y bus and Z bus. Figure 3.11 Comparison between indices calculated using information of Y bus and Z bus As observed in Figure 3.11, there are no differences between using Y bus or Z bus to determine Thévenin equivalents and therefore to calculate voltage stability margins base on them. This particular case illustrate the behavior of the linear index in different load locations of the IEEE 9-bus test system. It is important to note that the voltage stability margins from Thévenin equivalents derived from the Z bus, continue having the same set of assumptions of its initial development which uses information of the Y bus. Those assumptions are listed below: Generators does not reach any reactive power limit Loads does not change its power factor under its increase

44 33 Generators increase its active power output in proportion to load increase Finally, having the expressions to obtain Thévenin equivalents derived from the Z bus matrix, one can focus the attention into two properties of linear algebra that can be applied on this matrix in order to improve the speed of calculations required to its update. This update is required for real-time monitoring of voltage stability of the system when the topology change. This subjects will be present in the following chapters.

45 34 CHAPTER 4. USING THE MATRIX INVERSION LEMMA AND KRON REDUCTIONS FOR FAST CALCULATION OF THÉVENIN EQUIVALENTS 4.1 Introduction One of the main advantages of determining Thévenin equivalents to estimate VSMs based on the information of the Z bus of the system is the possibility to easily update the information of the network to calculate the VSMs, thereby avoiding inversion of matrices. This feature is a valuable capability for real-time applications because as discussed in Chapter 2, a voltage stability phenomenon is particularly important under n k contingencies into the system. This chapter will present two mathematical tools that can be used to quickly update the Z bus of the system when an n 1 contingency occurs and VSMs need to be updated. These tools are the Woodbury matrix identity (or matrix inversion lemma) and the Kron reduction technique. The use of these tools is well known in power systems so their application to voltage stability estimation will be illustrated. Additionally, the computational complexity associated with the matrix operations required to use those tools will be discussed. 4.2 The matrix inversion lemma As presented by Bergen and Vittal (2000), for an n n symmetric matrix Y, which will represent the admittance matrix of a power system, the inverse of a modified matrix [Y + µa k a T k ] 1 can be obtained as follows: [Y bus + µa k a T k ] 1 = Y 1 bus γb kb T k (4.1)

46 35 where µ is a scalar, a k is an n 1 vector, and b k and γ are given by the following expressions: b k = Y 1 bus a k (4.2) γ = (µ 1 + a T k b k) 1 (4.3) This formulation is derived from the Sherman-Morrison-Woodbury theorem, which states the following: Theorem 1 If A B(H) and G B(K) both be invertible, and Y, Z B(K,H), then A + Y GZ T is invertible iff G 1 + Z T A 1 Y is invertible. In which case, (A + Y GZ T ) 1 = A 1 A 1 Y (G 1 + Z T A 1 Y ) 1 Z T A 1 In this formulation, the operator Y GZ T updates the initial matrix A. This formula becomes as a practical application for network analysis such as those required to update the admittance matrix of a power system due to connections or disconnections of power equipment in the grid. A proof of the matrix inversion lemma is presented in Appendix A. In Theorem 1, if G is a scalar, Y and Z T become n 1 and 1 n vectors, and if A = Y bus, Y = a k, G = u, and Z T = a T k, Equation (4.1) can be derived from the following: [Y bus + µa k a T k ] 1 = Y 1 bus Y 1 bus a k(µ 1 + a T k Y 1 bus a k) 1 a T k Y 1 bus (4.4) It can be noticed that Equation (4.1) allows the determination of the updated impedance matrix of the system without performing inversions of its admittance matrix. It means that assuming that the inicial impedance matrix of the system is available, future updates of this matrix due to topology changes can be performed just doing matrix multiplications. 4.3 Admittance matrix modifications This study aims to update the Z bus of the system when n k contingencies occur and use this information to determine VSMs in real time. To achieve this goal, the matrix inversion lemma is a powerful tool to quickly update the inverse of the admittance matrix when this matrix changes because of n 1 contingencies.

