LÝ THUYẾT VÀ THỰC HÀNH VỀ ĐÁNH GIÁ TRẠNG THÁI HỆ THỐNG ĐIỆN (Power System State Estimation Theory and Implementation)

Offering an uptodate account of the strategies utilized in state estimation of electric power systems, this text provides a broad overview of power system operation and the role of state estimation in overall energy management. It uses an abundance of examples, models, tables, and guidelines to clearly examine new aspects of state estimation, the testing of network observability, and methods to assure computational efficiency.Includes numerous tutorial examples that fully analyze problems posed by the inclusion of current measurements in existing state estimators and illustrate practical solutions to these challenges.Written by two expert researchers in the field, Power System State Estimation extensively details topics never before covered in depth in any other text, including novel robust state estimation methods, estimation of parameter and topology errors, and the use of ampere measurements for state estimation. It introduces various methods and computational issues involved in the formulation and implementation of the weighted least squares (WLS) approach, presents statistical tests for the detection and identification of bad data in system measurements, and reveals alternative topological and numerical formulations for the network observability problem.

Power System State Estimation Theory and Implementation

Ali Abur Antonio Gomez Exposito

M A R C E L

MARCEL DEKKER, INC NEW YORK - BASEL

Although great care has been taken to provide accurate and current information, neither theauthor(s) nor the publisher, nor anyone else associated with this publication, shall be liablefor any loss, damage, or liability directly or indirectly caused or alleged to be caused by thisbook The material contained herein is not intended to provide specific advice or recom-mendations for any specific situation.

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1 Power Distribution Planning Reference Book, /V Aee MW/s

2 Transmission Network Protection: Theory and Practice, V G.Pa/fnan/rar

3 Electrical Insulation in Power Systems, /V /V Ma///r, /L /4 /4ra/ny, andM / Qt/resn/

/=)/-4 Electrical Power Equipment Maintenance and Testing, Fat// G///

5 Protective Relaying: Principles and Applications, Second Edition, J.Aesv/ls 5/ac/rAt/rn

6 Understanding Electric Utilities and De-Regulation, Aorr/n Pn/Y/psonand /V Aee M^7//s

7 Electrical Power Cable Engineering, M7///am/4 7*nt/e

8 Electric Systems, Dynamics, and Stability with ArtificialIntelligence Applications, James /3 Memo/? and Monamed F F/-

9 Insulation Coordination for Power Systems, /Sndretv /?

10 Distributed Power Generation: Planning and Evaluation, /V Aeet/V////s and t/Ma/fer G ScoM

1 1 Electric Power System Applications of Optimization, James A.Momoh

1 2 Aging Power Delivery Infrastructures, /V Aee M/////S, Gregory V.M/e/c/?, and /?anda// /? Scn/yeAer

13 Restructured Electrical Power Systems: Operation, Trading, andVolatility, Mo/?am/nacf Snan/cfenpot//* and Mtvwaffa^ /4/omous/?

14 Electric Power Distribution Reliability, /?/cnardF Frown

1 5 Computer-Aided Power System Analysis, Ramasamy /Vafa/*a/an

1 6 Power System Analysis: Short-Circuit Load Flow and Harmonics,

Ex-19 Dielectrics in Electric Fields, Gort/r G /?a/tv

20 Protection Devices and Systems for High-Voltage Applications,Wad/rn/r Givrey/cn

21 Electrical Power Cable Engineering: Second Edition, Revised andExpanded, lAW/am /4 7*nue

22 Vehicular Electric Power Systems: Land, Sea, Air, and Space hicles, /4//'fmad/^ Me^rdadFnsan/, and Jonn M M///er

Ve-23 Power Distribution Ptanning Reference Book: Second Edition, vised and Expanded, AV Aee MW/s

Re-24 Power System State Estimation: Theory and implementation, /4///)At//* a;7Gf/4/7foA?/b Gomez Fxpds/Yo

ADDITIONAL VOLUMES IN PREPARATION

To Our Parents

One of the major causes of the New York power outage of 1987 was mately traced to incorrect information about the status of a circuit in thesystem The operation of a major new market, such as the PJM market,would be nearly impossible without the capabilities afforded by state es-timation It is not yet known to what extent the blackout of 2003 mayhave been in part caused by missing information Undoubtedly, thus, thetheme of this book is an important one From its origins as a mathematicalcuriosity in the 1970's to its limited use during the 1980's to its expandedbut not yet central role in the operation of the system in 1990's, nowa-days state estimation has become nothing less than the cornerstone uponwhich a modern control center for a power system is built Furthermore,

ulti-to the extent that markets must be integrated with reliable system tion, state estimation has acquired a whole new role: it is the foundationfor the creation and operation of real time markets in power systems, andthus the foundation for all markets, real time or not, since ultimately allmarkets must derive their valuations from real time information Amongthe most important properties of a properly operated market is somethingthat I shall call "auditability," that is, the ability to go back and verifywhy certain things were done the way they were Without an accurateand ongoing knowledge of the status of every Row and every voltage in thesystem at all times, it would be impossible to "go back" and explain why,for example, prices were what they were at a particular time

opera-This book, written by two of the most prominent researchers in theHeld, brings a fresh perspective to the problem of state estimation Thebook offers a blend of theory and mathematical rigor that is unique andvery exciting In addition to the more traditional topics associated withweighted least squares estimation (including such & r^wewr topics as baddata detection and topology estimation), this book also brings forth severalnew aspects of the problem of state estimation that have not been presented

in a systematic manner prior to this effort Most notable among these arethe chapters on robust estimation and the work on ampere measurements,

to name just two In this sense the book distinguishes itself from the otherstate estimation book known to this writer, the book by the late great AlcirMonticelli In such way this book is a great complement to the efforts ofMonticelli.

