Inter-area Oscillations in Power Systems A Nonlinear and Nonstationary Perspective

Inter-area Oscillations in Power Systems

Power Electronics and Power Systems

Series Editors:Series Editors: M. A. PaiUniversity of Illinois at Urbana-ChampaignUrbana, Illinois

Alex StankovicNortheastern UniversityBoston, Massachusetts

Inter-area Oscillations in Power Systems: A Nonlinear and Nonstationary PerspectiveArturo Roman Messina, ed.ISBN 978-0-387-89529-1

Robust Power System Frequency ControlHassan BevraniISBN 978-0-387-84877-8

Synchronized Phasor Measurements and Their ApplicationsA.G. Phadke and J.S. ThorpISBN 978-0-387-76535-8

Digital Control of Electical DrivesSlobodan N. VukosavicISBN 978-0-387-48598-0

Three-Phase Diode Rectifiers with Low HarmonicsPredrag PejovicISBN 978-0-387-29310-3

Computational Techniques for Voltage Stability Assessment and ControlVenkataramana AjjarapuISBN 978-0-387-26080-8

Real-Time Stability in Power Systems: Techniques for Early Detection of the Risk of BlackoutSavu C. Savulesco, ed.ISBN 978-0-387-25626-9

Robust Control in Power SystemsBikash Pal and Balarko ChaudhuriISBN 978-0-387-25949-9

Applied Mathematics for Restructured Electric Power Systems: Optimization, Control, and Com-putational IntelligenceJoe H. Chow, Felix F. Wu, and James A. Momoh, eds.ISBN 978-0-387-23470-0

HVDC and FACTS Controllers: Applications of Static Converters in Power SystemsVijay K. SoodISBN 978-1-4020-7890-3

Power Quality Enhancement Using Custom Power DevicesArindam Ghosh and Gerard LedwichISBN 978-1-4020-7180-5

Computational Methods for Large Sparse Power Systems Analysis: An Object Oriented ApproachS.A. Soman, S.A. Khaparde, and Shubha PanditISBN 978-0-7923-7591-3

Continued after Index

Arturo Roman MessinaEditor

Inter-area Oscillationsin Power Systems

ANonlinear andNonstationary Perspective

1 3

Editor

Arturo Roman MessinaCentro de Investigacion yde Estudios Avanzadosdel IPN

Guadalajara, [email protected]

ISBN 978-0-387-89529-1 e-ISBN 978-0-387-89530-7DOI 10.1007/978-0-387-89530-7

Library of Congress Control Number: 2008939222

# Springer ScienceþBusiness Media, LLC 2009All rights reserved. This workmay not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer ScienceþBusinessMedia, LLC, 233 Spring Street, NewYork,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

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Preface

The study of complex dynamic processes governed by nonlinear and

nonstationary characteristics is a problem of great importance in the

analysis and control of power system oscillatory behavior. Power system

dynamic processes are highly random, nonlinear to some extent, and

intrinsically nonstationary even over short time intervals as in the case of

severe transient oscillations in which switching events and control actions

interact in a complex manner.Phenomena observed in power system oscillatory dynamics are diverse and

complex. Measured ambient data are known to exhibit noisy, nonstationary

fluctuations resulting primarily from small magnitude, random changes in load,

driven by low-scale motions or nonlinear trends originating from slow control

actions or changes in operating conditions. Forced oscillations resulting from

major cascading events, on the other hand, may contain motions with a broad

range of scales and can be highly nonlinear and time-varying.Prediction of temporal dynamics, with the ultimate application to real-time

system monitoring, protection and control, remains a major research challenge

due to the complexity of the driving dynamic and control processes operating

on various temporal scales that can become dynamically involved. An

understanding of system dynamics is critical for reliable inference of the

underlying mechanisms in the observed oscillations and is needed for the

development of effective wide-area measurement and control systems, and for

improved operational reliability.Complex power system response data can contain nonlinear and possibly

strong local trends, noise, and may exhibit sudden variations and other

nonlinear effects associated with large and abrupt changes in system topology

or operating conditions that make the extraction of salient features difficult.

Accounting for nonlinear and time-varying features can not only provide a

better description of the data but can also reveal crucial information on

system’s oscillatory behavior such as modal properties and moving patterns.