47 36 Figure 4.1 Simple interconected power system To explain the concept of the updated Y bus, consider that a disconnection of the transmission line between buses 1 and 3 in the system shown in Figure 4.1, occurs. The admittance matrix of the system can be expressed by the following: Y 11 Y 12 Y 13 Y bus = Y 21 Y 22 Y 23 Y 31 Y 32 Y 33 (4.5) If this system changes by the disconnection of the line between buses 1 and 3, the new admittance matrix of the system, Y n bus, will be given by the following: Y 11 y 3 Y 12 Y 13 + y 3 Y n bus = Y 21 Y 22 Y 23 Y 31 + y 3 Y 32 Y 33 y 3 (4.6) where y 3 is the admittance of the line between buses 1 and 3 that should be subtracted from the diagonal elements Y 11 and Y 33, and added to the off-diagonal elements Y 13 and Y 31. Rearranging Equation (4.6), Y n bus can written as follows: Y 11 Y 12 Y 13 y 3 0 y Y n bus = Y 21 Y 22 Y = Y bus y Y 31 Y 32 Y 33 y 3 0 y (4.7) Defining a vector a 3 as [1, 0, 1] T, Equation (4.7) can be expressed as follows: Y n bus = Y bus y 3 a 3 a T 3 (4.8)

48 37 Equation (4.8) shows that in general, for any line k added to or removed from a power system described as in Equation (4.5), the updated admittance matrix can be obtained as follows: Y n bus = Y bus + y k a k a T k (4.9) where y k is the admittance of the element that will modify the initial Y bus, being positive if the element is added to the system or negative if removed from it. a k is an n 1 vector with 1 at position i and -1 at position j, corresponding to the buses where the element is added to or removed from the system. Equation (4.9) describes the modified Y bus matrix of the system in the form Y +µa k a T k, and its inverse can be easily calculated using the matrix inversion lemma described in Section 4.2. Then: Z n bus = [Y bus + y k a k a T k ] 1 = Z bus + γb k b T k (4.10) where: Z bus = Y 1 bus (4.11) b k = Y 1 bus a k (4.12) γ = ( y 1 k + a T k b k) 1 (4.13) Moreover, as γ can be written as follows: γ = ( y 1 k + a T k b k) 1 = ( y 1 k + a T k Z busa k ) 1 (4.14) the term a T k Z busa k can be expanded in the following form: Z Z 1i... Z 1j... Z 1n [ ] Z i1... Z ii... Z ij... Z in 1 bus i a T k Z busa k = Z j1... Z ji... Z jj... Z jn 1 bus j Z n1... Z ni... Z nj... Z nn 0

49 38 0. ] 1 = [Z i1 Z j1... Z ii Z ji... Z ij Z jj... Z in Z jn = Z ii Z ji Z ij + Z jj = Z ii 2Z ji Z jj (4.15) Then, the expression for γ becomes: γ = ( y 1 k + a T k Z busa k ) 1 = ( y 1 k + Z ii 2Z ji Z jj ) 1 = (z k + Z ii 2Z ji Z jj ) 1 (4.16) where z k = 1/ y k is the impedance of the element added to or removed from the initial system. The main advantage of this approach is that if the impedance matrix of the system Z bus is provided as an input for VSM estimations, the calculations required to update these estimations when the topology of the system change, do not involve matrix inversions. This is computationally convenient for application in real-time systems. Nevertheless, should be considered that the MIL works only for n 1 contingencies into the system. Therefore, if more contingencies are present, multiple calculations of the MIL must be performed. 4.4 Updating the impedance matrix considering shunt elements When high-voltage systems are studied, the classical model considered for transmission lines, is the so-called pi-model. This model takes into account the line capacitance, that is a leakage (or charging) path for the AC line currents resulting from the differences of potential between the conductors as well as between the conductors and the ground. The majority of software tools used for the analysis of power systems include within the parameters of the line its capacitance. Therefore, the admittance matrix obtained from these

50 39 tools will include the effect of line charging, and the update of this matrix should consider its effect. When the admittance matrix of a power system is constructed considering the shunt capacitance of their transmission lines, the updated impedance matrix obtained in Equation (4.10) will contain the effect of these shunt elements. Therefore, one more computation is then necessary to properly update the impedance matrix of the system when transmission lines are added to or removed from the initial system. In this work, Kron reduction is proposed to complement the update of the impedance matrix of the system when the capacitance of the line is considered. This methodology can be expanded to the analysis of the effect of other shunt elements in the grid. It is important to notice that connections and disconnections of this equipment, can be presented during real long-term voltage stability issues. To explain how Kron reductions work, the methodology presented by Bergen and Vittal (2000) will be followed. To do so, Equation (4.17), which is derived step by step in Appendix B, should be studied. This equation represents the addition of a new impedance (z k ) between an existing node (node i) and a new node in the system (node k): Z n bus = Z Z i (4.17) Z ii + z k Z T i where Z n bus is the modified impedance matrix, Z i is the ith column of Z and Z ii is the iith element of Z. The relationship between the new impedance matrix and the associated voltages and currents (V ref and I ref ), can be written as follows: V = Z Z i I Z ii + z k V ref Z T i I ref (4.18) As this study focuses on removing the shunt capacitances of the transmission lines that are modeled as parameters connected to the ground, V ref is 0 and Equation (4.18) becomes: V = Z Z i I (4.19) 0 Z ii + z k Z T i I ref