The readers of the book will also find it quite pleasing to have a nicereview of a number of topics relating to efficient computation The bookprovides excellent material for those wishing to review the topic of efficientcomputation and sparsity in general Proper attention is paid throughoutthe book to computational efficiency issues Given that computationalefficiency is the key to making state estimation work in the first place, theimportance of this topic cannot be understressed

Although the bibliography associated with every chapter and with theappendix is short, it is all quite pertinent and very much to the point

In this sense, the readers can get focused and rapid access to additionaloriginal material should they wish to investigate a topic further

I am particularly pleased to have had the opportunity to comment onboth the theme of the book and the book itself, since the authors of thisbook are unquestionably respected leaders in the field and are themselvesthe originators of many of the ideas that are in present use throughout theHeld of state estimation and beyond I am sure readers will share with methese sentiments after reading this book

Fernando L Alvarado

Power system state estimation is an area that matured in the past threedecades Today, state estimators can be found in almost every power sys-tem control center While there have been numerous papers written onmany different aspects of state estimation, ranging from its mathemati-cal formulation to the implementation and start-up issues at the controlcenters, relatively few books have been published on this subject

This book is the product of a long-term collaboration between the thors, starting from the summer of 1992 when they worked at the University

au-of Seville on a joint project that was sponsored by the Ministry au-of Scienceand Education of the Spanish Government Since then, they have spenttwo summers working together on different projects related to state esti-mation and continued their collaboration They each taught regular andshort courses on this topic and developed class notes, which make up most

of the material presented in this book

The chapters of the book are written in such a way that it can be used as

a textbook for a graduate-level course on the subject However, it may also

be used as a supplement in an undergraduate-level course in power systemanalysis Professionals working in the Reid of power systems may also findthe chapters of the book useful as self-contained references on specific issues

of interest

The book is organized into nine chapters and two appendices The ductory chapter provides a broad overview of power system operation andthe role of state estimators in the overall energy management system con-figurations The second chapter describes the modeling of electric networksduring steady state operation and formulates one of the most commonlyused state estimation methods in power systems, namely the weighted leastsquares (WLS) method Application of the WLS method to power systemstate estimation presents several challenges ranging from numerical insta-bilities to the handling of measurements with special constraints Chapter

intro-3 presents various techniques for addressing these problems Network servability is analyzed in Chapter 4, where a brief review of networks and

ob-graphs is foHowed by the description of alternative methods for networkobservability determination Chapter 5 is concerned with detecting andidentifying incorrect measurements In this chapter, it is assumed that theWLS method is used for state estimation and bad data processing takesplace after the convergence of the WLS state estimator In Chapter 6, thetopic of robust estimation is introduced and some robust estimation meth-ods which have already been investigated for power system applicationsare presented Chapter 7 is about different methods of estimating trans-mission line parameters and transformer taps These network parametersare typically assumed to be perfectly known, despite the fact that errors

in them significantly affect the state estimates The problem of topologyerror identification is the topic of Chapter 8 Topology errors cause stateestimators to diverge or converge to incorrect solutions The challenges indetecting and identifying such errors and methods of overcoming them arepresented in this chapter Finally, Chapter 9 discusses the use of amperemeasurements and various issues associated with their presence in the mea-surement set The book also has two appendices, one on basic statisticsand the other on sparse linear equations

All chapters, except for the first one, end with some practice problems.These may be useful if the book is adopted for teaching a course at either thegraduate or undergraduate level The first five chapters are recommended

to be read in the given order since each one builds on the previously coveredmaterial However, the last four chapters can be covered in any arbitraryorder

Parts of the work presented in this book have been funded by theUnited States National Science Foundation projects ECS-9500118 and ECS-

8909752 and by the Spanish Government, Directory of Scientific and nical Investigations (DGICYT) Summer Research Grants No SAB 95-0354and SAB 92-0306, and Research Project No PB94-1430

Tech-It has been a pleasure to work with our many graduate students whohave contributed to the development and implementation of some of theideas in this book Specifically, we are happy to acknowledge the contri-butions made by Esther Romero, Francisco Gonzalez, Antonio de la Villa,Mehmet Kemal Celik, Hongrae Kim, Fernando Hugo Magnago and Bei Gou

in their respective research projects

Finally, we are also grateful for the constant encouragement and port that we have received from our spouses, Aysen and Cati, during thepreparation of this book

sup-Ali AburAntonio Gomez Exposito

Foreword (Fernando L Alvarado)