By tracking the evolving dynamics of the underlying oscillations, the onset of

system instability can be determined and the critical stages for analysis and

control can be identified.

v

Recent years have seen a flourishing of activity in various techniques for theanalysis of power system dynamic behavior. Foremost among linear analysistools, Prony’s method has been widely applied to estimate small-signal dynamicproperties frommeasured and simulated data. Applications of linear techniquesin the context of power oscillations include, for example, modal extraction fromringdowns, the analysis of dynamic tests, and the identification of transferfunctions. Ongoing research into the study of modal behavior in the presenceof high noise levels and possibly nonstationary situations has resulted invariations to these approaches that extend their practical use to the realm ofnear-real-time stability assessment and control, and has stimulated thedevelopment of enhanced monitoring systems. This in turn, has sparked aresurgence of interest in the development of new algorithms that use theavailable online information to estimate modal properties.

Advances in signal processing algorithms, along with continuously growingcomputational resources and monitoring systems are beginning to makefeasible the analysis and characterization of transient processes using real-time information. Much of the recent work has been driven by interest in nearreal-time estimation of electromechanical modal properties from measuredambient data. This effort has resulted in various signal processing methodswith the capability of tracking the evolving dynamics of critical system modes.

Complementary, time–frequency analysis techniques that explicitlyacknowledge and incorporate nonlinearity or nonstationarity in both the timeand frequency domain are emerging as subjects of research and application inengineering investigations. Adaptive, nonlinear time-varying methods with theability to capture the temporal evolution of critical modal parameters, promise toenhance our understanding of the physical mechanisms that underlie systemoscillatory dynamics and have the potential to be applied to more generaltransient oscillations, governed by multiscale, time-varying processes.

A significant element of this major thrust is the development of wide-areameasurement systems. Extracting the salient features of interest from a widelydispersed and usually large number of system observations is a complexproblem. In the analysis of large models, where a significant amount ofobservational data is available, the development of data-based statisticalmodels with the capacity to process the vast wealth of information andextract relevant, physically independent patterns is appealing. For many ofthe above developments, a complete framework for temporal characterizationof system behavior, however, is still evolving.

The combined utilization of temporal, modal information and advancedmeasurement and control techniques holds also enormous potential to providecritical information for early detection, mitigation, and avoidance of large-scalecascading failures and could form the basis of smart, wide-area automatedanalysis and control systems. Analysis and characterization of time-synchronized system measurements requires mathematical tools that areadaptable to the varying system conditions, accurate and fast, while reducingthe complexity of the data to make them comprehensible and useful for control

vi Preface

and real-time decisions. Experience with the analysis of complex inter-areaoscillations from measured data, shows that issues such as noise, time-varyingbehavior, data measurement errors, and nonlinear effects have to be addressedif these tools are to be of practical use. Further, the applicability of thesetechniques to both, ambient from online system measurements and large-scaletransient oscillations has to be fully investigated because some techniques arebetter suited for a specific type of behavior.

This book deals with the development and application of advancedmeasurement-based signal processing techniques to the study, characterization,and control of complex transient processes in power systems. Recent advances inunderstanding, modeling and controlling system oscillations are reviewed.Specific attention is given to the modeling and control of complex time-varying(and possibly nonlinear) power system transient processes which have not beenpresent in previous work. Techniques that explicitly address and treatnonlinearity and nonstationarity are given and efficient methods to generatetime-varying system approximations from both measured and simulated dataare discussed. Attention is also given to the vital new ideas of dynamic securityassessment in real-time implementations and the development of smart, wide-area measurement and control systems incorporating FACTS (flexible ACtransmission system) technology. Application examples include the analysis ofreal data collected on grids in western North America, Australia, Italy, andMexico. These studies are expected to stimulate the interest of otherresearchers, toward the investigation of complex nonstationary power systemoscillations andmay form the basis of more advanced computational algorithms.

The book is organized into eight chapters written by leading researchers whoare major contributors to knowledge in this field.

Chapter 1 demonstrates and examines the performance of several methods forestimating small-signal dynamic properties from measured responses. Thetheoretical basis for these methods is described as well as application,properties, and performance. Examples include computer simulations andactual system experiments from the western North American power grid.Analysis goals center on estimating the modal properties of the systemincluding modal frequency, damping, and shape.

Chapter 2 revisits some of the fundamental assumptions of the recentlyintroduced Hilbert–Huang transform. The ability of empirical modedecomposition (EMD) to yield monocomponent intrinsic mode functions isexamined in the context of power system oscillations. Some enhancements tothe EMD are proposed to enhance its ability to better discriminate betweenclosely spaced frequency components. Additionally, frequency demodulation issuggested, to extract physically relevant instantaneous frequency from theHilbert transform. Synthetic data as well as real life data are used todemonstrate the validity of the enhancements.