51 40 Then, to eliminate I ref from Equation (4.19) (Kron reduction), the second row of this equation is multiplied by Z i (Z ii + z k ) 1, and this multiplied second row is added to the first row, obtaining the following: V = Z Z i(z ii + z k ) 1 Z T i 0 I 0 Z T i Z ii + z k I ref (4.20) V = (Z Z i (Z ii + z k ) 1 Z T i )I (4.21) Hence, the new Z bus obtained by performing Kron reduction is as follows: Z n = Z Z i (Z ii + z k ) 1 Z T i (4.22) where z k = 1/(y/2) and y/2 is the shunt susceptance of the line removed from the system. This procedure should be performed twice to add/remove the shunt susceptances at both ends of the added/removed line. 4.5 Computational efforts for using matrix inversion lemma and Kron reductions The main advantage of using MIL and Kron reductions to calculate Thévenin equivalents is that the calculations required to update the impedance matrix of the system are the addition of matrices and the multiplication of a vector by a matrix. The computational complexity of these operations is considerable lower compared with inversion. Consequently, the speed of the calculations required to update the impedance matrix of the system and Thévenin equivalents, is substantially increased compared with previous approaches that need perform inversion of matrices. In order to give an idea of the computation times that could be saved by applying the methodology proposed in this work, the use of big -O notation functions for multiplication and inversion of matrices will be compared. The big-o notation defines a function O in terms of a variable n to express an estimation of the computations complexity. The variable n is the number of elements in the operation (Table 4.1). For the case under study in this thesis, n

52 41 corresponds to number of buses of the system analyzed and the values obtained from the big-o functions are given in computational time units. Table 4.1 Complexity of operations using big-o notation (Wikipedia (2006)) Operation Input Output Complexity Matrix multiplication Two n n matrices One n n matrix O(n 3 ) Matrix multiplication One n m matrix One n p matrix O(nmp) & one m p matrix Matrix inversion One n n matrix One n n matrix O(n 3 ) Determinant One n n matrix One number O(n!) Back substitution Triangular matrix n solutions O(n 2 ) According with the information in Table 4.1, the multiplications required to update the impedance matrix of the system when an n 1 contingency takes place, could result in a polynomial of degree 2 by using Equations 4.10 and In contrast, if the initial approach of using information of admittance matrix of the system were used, inversion of matrices would be required and this operation would be defined by a polynomial of degree 3. Furthermore, even with the use of more refined algorithms, the inversion of matrices will result in polynomials of degree which is still higher than the one required for multiplications. Polynomials of degrees 2, and 3 are presented in Figure 4.2 to illustrate the big-o notation concept. From this figure, can be observed how a random computer processing time of would allow the multiplication of vectors of order 5, In contrast, if the inversion is performed, the dimension of the matrices required to maintain this computational processing time is reduced to 1, 311 if there are used refined inversion methods or up to 292 by using conventional inversion methods. As a further matter, the advantages in computational time processing obtained by using the methodology proposed in this work instead of use inversion of matrices, increases with the increase of the size of the power system. This aspect, converts this approach in a very suitable way to implement real-time applications based on the information contained in the impedance matrix of the system.

53 42 Figure 4.2 Computation complexity of different big-o functions

54 43 CHAPTER 5. UPDATING THÉVENIN EQUIVALENTS IN REAL-TIME 5.1 Introduction This chapter will present the details of the methodology proposed to implement the use of matrix inversion lemma and Kron reductions to obtain Thévenin equivalents in real-time. This methodology is based on the use of the impedance matrix that is assumed available as an input of the process as will be described in Section 5.2. Then, two subprocesses that are part of the approach will be described in Sections 5.3 and 5.4. Finally, general comments of this methodology will be provided in Section General description of the process The approach proposed in this work suggests the use of the impedance matrix of the system instead of its admittance matrix to compute the Thévenin equivalents that will be used to perform long-term voltage stability estimations. As indicated in Chapter 4, the use of the impedance matrix allows a quick updating of the information of the system when topological changes take place on it by using the matrix inversion lemma and Kron reductions. A summary of the approach proposed is illustrated in Figure 5.1. As shown in the left-side flow chart in Figure 5.1, the process begins by considering the impedance matrix of the system as an input. This matrix can be obtained from any network application of the real-time platforms available in the power system such as SCADA/EMS or synchrophasors systems. The impedance matrix can also be introduced as a parameter calculated off-line and then been updated with real-time information of the topology of the system. In this latest case, the inversion of the offline admittance matrix could be required.