Preface

1 Introduction

1.1 Operating States of a Power System

1.2 Power System Security Analysis

2.3 Building the Network Model

2.4 Maximum Likelihood Estimation

2.4.1 Gaussian (Normal) Probability Density Function2.4.2 The Likelihood Function

2.5 Measurement Model and Assumptions

2.6 WLS State Estimation Algorithm

2.6.1 The Measurement Function, A(a^)2.6.2 The Measurement Jacobian, R2.6.3 The Gain Matrix, G

2.6.4 Cholesky Decomposition of (72.6.5 Performing the Forward/Back Substitutions2.7 Decoupled Formulation of the

WLS State Estimation2.8 DC State Estimation Model

2.9 Problems

3 Alternative Formulations of the WLS State Estimation 3.1 Weaknesses of the Normal Equations Formulation

3.2 Orthogonal Factorization

3.3 Hybrid Method

3.4 Method of Peters and Wilkinson

3.5 Equality-Constrained WLS State Estimation

3.6 Augmented Matrix Approach

3.7 Blocked Formulation

3.8 Comparison of Techniques

3.9 Problems

References

4 Network Observability Analysis

4.1 Networks and Graphs

4.1.1 Graphs 4.1.2 Networks 4.2 NetworkMatrices

4.2.1 Branch to Bus Incidence Matrix 4.2.2 Fundamental Loop to Branch Incidence Matrix 4.3 LoopEquations

4.4 Methods of Observability Analysis

4.5 Numerical Method Based on the Branch Variable

Formula-tion

4.5.1 New Branch Variables 4.5.2 Measurement Equations 4.5.3 Linearized Measurement Model 4.5.4 Observability Analysis

4.6 Numerical Method Based on the Nodal Variable Formulation 4.6.1 Determining the Unobservable Branches

4.6.2 Identification of Observable Islands 4.6.3 Measurement Placement to Restore Observability

4.7 Topological Observability Analysis

Method 4.7.1 Topological Observability Algorithm 4.7.2 Identifying the Observable Islands 4.8 Determination of Critical Measurements

4.9 Measurement Design

4.10 Summary

4.11 Problems

References

5 Bad Data Detection and Identification

5.1 Properties of Measurement Residuals

5.2 Classification of Measurements

5.3 Bad Data Detection and IdentiRability

5.4 Bad Data Detection

5.4.1 Chi-squares x^ Distribution5.4.2 Use of x^ Distribution for Bad Data Detection5.4.3 x^-Test for Detecting Bad Data in WLS State Esti-

mation

5.4.4 Use of Normalized Residuals for Bad DataDetection

5.5 Properties of Normalized Residuals

5.6 Bad Data Identification

5.7 Largest Normalized Residual (r^aa) Test

5.7.1 Computational Issues5.7.2 Strengths and Limitations of the r^ag Test5.8 Hypothesis Testing Identification (HTI)

5.8.1 Statistical Properties of eg5.8.2 Hypothesis Testing5.8.3 Decision Rules5.8.4 HTI Strategy Under Fixed /35.9 Summary

5.10 Problems

Reference

6 Robust State Estimation

6.1 Introductio

6.2 Robustness and Breakdown Points

6.3 Outliers and Leverage Points

6.3.1 Concept of Leverage Points6.3.2 Identification of Leverage Measurements6.4 M-Estimators

6.4.1 Estimation by Newton's Method6.4.2 Iteratively Re-weighted Least SquaresEstimation

6.5 Least Absolute Value (LAV) Estimation

6.5.1 Linear Regression6.5.2 LAV Estimation as an LP Problem6.5.3 Simplex Based Algorithm

6.5.4 Interior Point Algorithm6.6 Discussion

6.7 Problems

References

7 Network Parameter Estimation

7.1 Introduction

7.2 Influence of Parameter Errors on State

Estimation Results7.3 Identification of Suspicious Parameters

7.4 Classification of Parameter Estimation

Methods

7.5 Parameter Estimation Based on Residua! Sensitivity Analysis

7.6 Parameter Estimation Based on State

Vector Augmentation

7.6.1 Solution Using Conventional Normal Equation

7.6.2 Solution Based on Kalman Filter Theory7.7 Parameter Estimation Based on Historical Series of Data7.8 Transformer Tap Estimation

7.9 Observability of Network Parameters

8.2 Types of Topology Errors

8.3 Detection of Topology Errors

8.4 Classification of Methods for Topology Error Analysis

8.5 Preliminary Topology Validation

8.6 Branch Status Errors

8.6.1 Residual Analysis8.6.2 State Vector Augmentation8.7 Substation Configuration Errors

8.7.1 Inclusion of Circuit Breakers in the Network Model8.7.2 WLAV Estimator

8.7.3 WLS Estimator8.8 Substation Graph and Reduced Model

8.9 Implicit Substation Model: State and

Status Estimation8.10 Observability Analysis Revisited

8.11 Problems

References

9 State Estimation Using Ampere Measurements

9.1 Introduction

9.2 Modeling of Ampere Measurements

9.3 Difficulties in Using Ampere

Measurements

9.4 Inequality-Constrained State Estimation

9.5 Heuristic Determination of F-# Solution Uniqueness

9.6 Algorithmic Determination of Solution

Uniqueness9.6.1 Procedure Based on the Residual Covariance Matrix9.6.2 Procedure Based on the Jacobian Matrix

9.7 Identification of Nonuniquely Observable Branches

9.8 Measurement Classification and Bad Data Identific

9.8.1 LS Estimation9.8.2 LAV Estimation9.9 Problems

ReferencesAppendix A Review of Basic Statistics

A.I Random Variables

A.2 The Distribution Function (d.f.), F(x)

A.3 The Probability Density Function (p.d.f), f(x)

A.4 Continuous Joint Distributions

A.5 Independent Random Variables

A.6 Conditional Distributions

A.7 Expected Value

A.8 Variance

A.9 Median

A.10 Mean Squared Error

A.11 Mean Absolute Error

A.12 Covariance

A.13 Normal Distribution

A.14 Standard Normal Distribution

A.15 Properties of Normally Distributed Random VariablesA.16 Distribution of Sample Mean