Chapter 3 discusses some refinements to the Hilbert–Huang technique toanalyze time-varying multicomponent oscillations. Improved masking signaltechniques for the EMD are proposed and tested on measured data of a real

Preface vii

event in northern Mexico. Based on this framework, a novel approach to thecomputation of instantaneous damping is suggested and a local implementationof the Hilbert transform is also described. The accuracy of the method isdemonstrated by comparisons to Prony and Fourier analysis.

Chapter 4 investigates the applicability of Hilbert–Huang analysis techniqueto extract modal information in the presence of noise and possibly nonstationarysituations. Application of Hilbert analysis is examined relative to the moreestablished Prony analysis, with particular reference to the considerablestructural differences which exist between the two methods. Factors affectingthe performance of the techniques including noise tolerance, performance in thecase of closely spaced frequency components and changes in the underlyingsystem dynamics are discussed and investigated using synthetic and measureddata.

In Chapter 5 a real-time centralized controller for addressing small-signalinstability related events in large electric power systems is proposed. Using wide-area monitoring schemes to identify the emergence of growing or undampedoscillations related to interarea and/or local modes, rules are developed forincreasing multi-Prony method’s observability and dependability. Thisinformation is then utilized to initiate static VAR compensation controls toenhance the damping of a critical mode; the algorithms are tested in a two-areapower system and in a large-scale simulation example.

Chapter 6 discuses the use of multivariate data analysis techniques to extractand identify dynamically independent spatiotemporal patterns from time-synchronized data. By seeing the snapshots of system data as a realizationof random fields generated by some kind of stochastic process, a statisticalapproach to investigate propagating phenomena of different spatial scales andtemporal frequencies is proposed and tested on real noisy measurements fromtheMexican system. Themethod provides accurate estimation of nonstationaryeffects, modal frequency, time-varying shapes, and time instants of intermittenttransient behavior.

Chapter 7 proposes new techniques for detection and estimation ofnonstationary power transients. Attention is focused on two aspects of smallsignal models: the detection of change in the system and the identification of thenew operating parameters. Techniques to detect significant changes in systemdynamics by analyzing the dynamic response to continual load changes basedon detection theory are proposed. Approaches based on time–frequencyanalysis techniques are then used to yield improved modal estimates innonstationary environments. Applications to measurement data from theAustralian connected system are presented.

Finally, Chapter 8 discusses the development of advanced monitoring andcontrol approaches for enhancing power system security. The monitoringstructure is based on wavelet analysis of wide-area measurements systemstargeted to extract the critical damping of critical oscillation modes. Ahierarchical response-based control strategy that may incorporate FACTStechnologies and special protection systems is developed and tested on a

viii Preface

dynamic model of the Italian interconnected system to provide effectivestabilization of critical modes.

The book is the first comprehensive, systematic account of current analysismethods in power system oscillatory dynamics in both time and frequencydomains ranging from modal analysis, to data-driven time-series models andstatistical approaches. The procedures can be used in various disciplines otherthan power engineering, including signal and time analysis, processidentification and control, and data compression and has wide applications tomany important problems covering engineering, biomedical, physical,geophysical, and climate data.

This is a book intended for advanced undergraduate and graduate courses,as well as for researchers, utility engineers, and advanced teaching in the fieldsof power engineering, signal processing, and identification and applied control.

Guadalajara, Mexico A.R. Messina

Preface ix

Acknowledgments

The editor is grateful to the contributing authors. He is also thankful to anumber of colleagues who provided the thrust for this work. Among them areJuan J. Sanchez-Gasca (GE, Schenectady, NY), Mike Gibbard (University ofAdelaide, Australia), and Brian Cory (Imperial College, UK). Their numerouscomments and observations throughout the years are a highly appreciatedcontribution.

It is also a pleasure to acknowledge the support of Ms Katelyn Stanne,Springer US, who proofread different versions of the manuscript and guidedthe editor during the editorial work.

xi

Contents

1 Signal Processing Methods for Estimating Small-Signal Dynamic

Properties from Measured Responses . . . . . . . . . . . . . . . . . . . . . . . . . 1Daniel Trudnowski and John Pierre

2 Enhancements to the Hilbert–Huang Transform for Application to

Power System Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Nilanjan Senroy

3 Variants of Hilbert–Huang Transform with Applications to Power

Systems’ Oscillatory Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Dina Shona Laila, Arturo Roman Messina, and Bikash Chandra Pal

4 Practical Application of Hilbert Transform Techniques in Identifying

Inter-area Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101T. J. Browne, V. Vittal, G.T. Heydt, and Arturo Roman Messina

5 A Real-Time Wide-Area Controller for Mitigating Small-Signal

Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Jaime Quintero and Vaithianathan (Mani) Venkatasubramanian