55 44 Start Calculate VSMs Modify Zbus with MIL and Kron reduction Ask for Zbus Organize Zbus: ZGG, ZGL,ZGZ, ZLL, ZLG ZLZ, ZZZ, ZZL, ZZG Find vector aij (i,j: terminals of line or transformer outaged) Ask for voltages at load and generation buses and for power at load buses. Calculate matrices ZL and K Obtain yij (yij: admitance of line or transformer outaged) Calculate VSMs Calculate Eeq and ZTh Modify Zbus by using MIL n-1? Yes Modify Zbus with MIL and Kron reduction No Real time calculations Compute margin Display VSM End The outage corresponds to a transmission line? Yes Apply Kron reduction to eliminate shunt capacitances No Zbus modified (it means a transformer outage) End Figure 5.1 Flow chart for fast calculation of Thévenin equivalents for real-time steady state voltage stability estimation

56 45 By having the impedance matrix, the following step is gathering the voltages at every load bus of the system in phasorial form. As in the case of the impedance matrix, those voltages of the network can be obtained from SCADA/EMS systems, state estimator or synchrophasor applications in real-time. Then, the routine to calculate VSM can start (process Calculate VSMs in Figure 5.1). Once the VSMs has been obtained, they must be updated periodically considering the sample time of the real-time systems (typically every 4 seconds). If there are no topological changes in the network, the update of the VSMs is performed only by updating the information of voltages in the system. However, if there is topological change in the network (n 1 contingency), prior to update the information of voltages, the impedance matrix of the system must be updated. If this is the case, the process Modify Z bus with MIL and Kron reduction in Figure 5.1 will take place. 5.3 Subprocess: Calculate VSMs This subprocess is presented in the central flow chart of Figure 5.1. For illustrative purposes, the equations corresponding to the linear index that was presented in Section will be used along the rest of the methodology. The subprocess starts by organizing the impedance matrix of the system in such a way that generator buses, zero-power buses and load buses are placed separately between them. By doing so, the following submatrices are obtained: Z GG : matrix with the impedances between generator buses. Z GZ : matrix with the impedances between generator buses and zero-power buses. Z GZ = Z T ZG Z GL : matrix with the impedances of the branches between load buses and generator buses. Z GL = Z T LG Z ZZ : matrix with the impedances between zero-load buses. Z ZL : matrix with the impedances of the branches between zero-power buses and load buses. Z ZL = Z T LZ

57 46 Z LL : matrix with the impedances between zero-load buses. The impedance matrix of the system will look as in equation Equation (5.1) Z GG Z GZ Z GL Z ZG Z ZZ Z ZL Z LG Z LZ Z LL (5.1) By having the sub matrices of the impedance matrix of the system, the parameters K and Z L that are required to obtain the Thévenin equivalent of the grid can be obtained applying equations Equations (3.48) and (3.49). These equations are repeated here for convenience: K = Z LG Z 1 GG Z L = (Z LL Z LG Z 1 GG Z GL) Then, the Thévenin equivalents of the network at each load location are calculated as follows: E eqj = Z T hj = n G i=1 n L i=1 K ji V Gi (5.2) Z Lji S L i S L j V L j V L i (5.3) Now, the linear index of the system can be computed applying Equation (3.40) M = min(m 1, M 2,..., M j ) where M j is the linear index at every load bus as indicated in equation Equation (3.39) Finally, the VSM of the system is given by: M j = 1 V R j E eqj V Rj V crj 2 P max = P 1 M (5.4) The VSM is proposed to be displayed in the control center as real-time measurement of the status of the system in terms of voltage stability that can help to increase the situational awareness of operator.

58 Subprocess: Modify the Z bus with MIL and Kron reductions This subprocess is triggered when there are disconnections in the system such as transmission lines, transformers, shunt reactors and shunt capacitors. This kind of contingencies makes necessary the update of the impedance matrix of the system for a proper estimation of the parameters required to obtain Thévenin equivalents. The first step in this sub process is to determine the buses between where the disconnection of the element takes place. As indicated to initially update the topology of the system, this information can also be obtained of the real-time monitoring system applications. With the information of the disconnection, the vector a ji can be constructed as follows: 0. 1 bus i. 1 bus j. 0 Where i and j, are the buses where the disconnection took place. Once identified the outage in the system and its correspondent vector, the admittance of the disconnected element must be queried (parameter y ij ). Then, the matrix inversion lemma can be applied to obtain the updated impedance matrix of the system as follows: Z n bus = [Y bus + y k a k a T k ] 1 = Z bus + γb k b T k (5.5) where: Z bus = Y 1 bus (5.6) b k = Y 1 bus a k (5.7) γ = (z k + Z ii 2Z ji Z jj ) 1 (5.8) where z k = 1/ y k.