A.17 Likelihood Function and Maximum

Likelihood EstimatorA.17.1 Properties of MLE'sA.18 Central Limit Theorem for the Sample Mean

Appendix B Review of Sparse Linear Equation SolutionB.I Solution by Direct Methods

B.2 Elementary Matrices

B.3 LU Factorization Using Elementary Matrices

B.3.1 Grout's AlgorithmB.3.2 Dooh'ttle's AlgorithmB.3.3 Factorization of Sparse Symmetric MatriceB.3.4 Ordering Sparse Symmetric MatricesB.4 Factorization Path Graph

B.5 Sparse Forward/Back Substitutions

B.6 Solution of Modified Equations

B.6.1 Partial RefactorizationB.6.2 Compensation

B.7 Sparse Inverse

B.8 Orthogonal Factorization

B.9 Storage and Retrieval of Sparse Matrix ElementsB.10 Inserting and/or Deleting Elements in a Linked ListB.10.1 Adding a Nonzero Element

B.10.2 Deleting a Nonzero ElementReferences

at these substations The output voltages of generators typically do notexceed 30-kV Hence, transformers are used to increase the voltage levels

to levels ranging from 69-kV all the way up to 765-kV at the generatorterminals for efficient power transmission High voltage is preferred atthe transmission system for different reasons one of which is to minimizethe copper losses that are proportional to the ampere Rows along lines

At the receiving end, the transmission systems are connected to the transmission or distribution systems which are operated at lower voltagelevels ranging from 115-KV to 4.16-KV Distribution systems are typicallyconfigured to operate in a radial configuration, where feeders stretch fromdistribution substations and form a tree structure with their roots at thesubstation and branches spreading over the distribution area

sub-1.1 Operating States of a Power System

The operating conditions of a power system at a given point in time can bedetermined if the network model and complex phasor voltages at every sys-tem bus are known Since the set of complex phasor voltages fully specifiesthe system, it is referred to as the static state of the system According to[1], the system may move into one of three possible states, namely normal,emergency and restorative, as the operating conditions change

A power system is said to operate in a normal state if all the loads in thesystem can be supplied power by the existing generators without violating

any operational constraints Operational constraints include the limits onthe transmission line flows, as well as the upper and lower limits on busvoltage magnitudes A normal state is said to be secwre if the system canremain in a normal state following the occurrence of each contingency from

a list of critical contingencies Common contingencies of interest are mission line or generator outages due to unexpected failures of equipment

trans-or natural causes such as sttrans-orms Otherwise, the ntrans-ormal state is classified asmsecwe where the power balance at each bus and all operating inequalityconstraints are still satisfied, yet the system remains vulnerable with re-spect to some of the considered contingencies If the system is found to be

in a normal but msecwe operating state then, preventive actions must betaken to avoid its move into an emergency state Such preventive controlscan be determined typically by the help of a security constrained optimalpower flow program which accounts for a list of critical contingencies.Operating conditions may change significantly due to an unexpectedevent which may cause the violation of some of the operating constraints,while the power system continues to supply power to all the loads in thesystem In such a situation the system is said to be operating in an emer-gency state Emergency state requires immediate corrective action to betaken by the operator so as to bring the system back to a normal state.While the system is in the emergency state, corrective control measuresmay be able to avoid system collapse at the expense of disconnecting variousloads, lines, transformers or other equipment As a result, the operatinglimit violations may be eliminated and the system may recover stabilitywith reduced load and reconfigured topology Then, the load versus gener-ation balance may have to be restored in order to start supplying power toall the loads Such an operating state is called the restorative state, and theactions to be taken in order to transform it into a normal state are referred

to as restorative controls The state diagram in Figure 1.1 illustrates thepossible transitions between the different operating states defined above

1.2 Power System Security Analysis

Power systems are operated by system operators from the area controlcenters The main goal of the system operator is to maintain the system inthe normal secure state as the operating conditions vary during the dailyoperation Accomplishing this goal requires continuous monitoring of thesystem conditions, identification of the operating state and determination

of the necessary preventive actions in case the system state is found to bemsecwe This sequence of actions is referred to as the security analysis ofthe system

The first stop of security analysis is to monitor the current state of thesystem This involves acquisition of measurements from all parts of the

NORMAL STATE

SECUREorINSECURE

RESTORATIVESTATE

PARTIAL ORTOTAL BLACKOUT

EMERGENCYSTATEOPERATIONAL LIMITSARE VIOLATED

Figure 1.1 State Diagram for Power System Operation

system and then processing them in order to determine the system state.The measurements may be both of analog and digital (on/off status ofdevices) type Substations are equipped with devices called remote terminalunits (RTU) which collect various types of measurements from the fieldand are responsible for transmitting them to the control center Morerecently, the so-called intelligent electronic devices (IED) are replacing orcomplementing the existing RTUs It is possible to have a mixture of thesedevices connected to a local area network (LAN) along with a SCADAfront end computer, which supports the communication of the collectedmeasurements to the host computer at the control center The SCADAhost computer at the control center receives measurements from all themonitored substations' SCADA systems via one of many possible types ofcommunication links such as fiber optics, satellite, microwave, etc Figure1.2 shows the configuration of the EMS/SCADA system for a typical powersystem