6 Complex Empirical Orthogonal Function Analysis of Power System

Oscillatory Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159P. Esquivel, E. Barocio, M.A. Andrade, and F. Lezama

7 Detection and Estimation of Nonstationary Power Transients . . . . . . 189Gerard Ledwich, Ed Palmer, and Arindam Ghosh

8 Advanced Monitoring and Control Approaches for Enhancing Power

System Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Sergio Bruno, Michele De Benedictis, and Massimo La Scala

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

xiii

Contributors

M.A. Andrade Department of Electrical Engineering, Universidad Autonoma

de Nuevo Leon, Monterrey, Mexico, [email protected]

E. Barocio Department of Electrical Engineering, University of Guadalajara,

Guadalajara, Mexico, [email protected]

Michele De Benedictis Dipartimento di Elettrotecnica ed Elettronica (DEE),

Politecnico di Bari, Bari, Italy, [email protected]

T.J. Browne Ira A. Fulton School of Engineering, Department of Electrical

Engineering, Arizona State University, Tempe, AZ, USA, [email protected]

Sergio Bruno Dipartimento di Elettrotecnica ed Elettronica (DEE),


P. Esquivel Department of Electrical and Computer Engineering, The Center

for Research and Advanced Studies, Cinvestav, Mexico

[email protected]

Arindam Ghosh Faculty of Built Environment and Engineering, Queensland

University of Technology, Brisbane, Australia, [email protected]

G.T. Heydt Ira A. Fulton School of Engineering, Department of Electrical

Engineering, Arizona State University, Tempe, AZ, USA, [email protected]

Dina Shona Laila Department of Electrical and Electronic Engineering,

Imperial College, London, UK, [email protected]

Massimo La Scala Dipartimento di Elettrotecnica ed Elettronica (DEE),


Gerard Ledwich Faculty of Built Environment and Engineering, Queensland

University of Technology, Brisbane, Australia, [email protected]

F. Lezama Department of Electrical and Computer Engineering, The Center for

research and Advanced Studies, Cinvestav, Mexico, [email protected]

xv

Arturo Roman Messina Department of Electrical and Computer Engineering,The Center for Research and Advanced Studies, Cinvestav, Guadalajara,Mexico, [email protected]

Bikash Chandra Pal Department of Electrical and Electronic Engineering,Imperial College, London, UK, [email protected]

Ed Palmer Faculty of Built Environment and Engineering, QueenslandUniversity of Technology, Brisbane, Australia, [email protected]

John Pierre Electrical and Computer Engineering Department, Universityof Wyoming, Laramie, WY, USA, [email protected].

Jaime Quintero Faculty of Engineering, Universidad Autonoma de Occidente,Cali-Valle, Colombia, [email protected]

Nilanjan Senroy Department of Electrical Engineering, Indian Instituteof Technology, New Delhi, India, [email protected]

Daniel Trudnowski Electrical Engineering Department, Montana Tech of theUniversity of Montana, Butte, MT, USA, [email protected]

V. Vittal Ira A. Fulton School of Engineering, Department of ElectricalEngineering, Arizona State University, Tempe, AZ, USA, [email protected]

Vaithianathan (Mani) Venkatasubramanian School of Electrical Engineeringand Computer Science, Washington State University, Pullman, WA 99164USA, [email protected]

xvi Contributors

Chapter 1

Signal Processing Methods for Estimating

Small-Signal Dynamic Properties

from Measured Responses

Daniel Trudnowski and John Pierre

Abstract Power system small-signal electromechanical dynamic properties areoften described using linear system concepts. The underlying hypothesis is thatsmall motions of the system can be described by a set of ordinary differentialequations. Modal analysis of these governing equations provides considerableinsight into the stability properties of the system. Over the past two decades,many signal processing techniques have been developed to conduct modalanalysis using only time-synchronized actual system measurements. Some tech-niques are appropriate for transient signals, others are for ambient signalconditions, and some are for conditions where a known probing signal isexciting the system. In this chapter, an overview of many of the more successfulanalysis techniques is presented. The theoretical basis for these methods isdescribed as well as application properties and performance. Examples includecomputer simulations and actual system experiments from the western NorthAmerican power system. Analysis goals center on estimating the modal proper-ties of the system including modal frequency, damping, and shape.