59 48 Additionally, if the disconnected element is a transmission line, Kron reductions are required in order to subtract from the updated matrix the effect of the shunt capacitances associated with this elements. To do so, Equation (4.22) must be performed twice (once for every shunt capacitance associated) as follows: Z n = Z Z i (Z ii + z k ) 1 Z T i where z k = 1/(y/2) and y/2 is the shunt susceptance of the outaged line. 5.5 General comments The use of the matrix inversion lemma to update the impedance matrix of the system, is applicable for n 1 contingencies of the elements that are contained in the impedance matrix of the system such as transmission lines, transformers, shunt capacitors o reactors. These contingencies as proposed in this work, are related with connection and disconnection of the mentioned power equipment. Therefore, if multiple contingencies are present in the system (n k contingencies), multiple updates should be performed in the impedance matrix, considering just one element connection or disconnection at the time. It is important to note that in this approach, the fast updating of the impedance matrix of the system is applied for the determination of Thévenin equivalents of a multi-load system that can be used for long-term voltage stability analysis. However, this methodology could be used for other applications for real-time operation of power systems that requires either fast update of the impedance matrix of the system or Thévenin equivalents based on the information of this matrix. The approach proposed intend the fast calculation of VSMs in real-time to be used in control centers to increase the situational awareness of operators. Though, other additional applications of this indicator of voltage stability can be studied. For example, the definition of automatic schemes of load shedding or reactive power control, designed to increase the voltage stability margins of the system during its real-time operation.

60 49 CHAPTER 6. APPLICATIONS AND RESULTS In order to demonstrate the validity of the approach proposed in this work, a set of simulations and calculations were performed using the IEEE 9-bus and in the IEEE 118-bus test systems. The IEEE 9-bus test system will be used to illustrate the convenience of use the updated Z bus to calculate Thévenin equivalents and then use those equivalents to determine voltage stability margins. The IEEE 118-bus system will be used to validate the applicability of this approach in bigger power systems. For both test systems, normal conditions (no contingencies) and n k contingencies will be analyzed. The IEEE 9-bus test system is presented in Figure 6.1. Figure 6.1 IEEE 9-bus test system The assumptions for the simulations are as follows: The impedance of the load is increased in all the buses proportionally The output power of generators is increased proportionally according to the load increase The power factor of the loads is constant during load increases

61 50 Generators does not reach its limits of reactive power Voltage collapse is determined when power flow simulations using the Newton-Raphson method does not converge The first simulation performed was a comparison of different voltage stability indices as presented in Figure 6.2. To do so, the load of the IEEE 9-bus test system will be increased in steps of about 25 MW. For every step, Index 1 and Index 2 as described by Equations (3.28) and (3.40), are presented. In addition, the expression 1 Z T h /Z load is presented to illustrate the coherency of the results of the indices and the basic matching impedance criteria explained in section Section 2.2. Indeed, can be appreciated how when the impedance of the load (Z load ) equals the impedance of the system (given by a Thévenin equivalent impedance Z T h ), the expression 1 Z T h /Z load is zero and coincides with the zero value given by Index 1 and Index 2 for the point of instability of the system. Finally, this graph includes the P-V curve of the system for this increasing of load conditions. This curve, helps to verify that effectively the voltage collapse which corresponds to a dramatic reduction in the voltage magnitude of about 0.6 p.u., occurs when indices are zero. Figure 6.2 Indices of voltage stability for IEEE 9-bus test system using information of Z bus

62 51 The results presented in Figure 6.2 are obtained using the information of the Z bus of the system that were used to obtain their Thévenin equivalents and the correspondent indices. It can be appreciated that for all the cases, the results are coherent with the theory discussed along this work. One additional observation on this results, is that those are correspond to the bus #7 in the IEEE 9-bus system. As was described in sections Sections and 3.3.4, for the methodologies studied in this work, the VSMs of the entire power system corresponds to the minimum VSMs at load locations. In order to compare the accuracy of Index 1 and Index 2 to estimate the maximum power of the system before losing its stability, the equations for VSM1 and VSM2 will be used (Equations (3.23) and (3.42)). Then, as shown in Figure 6.2, the maximum power of the load of the system to maintain stability is around 800 MW. The indices presented in this figure, can be used to calculate VSMs and to determine the errors associated with this calculations. Table 6.1 shows a comparison between errors obtained by determining VSMs based on Thévenin equivalents calculated by using the Z bus of the system. Table 6.1 Errors in estimation of voltage stability margins based on Z bus System Normal System load (MW) VSM1 (based on Index 1) VSM2 (based on Index 2) MW Error MW Error % % % % % % The results presented in Table 6.1 shown that VSM 1 and 2 has similar performance in terms of accuracy to predict the maximum amount of power the system can stand without losing voltage stability. It can be appreciated that errors in estimating the VSM of the system at different load conditions, remains below of 3%. Can also be observed that the higher the load of the system the better the estimation of the VSMs. The reason for this is that the indices for voltage stability are defined in such a way that they takes values close to cero, as long as the system moves to a voltage collapse.