Measurements received at the control center will include line powerHows, bus voltage and line current magnitudes, generator outputs, loads,circuit breaker and switch status information, transformer tap positions,and switchable capacitor bank values These raw data and measurementsare processed by the state estimator in order to filter the measurement noiseand detect gross errors State estimator solution will provide an optimalestimate of the system state based on the available measurements and onthe assumed system model This will then be passed on to all the energymanagement system (EMS) application functions such as the contingencyanalysis, automatic generation control, load forecasting and optimal powernow, etc The same information will also be available via a LAN connection

ANALYSIS FUNCTIONS

LocaiArea Network

FUNCTtONSA

Monitored Devices Substation Figure 1.2 EMS/SCAOA system configuration.

to the corporate offices where other planning and analysis functions can beexecuted off-line

Initially, power systems were monitored only by supervisory control tems These are control systems which essentially monitor and control thestatus of circuit breakers at the substations Generator outputs and the sys-tem frequency were also monitored for purposes of Automatic GenerationControl (AGC) and Economic Dispatch (ED) These supervisory controlsystems were later augmented by real-time system-wide data acquisitioncapabilities, allowing the control centers to gather all sorts of analog mea-surements and circuit breaker status data from the power system This led

sys-to the establishment of the first Supervisory Control and Data Acquisition(SCADA) Systems The main motivation behind this development was thefacilitation of security analysis Various application functions such as con-tingency analysis, corrective real and reactive power dispatch could not beexecuted without knowing the real-time operating conditions of the system.However, the information provided by the SCADA system may not always

be reliable due to the errors in the measurements, telemetry failures, munication noise, etc Furthermore, the collected set of measurements maynot allow direct extraction of the corresponding A.C operating state of thesystem For instance, bus voltage phase angles are not typically measured,and not all the transmission line flows are available Besides, it may not beeconomically feasible to telemeter all possible measurements even if theyare available from the transducers at the substations

com-1.3 State Estimation

The foregoing concerns were first recognized and subsequently addressed

by Fred Schweppe, who proposed the idea of state estimation in power tems [2, 3, 4] Introduction of the state estimation function broadened thecapabilities of the SCADA system computers, leading to the establishment

sys-of the Energy Management Systems (EMS), which would now be equippedwith, among other application functions, an on-line State Estimator (SE)

In order to identify the current operating state of the system, stateestimators facilitate accurate and efficient monitoring of operational con-straints on quantities such as the transmission line loadings or bus voltagemagnitudes They provide a reliable real-time data base of the system,including the existing state based on which, security assessment functionscan be reliably deployed in order to analyze contingencies, and to determineany required corrective actions

The state estimators typically include the following functions:

* Topology processor: Gathers status data about the circuit breakersand switches, and configures the one-line diagram of the system

* Observability analysis: Determines if a state estimation solution forthe entire system can be obtained using the available set of mea-surements Identifies the unobservable branches, and the observableislands in the system if any exist

< State estimation solution: Determines the optimal estimate for thesystem state, which is composed of complex bus voltages in the en-tire power system, based on the network model and the gatheredmeasurements from the system Also provides the best estimates forall the line Hows, loads, transformer taps, and generator outputs

* Bad data processing: Detects the existence of gross errors in the surement set Identifies and eliminates bad measurements providedthat there is enough redundancy in the measurement configuration

mea-< Parameter and structural error processing: Estimates various work parameters, such as transmission line model parameters, tapchanging transformer parameters, shunt capacitor or reactor param-eters Detects structural errors in the network configuration andidentifies the erroneous breaker status provided that there is enoughmeasurement redundancy

net-Thus, power system state estimator constitutes the core of the on-linesecurity analysis function It acts like a filter between the raw measurementsreceived from the system and all the application functions that require themost reliable data base for the current state of the system Figure 1.3

describes the data and functional interfaces between the various tion functions involved in the on-line static security assessment procedure.Raw measurements which include the switch and circuit breaker positions

applica-in the substations, are processed by the topology processor, which applica-in turngenerates a bus/branch model of the power system This model not only in-cludes all buses within the area of the control center EMS, but also selectedbuses from the neighboring systems The information and measurementsobtained from the neighboring systems are used to build and update theexternal system model Furthermore, there may be unobservable pocketswithin one's own area due to temporary loss of telemetry, rejected baddata or other unexpected failures Such areas whether physically locatedwithin the control area or part of the external system, will be estimated viathe use of pseudo measurements Pseudo measurements can be generatedbased on short term load forecasts, generation dispatch, historical records

or other similar approximation methods Naturally, they are assigned highvariances (low weights) or they can be forced to be critical measurements

by design Definition and properties of a critical measurement will be cussed in detail in chapter 5 In addition, there may be passive buses with

dis-no generation or load, having net zero real and reactive power injection.Such bus injections, even though not measured, can be used as error freemeasurements in the state estimation formulation and referred to as "vir-tual" measurements The results obtained by the state estimator will bechecked in order to classify the system state into one of the three categoriesshown in Figure 1.1 If it is found to be in the normal state, then contin-gency analysis will be carried out to determine the system security against aset of predetermined contingencies In case of insecurity, preventive controlactions have to be calculated via the use of a software tool such as a securityconstrained optimal power flow Implementing these preventive measureswill move the system into the desired normal and secwe state Figure 1.3

also indicates the emergency and restorative control actions which will bedeployed under a&nonnaZ operating conditions, however these topics arebeyond the scope of this book and will not be discussed any further

1.4 Summary

Power systems are continuously monitored in order to maintain the ating conditions in a normal and secure state State estimation function isused for this purpose It processes redundant measurements in order to pro-vide an optimal estimate of the current operating state State estimationproblem has been investigated by several researchers since its introduc-tion in the late 1960s Being an on-line function, computational issues re-lated to speed, storage and numerical robustness of the solution algorithmshave been carefully studied Measurement configuration and its effect on

oper-state estimation have been addressed by the developed observability ysis methods State estimators also function as filters against incorrectmeasurements, data and other information received through the SCADAsystem Hence, the subject of bad data processing has been investigatedand detection/identification algorithms for errors in analog measurementshave been developed Special methods also exist for the identification ofthose errors related to the topology information and/or network parame-ters On the other hand, the use of ampere measurements present someproblems which do not exist in their absence from the measurement set.

anal-In the following chapters, these issues will be presented in more detail andmethods which are developed to address them will be described

[3] Schweppe F.C and Rom D.B., "Power System Static-State tion, Part II: Approximate Model", IEEE Transactions on Power Ap-paratus and Systems, Vol.PAS-89, January 1970, pp.125-130.

Estima-[4] Schweppe F.C., "Power System Static-State Estimation, Part III: plementation" , IEEE Transactions on Power Apparatus and Systems,Vol.PAS-89, January 1970, pp 130-135

Im-[5] Fink L.H and Carlsen K., "Operating under Stress and Strain", IEEESpectrum, March 1978

[6] N Balu et al "On-line Power System Security Analysis", Proc of theIEEE, vol 80(2), pp 262-280

of such errors will be separately discussed later on in chapters 7 and 8.

2.2 Component Modeling and Assumptions

Power system is assumed to operate in the steady state under balancedconditions This implies that all bus loads and branch power flows will

be three phase and balanced, all transmission lines are fully transposed,and all other series or shunt devices are symmetrical in the three phases.These assumptions allow the use of single phase positive sequence equivalentcircuit for modeling the entire power system The solution that will beobtained by using such a network model, will also be the positive sequencecomponent of the system state during balanced steady state operation As

in the case of the power flow, all network data as well as the networkvariables, are expressed in the per unit system The following componentmodels will thus be used in representing the entire network

Figure 2.1 Equivaient circuit for a transmission tine

2.2.2 Shunt Capacitors or Reactors

Shunt capacitors or reactors which may be used for voltage and/or reactivepower control, are represented by their per phase susceptance at the corre-sponding bus The sign of the susceptance value will determine the type ofthe shunt element It will be positive or negative corresponding to a shuntcapacitor or reactor respectively

2.2.3 Tap Changing and Phase Shifting Transformers

Transformers with off-nominal but in-phasc taps, can be modeled as seriesimpedances in scries with ideal transformers as shown in Figure 2.2 The

two transformer terminal buses m and /c are commonly designated as theimpedance side and the tap side bus respectively.

Figure 2.2 Equivatent circuit for an off-nominat tap transformer

The nodal equations of the two port circuit of Figure 2.2 can be derived

by first expressing the current flows ^^ and i^ at each end of the seriesbranch R + jJf Denoting the admittance of this branch ^ — m by y, theterminal current injections will be given by:

(2.1)

Substituting for ^rn and

the final form will be obtained as follows:

(2.2)

where a is the in phase tap ratio Figure 2.3 shows the corresponding twoport equivalent circuit for the above set of nodal equations

Figure 2.3 Equivatent circuit of an in-phase tap changer

For a phase shifting transformer where the off-nominal tap value a, iscomplex, the equations will slightly change as:

yielding the following set of nodal equations:

(2.3)

Note the loss of reciprocity as the admittance matrix is no longer cal Therefore, a passive equivalent circuit such as the one shown in Figure2.3 for the in-phase tap changer, can no longer be realized for the phaseshifting transformer However, the circuit equations can still be solved asbefore by only modifying the admittance matrix which is no longer sym-metrical

symmetri-2.2.4 Loads and Generators

Loads and generators are modeled as equivalent complex power injectionsand therefore have no effect on the network model Exceptions are con-stant impedance type loads which are included as shunt admittances at thecorresponding buses

2.3 Building the Network Model

The above-described component models can be used to build the networkmodel for the entire power system This is accomplished by writing a set

of nodal equations which are derived by applying KirchhofF's current law

at each bus Denoting the vector of net current injections by 7, and thevector of bus voltage phasors by V, these equations will take the followingform:

where

^ is the net current injection phasor at bus A;

v^ is the voltage phasor at bus A:

%m is the (A:,m)th element of K

Note that, as a convention, currents (or power) entering a bus will beassumed to be positive injections throughout the rest of the book Matrix Y

is referred to as the bus admittance matrix, and has the following properties:

1 It is in general complex, and can be written as G + j_B.

2 It is structurally symmetric It may also be numerically symmetricdepending upon the absence of certain network components such asphase shifters, with non-symmetrical nodal equations

topol-As an example, consider a two-port model of a transformer connectedbetween bus A; and m, having a series admittance of y^ and a tap ratio of

a, represented by the following nodal equations:

of the simplest subsystems is a two-port network such as the model of atransformer or a transmission line as shown in Figures 2.1 and 2.3

Example 2.1:

Consider the 4-bus power system whose one-line diagram is given in Figure2.4 Network data and the steady state bus voltages are listed below Thesusceptance of the shunt capacitor at bus 3 is given as 0.5 per unit

FromBus1122

ToBus2334

R

pu0.020.020.050.00

Xpu0.060.060.100.08

Total LineCharging Susceptance

0.200.250.000.00

Tapa -0.98

Tap SideBus -2

Bus No.