63 52 Now, the behavior of the VSMs will be presented for conditions where the system is affected by n 1 contingencies. Under n 1 contingencies, the IEEE 9-bus test system suffers a serious reduction of its voltage stability margin due to the change in its equivalent impedance. In this condition the voltage stability margin is about 75 MW for a base power of 315 MW, which means that the maximum power in the system to maintain voltage stability is 390 MW approximately. This represents a reduction of 85% in the VSM with respect to the case where the system has no contingencies (Figure 6.3). Figure 6.3 Voltage Stability Indices under an n 1 contingency Information in Figure 6.3 was calculated by updating the Z bus of the system, i.e. considering the Z bus of the outaged system (corresponding to the disconnection of the line between buses 4 and 7). However, if these indices were obtaining without updating the Z bus of the system, i.e. using the matrix for the system without contingencies, there is an overestimation of the VSMs. As shown in Figure 6.4, the maximum power to the load under n 1 contingency studies, is about 415 MW (corresponding to a VSM margin of 100 MW approximately for a

64 53 base load power of 315 MW). The overestimation of these VSMs is understood in the sense that can be wrongly expected more margin of load increase than the one that the system can really support. Other important finding at this point is that in spite of the overestimation on the VSM, the index used for this simulations (linear index) still shows a linear behavior along the load increase. Moreover, as the main contribution of this work is to propose a way to calculate Thévenin equivalents based on the updated information of the Z bus of the system, Figure 6.4 will present the behavior of linear index under different load increases. For illustrative purposed of the necesity of update the Z bus of the system the linear index calculated in every step of the methodology proposed in Chapter 5. Figure 6.4 Linear index (Index 2) under an n 1 contingency In Figure 6.4, can be appreciated that without considering the update of the Z bus of the system, the linear index for an n 1 contingency could estimate a voltage stability margin of about of 100 MW for a base power of 315 MW. It would mean that the system could stand 415 MW of load. However, as presented in Figure 6.3, the maximum power of the system under

65 54 this condition is about 390 MW. Therefore, the no updating of the Z bus of this test system for an n-1 contingency would represent an overestimation of 20 MW in determining its maximum power and consequently in determining its VSMs. As established previously, for a base power of 315 MW the voltage stability margin is 75 MW (which corresponds to a maximum power of 390 MW) so that, the 20 MW of overestimation in the VSM would correspond to a 33% of error in this estimation. Furthermore, can also be appreciated in this figure how the overestimation of the VSM is reduced to 6.7% by updating the Z bus of the system using the matrix inversion lemma and how there are no overestimation by using Kron reductions in addition to the matrix inversion lemma. Finally, the linear index by updating the both Z bus and Y bus is presented. Can be noticed that there are no differences between the calculation of this index using the approach proposed in this work (Z bus, MIL and Kron reductions) in comparison with the calculation using the approach initially proposed in the literature which is based on the use of elements of the Y bus. On the other hand to validate the applicability of the approach proposed in a bigger system, similar simulations were performed using the IEEE 118-bus test system. Appendix C presents the technical data of this test system (Figure C.1).In this case, similar to the IEEE 9-bus system, the voltage stability indices calculated with information of the Z bus are equal to those calculated with the Y bus (Figure C.2). Figure 6.5 presents the behavior of Thévenin equivalents for bus #60 in the IEEE 118-bus test system along the load increase in the system before voltage collapse. Different topological conditions are considered to evaluate the behavior of the Thévenin equivalents. Indeed, system normal or without contingencies, and some n 1 line contingencies illustrated. It can be appreciated a fairly constant behavior of the Thévenin equivalents. A small reduction in magnitude of the Thévenin equivalent impedance appears for high load conditions in the system. As shown in Figure 6.6, this reduction in the magnitude of the Thévenin impedance represents a conservative condition in the estimation of VSMs. It means that this behavior slightly affects the linearity of the linear index. However, this reduction in the magnitude of the Thévenin impedance represents a conservative condition in the estimation of VSMs. It means that this