1 2

34

Voltage Mag.

pu 1.0000

0.96290.95970.9742

Phase Angledegrees0.00-2.76-3.58-3.96

Figure 2.4 One-line diagram of a 4-bus power system

* Write the nodal equations for the 2-port 7r-model of the transformer nected between bus 2 and 4

con-* Form the bus admittance matrix, IK for the entire system

* Calculate the net complex power injections at each bus

Solution:

The nodal equations for the transformer branch will be obtained by tuting for y and a in Equation (2.2):

substi J 13.02 j 12.75J12.75 -j'12.50Bus admittance matrix for the entire system can be obtained by including onebranch at a time and expanding the above admittance matrix to a 4x4 matrix:

10.00-j'29.77 -5.00 + J15.00 -5.00 + J15.00 0-5.00j'15.00 9.00-j'35.91 -4.00 + j'8.00 j'12.75-5.00jl5.00 -4.00 + j'8.00 9.00-J22.37 0

0 j'12.75 0 -j'12.50Complex power injection at bus A; will be given by:

Substituting for :^ from the above noda] equation:

2.4 Maximum Likelihood Estimation

The objective of state estimation is to determine the most likely state ofthe system based on the quantities that are measured One way to accom-plish this is by maximum likelihood estimation (MLE), a method widelyused in statistics The measurement errors are assumed to have a knownprobability distribution with unknown parameters The joint probabilitydensity function for all the measurements can then be written in terms ofthese unknown parameters This function is referred to as the likelihoodfunction and will attain its peak value when the unknown parameters arechosen to be closest to their actual values Hence, an optimization problemcan be set up in order to maximize the likelihood function as a function ofthese unknown parameters The solution will give the maximum likelihoodestimates for the parameters of interest

The measurement errors are commonly assumed to have a Gaussian(Normal) distribution and the parameters for such a distribution are itsmean, ^ and its variance, o*^ The problem of maximum likelihood esti-mation is then solved for these two parameters The Gaussian probabilitydensity function (p.d.f.) and the corresponding probability distributionfunction (d.f.) will be reviewed below briefly before describing the maxi-mum likelihood estimation method

2.4.1 Gaussian (Normal) probability density function

The Normal probability density function for a random variable 2 is definedas:

27rcr

where z : random variable

/n : mean (or expected value) of 2 = E(z)o* : standard deviation of 2

The function /(z) will change its shape depending on the parameters /i and

cr However, its shape can be standardized by using the following change

of variables:

which yields:

- ^) = = 1.0Hence, the new function becomes:

2.4.2 The likelihood function

Consider the joint probability density function which represents the ability of measuring m independent measurements, each having the sameGaussian p.d.f The joint p.d.f can simply be expressed as the product ofindividual p.d.f's if each measurement is assumed to be independent of therest:

r — joy f /2\ —

**-* — iu& JmA^/ —

MLE will maximize the likelihood (or log-likelihood) function for a givenset of observations 21,23, , 2^, Hence, it can be obtained by solving thefollowing problem:

maximize log 7^(2)OR

minimize

This minimization problem can be re- written in terms of the res^&ta^

of measurement !, which is defined as:

where the mean /^, or the expected value -E(2t) of the measurement 2^ can

be expressed as 7^($;), a nonlinear function relating the system state vector

z to the zth measurement Square of each residual r^ is weighted by W^

= cr^, which is inversely related to the assumed error variance for thatmeasurement Hence, the minimization problem of Equation (2.6) will beequivalent to minimizing the weighted sum of squares of the residuals orsolving the following optimization problem for the state vector 2:

(2.8)

The solution of the above optimization problem is called the

gttares (WLS) estimator for 2 A review of the measurement modeland the associated assumptions will be given next, before discussing thenumerical solution methods

2.5 Measurement Model and Assumptions

Consider the set of measurements given by the vector 2:

e*^ = [ei, 62, , e^] is the vector of measurement errors

The following assumptions are commonly made, regarding the statisticalproperties of the measurement errors:

* Measurement errors are independent, i.e E^e.,] = 0

Hence, Cou(e) = E[e - e^] = R = diag { cr^, <r^, - - - , o*^ }

The standard deviation cr^ of each measurement ^ is calculated to reflectthe expected accuracy of the corresponding meter used

The WLS estimator will minimize the following objective function:

(2.10)

At the minimum, the first-order optimality conditions will have to besatisfied These can be expressed in compact form as follows:

where is the iteration index,

is the solution vector at iteration k,

G(x) is called the f?am ma^Wa; It is sparse, positive definite and ric provided that the system is fully observable The issue of observabilitywill be discussed in detail in Chapter 4 The matrix G(a:) is typically notinverted (the inverse will in general be a full matrix, whereas G(a:) itself isquite sparse), but instead it is decomposed into its triangular factors andthe following sparse linear set of equations are solved using forward/backsubstitutions at each iteration &:

symmet-[C(^)]AT*=+i = R^x^R-^ - 7t(^)] (2.12)

where Aa^+i = 3^+* — aA The set of equations given by Equation (2.12)

is also referred to as the Normal Equations

To Bus233

Resistance

R(pu)0.010.020.03

Reactance

X(pu)0.030.050.08

Total Susceptance

26s (pu)0.00.00.0

: Power Measurement: Voltage Magnitude Measurement

Figure 2.6 One-tine diagram and measurement configuration of a 3-bus power

2345678

Type

Pl2 P13 P2

$12

<?13 92

14

^

Value (pu)0.8881.173-0.5010.5680.663-0.2861.0060.968

-/Rii (pu)0.0080.0080.0100.0080.0080.0100.0040.004The state vector 3 wiM have 5 elements in this case (n = 5),

3^ = [02, $3,14,^,^1

#i = 0 is chosen as the arbitrary reference angle

2.6 WLS State Estimation Algorithm

WLS State Estimation involves the iterative solution of the Normal tions given by Equation (2.12) An initial guess has to be made for the statevector a:^ As in the case of the power How solution, this guess typicallycorresponds to the Hat voltage profile, where all bus voltages are assumed

equa-to be 1.0 per unit and in phase with each other

The iterative solution algorithm for WLS state estimation problem can

be outlined as follows:

1 Start iterations, set the iteration index A; = 0

2 Initialize the state vector z^, typically as a flat start

3 Calculate the gain matrix, G(3^).

4 Calculate the right hand side ^ = R(^f,R-i(2 - ^(a;^))

5 Decompose G(a^) and solve for AaA

6 Test for convergence, max ] Aa^ [< e?

7 If no, update 3^+* = a;^ + Aa^, /c = A; + 1, and go to step 3 Else,stop

The above algorithm essentially involves the following computations ineach iteration, A;:

1 Calculation of the right hand side of Equation (2.12)

(a) Calculating the measurement function, ^(a;^)

(b) Building the measurement Jacobian, R(a^)

2 Calculation of G(a^) and solution of Equation (2.12)

(a) Building the gain matrix, (?(a;^)

(b) Decomposing C(a;^) into its Cholesky factors

(c) Performing the forward/back substitutions to solve for Aa;^+^

2.6.1 The Measurement Function, ^(a^)

Measurements can be of a variety of types Most commonly used ments are the line power Bows, bus power injections, bus voltage magnitudesand line current now magnitudes These measurements can be expressed interms of the state variables either using the rectangular or the polar coordi-nates When using the polar coordinates for a system containing N buses,the state vector will have (27V — 1) elements, TV bus voltage magnitudesand (TV — 1) phase angles, where the phase angle of one reference bus isset equal to an arbitrary value, such as 0 The state vector a; will have thefollowing form assuming bus 1 is chosen as the reference:

measure-The expressions for each of the above types of measurements are givenbelow, assuming the general two-port 7r-model for the network branches,shown in Figure 2.7:

* Real and reactive power injection at bus i:

p = ^ ]T V, (Gij cos 6^ + ^ sin ^ )

j'eKj

^ = ^ ^ ^7 ( C^' sin (9^ - B^ cos 6^ )

Figure 2.7 Two-port 7r-modei of a network branch

Real and reactive power Bow from bus z to bus j:

+ Fi? ) - ^ !j cos 6^ + &^- sin 6^ )

- 6^ cos 0

Line current flow magnitude from bus ^ to bus j:

or ignoring the shunt admittance (<?^ +

in Figure 2.7

f^ is the set of bus numbers that are directly connected to bus :

2.6.2 The Measurement Jacobian, R

The structure of the measurement Jacobian # will be as follows:

38

36 0

3V 3V 3V 3V 3V

The expressions for each partition are given below:

* Elements corresponding to real power injection measurements:

* Elements corresponding to real power flow measurements:

?ij sin 0^' — &^- cos 6*ij)

— = — V^Vy^j'sin&tj—^-cos^j)

^ cos ^j + &ij sin ^,,)

^ ^j

Elements corresponding to reactive power flow measurements:

—-^ = - ^ V,- (g^ cos 6<ij + 6^- sin 6<^)

= — %(^r; sin^ — 6ij cos^

Elements corresponding to voltage magnitude measurements:

Elements corresponding to current magnitude measurements ing the shunt admittance of the branch) :

(ignor-Exampie 2.3:

Consider the same system and measurement configuration shown in example2.2 Assume flat start conditions, where the state vector is equal to:

001.01.01.0Then, the measurement Jacobian can be evaluated as follows, using the expres-sions given above:

dpi

d#2

* -30.040.910.0-14.1

<%s dH-17.2

-10.96.94.1

10.06.9-10.030.017.2-30.01.0

dl/2-10.014.1-30.040.91.0

<9%

*-6.9-4.1-17.2-10.9

Note that the dimension of # is 7n x n = 8 x 5, and it is a sparse matrix Itssparsity becomes more pronounced for large scale systems, where the number ofnonzeros per row stays fairly constant, irrespective of the system size

2.6.3 The Gain Matrix, G

Gain matrix is formed using the measurement Jacobian R and the surement error covariance matrix, R The covariance matrix is assumed to

mea-be diagonal having measurement variances as its diagonal entries Since G

is formed as:

it has the following properties:

1 It is structurally and numerically symmetric

2 It is sparse, yet less sparse compared to R

3 In general it is a non-negative definite matrix, i.e all of its values are non-negative It is positive definite for fully observablenetworks