66 55 behavior slightly affects the linearity of the linear index under these conditions which can lead to a underestimation of VSM. Figure 6.5 Thévenin equivalents for bus #60 in IEEE 118-bus system Additionally, Figure 6.6 shows the reduction of VSM once contingencies n 1 and n 1 1 occurs in the IEEE 118-bus system. For system normal, the maximum loadability before the voltage collapse can be about 6,400 MW. An n 1 contingency reduces this maximum by 500 MW approximately, which represents a reduction of 7.8% of the margin with respect to the maximum without contingency. Lastly, for an n 1 1 contingency, the maximum loadability of the system would be around 5,100 MW, which represents a reduction of 20% in the margin with respect to the maximum for the non-contingency case. Moreover, similar to explained in Figure 6.4, Figure 6.7 shows the importance of updating the Z bus of the system after contingencies. In this case an n 1 contingency in the IEEE 118-bus test system without updating the Z bus could cause an overestimation of the maximum power of the system of 450 MW by using the linear index. For this case the maximum power

67 56 Figure 6.6 Behavior of linear index (index 2) under different contingencies in IEEE 118-bus system without updating the Z bus is estimated in 6,250 MW while by updating the Z bus this maximum is 5,800 MW approximately. Finally, it can be appreciated that for both test systems studied, topological changes in the network due to equipment disconnections can seriously affect the estimation of VSMs. For both cases the update of the Z bus of the system to determine Thévenin equivalents is required for accurate estimation of VSMs. In consequence, the use of the Z bus to compute Thévenin equivalents for VSMs estimation is an appropriate way to compute this indicators. Moreover, for real-time applications in control centers, the approach proposed in this work to use the matrix inversion lemma and Kron reductions to quickly update the Thévenin equivalents, represent a suitable option to perform these computations in a fast and accurate way.

68 57 Figure 6.7 Voltage stability margin comparison by updating Z bus

69 58 CHAPTER 7. CONCLUSIONS The determination of Thévenin equivalents of the system is a powerful tool to analyze longterm voltage stability issues. Indeed, important approaches have been developed recently to improve the determination of those equivalents. These developments are mainly focused in find tools to refine the way to capture as accurate as possible, the behavior of the entire power system in a single equivalent.. The use of multi-load Thévenin equivalents to estimate steady-state VSMs has potential applications in real-time systems but require a fast update of its parameters when the system topology changes. This is especially important in long-term voltage stability events since they involve multiple disconnections of equipment accompanied with load increase. The obtention of multi-load Thévenin equivalents by using the impedance matrix of the system (Z bus ) is equivalent to obtain by using the admittance matrix of the system (Y bus ), which is the approach found in the literature review in this subject. The use of the Z bus to determine the parameters required to calculate multi-load Thévenin equivalents, allows the use of the matrix inversion lemma to update the information contained in that matrix, when the topology of the system changes due to the connection or disconnection of power equipment such as transmission lines, power transformers or series and shunt compensators. As this update only requires additions and multiplication of vectors by matrices, the computation times are much lower than those required by the initial approaches that involves inversion of matrices. Special consideration should be taken into account when the Z bus should be updated due to the connection or disconnection of transmission lines. In these cases, the application of matrix inversion lemma only removes the series impedance of the line from the Z bus. Line shunt capacitances should also be removed or added to Z bus to obtain a proper determination of the parameters required for the determination of Thévenin equivalents. To solve this issue, Kron

70 59 reductions are proposed in this work. Kron reductions can also be performed by simple matrix summations, keeping the speed of this approach. The approach proposed in this work contributes to the solution of one the current issues in determine VSMs in real-time, which is the consideration of connection and disconnection of power equipment during long-term voltage stability issues. This methodology is suitable to be implemented in real systems, because it is still using information that can be obtained from SCADA/EMS or wide-are synchrophasor monitoring systems.

71 60 CHAPTER 8. FUTURE WORK The next steps in this field can be the implementation of the methodology proposed in this work in real-time digital simulator (RTDS) systems, in order to investigate the details to construct the algorithms required for its application in real systems. Details related with, sample rates, communications requirements, performance of the algorithms under multiple simultaneous contingencies and selection of the source of information of the voltages of the system (from SCADA/EMS or synchrophasor systems), are visualized as very immediate issues to address. Other interesting subject for future research in long-term voltage stability analysis is the development of indices and margins considering synchronous generators reaching its limits of reactive power. As presented in Chapter 3, the approaches reviewed in this work does not address this kind of situations which are a realistic scenario when voltage stability issues are present in power systems. Finally, as the use of VSMs for real-time systems have a potential application in developing remedial action schemes (RAS), the determination of thresholds for the automatic actions embedded in those schemes and the identification of other uses of Thévenin equivalents, are also envisioned as part of the future work in this subject..

72 61 APPENDIX A. PROOF OF MATRIX INVERSION LEMMA Here is presented the matrix inversion lemma as described by Bergen and Vittal (2000). Supposing that is wished to find the inverse of M +µaa T, where µ is a scalar, a is an n-vector and matrix M is symmetrical with dimensions n n with a known inverse, M 1. Then is possible to use the following inversion formula (the result in Equation (A.1) is a special case of a more general result known as the Householderf ormula.): [M + µaa T ] 1 = M 1 γbb T (A.1) where b = M 1 a γ = (µ 1 + a T b) 1 (A.2) (A.3) Proof of matrix inversion lemma The proof is done multiplying the modified matrix by its inverse and showing that the result is the identity matrix, 1. [M + µa1 T ][1 1 γbb T ] = MM 1 γmbb T + µaa T M 1 µγaa T bb T = 1 + remainder Now, lets proof that the remainder = 0, dividing the remainder by γ: remainder γ = Mbb T + µa 1 γ at M 1 µaa T bb T = ab T + µa(µ 1 + a T b)b T µaa T bb T = 0 (A.4) Where Mb = a, from Equation (A.2), and γ 1 = µ 1 + a T b, from Equation (A.3).

73 62 APPENDIX B. BUS IMPEDANCE MATRIX MANIPULATION Here are presented four types of circuit modifications and the rules for calculating updated Z bus of the system, Z n bus in terms of Z bus as is described by Bergen and Vittal (2000). Modification 1 Add a branch with impedance z b from a new (r + 1)st bus (node) to the reference node. Rule 1 Z n bus is given by the following (r + 1) (r + 1) matrix. Z n bus = Z bus 0 0 z b (B.1) Modification 2 Add a branch z b from a new (r + 1)st node to the ith node Rule 2 If the ith column of Z bus is Z i and the iith element of Z bus is Z ii, then Z n bus = Z bus Z i Z i Z ii + z b (B.2) Where Z i is replication of the ith column of Z bus. Modification 3 Add a branch with impedance z b between (existing) ith node and the reference node.

74 63 Rule 3 Suppose that the ith column of Z bus is Z i and the iith element of Z bus is Z ii. Then by augmenting the old Z bus with an extra row and column as shown in Equation (B.2), the size of the matrix is (r + 1) (r + 1). The voltage at the (r + 1)th bus is the voltage at the reference node and is equal to zero. Hence, by Kron reducing the last row and column the expression for Z n kj can be written as: Z n kj = Z kj Z k(r+1)z r+1 j Z ii + z b (B.3) Modification 4 Add a branch z b between (existing) ith and jth nodes. Rule 4 Suppose that the ith and jth columns of Z bus are Z i and Z j and the iith, jjth and ijth elements of Z bus are Z ii, Z jj, Z ij, respectively. Then Z n bus = Z bus γbb T (B.4) where b = Z i Z j and γ = (z b + Z ii + Z jj 2Z ij ) 1. Note: By the rules of matrix multiplication, b 1 b 2 1 b 1 b 2... b 1 b r bb T b 2. ] b 2 b 1 b = [b b 2 b r 1 b 2... b r =.... b r b r b 1 b r b 2... b 2 r (B.5)

75 64 APPENDIX C. IEEE 118 BUS TEST SYSTEM DATA Table C.1 IEEE 118-bus test system - Bus data bus type Pd Qd Gs Bs area Vm Va base kv zone Vmax Vmin Continued on next page

76 65 Table C.1 (Continued) IEEE 118-bus test system - Bus data bus type Pd Qd Gs Bs area Vm Va base kv zone Vmax Vmin Continued on next page

77 66 Table C.1 (Continued) IEEE 118-bus test system - Bus data bus type Pd Qd Gs Bs area Vm Va base kv zone Vmax Vmin Continued on next page

78 67 Table C.1 (Continued) IEEE 118-bus test system - Bus data bus type Pd Qd Gs Bs area Vm Va base kv zone Vmax Vmin Table C.2 IEEE 118-bus test system - Generator data bus Pg Qg Qmax Qmin Vg mbase status Pmax Pmin Continued on next page

79 68 Table C.2 (Continued) IEEE 118-bus test system - Generator data bus Pg Qg Qmax Qmin Vg mbase status Pmax Pmin

80 69 Table C.3 IEEE 118-bus test system - Branch data fbus tbus r x b ratio angle status Continued on next page

81 70 Table C.3 (Continued) IEEE 118-bus test system - Branch data fbus tbus r x b ratio angle status Continued on next page

82 71 Table C.3 (Continued) IEEE 118-bus test system - Branch data fbus tbus r x b ratio angle status Continued on next page

83 72 Table C.3 (Continued) IEEE 118-bus test system - Branch data fbus tbus r x b ratio angle status Continued on next page

84 73 Table C.3 (Continued) IEEE 118-bus test system - Branch data fbus tbus r x b ratio angle status

85 74 Figure C.1 One-line diagram of IEEE 118-bus test system

86 75 Figure C.2 Comparison of linear index using Y bus and Z bus in IEEE 118-bus test system