1.1 Introduction

Time-synchronized measurements provide rich information for estimating apower system’s electromechanical modal properties via advanced signal proces-sing. This information is becoming critical for the improved operational relia-bility of interconnected grids. A given mode’s properties are described by itsfrequency, damping, and shape. Modal frequencies and damping are usefulindicators of power system stress, usually declining with increased load orreduced grid capacity. Mode shape provides critical information for opera-tional control actions. Over the past two decades, many signal processing

D. Trudnowski (*)Electrical Engineering Department, Montana Tech of the University of Montana,Butte, MT, USAe-mail: [email protected]

A.R. Messina, Inter-area Oscillations in Power Systems,Power Electronics and Power Systems, DOI 10.1007/978-0-387-89530-7_1,� Springer ScienceþBusiness Media, LLC 2009

1

techniques have been developed and tested to conduct modal analysis usingonly time-synchronized actual system measurements. Some techniques areappropriate for transient signals while others are for ambient signal conditions.

Many of the signal processing algorithms described in this chapter are thebasis for several evolving software tools. The majority of these tools are used toconduct engineering analysis of the grid in an off-line or post disturbancesetting [1]. More recently, online real-time software tools and applications areevolving [2] and will likely continue to be a research focus area for the powersystem community.

Near-real-time operational knowledge of a power system’s modal propertiesmay provide critical information for control decisions and thus enable reliablegrid operation at higher loading levels. For example, modal shapemay somedaybe used to optimally determine generator and/or load-tripping schemes toimprove the damping of a dangerously low damped mode. The optimizationinvolves minimizing load shedding and maximizing improved damping. Thetwo enabling technologies for such real-time applications are a reliable real-time-synchronized measurement system and accurate modal analysis signalprocessing algorithms.

In this chapter, an overview of many of the more successful analysis techni-ques is presented. The theoretical basis for these methods is described as well asapplication and performance properties. Examples include computer simula-tions and actual system experiments from the western North American powersystem (wNAPS). Analysis goals center on estimating the modal properties ofthe system including modal frequency, damping, and shape.

The chapter is organized as follows. Section 1.2 discusses system basics. Anoverview of mode estimation algorithms is provided in Section 1.3. Section 1.4discusses the use of probing signals to improve mode estimates. Section 1.5provides some examples. Model validation and estimation assessment is dis-cussed in Section 1.6, and Section 1.7 covers mode-shape estimation. Finally,conclusions are discussed in Section 1.8.

1.2 System Basics

Analyzing and estimating power system electromechanical dynamic effects area challenging problem because the system:

1. is nonlinear, high order, and time varying;2. contains many electromechanical modes of oscillation close in frequency;

and3. is primarily stochastic in nature.

Design of signal processing algorithms requires that one address each of theseissues. Fortunately, the system behaves relatively linear when at a steady-stateoperating point [3].

2 D. Trudnowski and J. Pierre

As has been established in one of the many excellent books that address theproperties and nature of electromechanical dynamics in power systems (e.g., see[4, 5]), electromechanical modes are typically classified as either local or inter-area in nature. Local modes occur when a single generator or plant swingsagainst the system while an inter-area mode occurs when several generators inan area swing against generators in another area. Because local modes arecharacterized by larger inertias and lower impedance paths, their frequenciestend to be higher. In general, local modes tend to be in the 1–2 Hz range whileinter-area modes tend to be in the 0.2–1.0 Hz range. Typically, the inter-areamodes are more troublesome.

Consistent with power system dynamic theory, we assume that a powersystem can be linearized about an operating point [4, 5]. The underlyingassumption is that small motions of the power system can be described by aset of ordinary differential equations of the form

_x tð Þ ¼ Ax tð Þ þ BLq tð Þ þ BEuE tð Þ

y tð Þ ¼ Cx tð Þ þDLq tð Þ þDEuE tð Þ þ m tð Þ(1:1)

where q is a hypothetical random vector perturbing the system, vector x con-tains all system states including generator angles and speeds, and t is time.Control actions that can be described as smooth functions of the state x areembedded in the system A matrix, and all other actions are represented by theexogenous input vector uE. These include set-point changes, low-level probingsignals (e.g., a low-level probing signal into a DC converter), and load pulsesthat are applied to examine system dynamics. Measurable signals are repre-sented by y which contains measurement noise m that includes effects frominstruments, communication channels, recording systems, and similar devices.In general, measurement noise has a relatively small amplitude when qualityinstrumentation is employed. Changes which are breaker actuatedmay producesystem topology changes that alter the system A matrix to various degrees.

The assumption for q is that it is a vector of small-amplitude randomperturbations typically conceptualized as noise-produced load switching. Ithas been hypothesized that the load switching is primarily integrated stationaryGaussian white noise with each element of q independent [6]. This assumption iscertainly open to more research.

An expanded perspective of the system is shown in Fig. 1.1 where yi is the ithelement of y [7]. Multiple-input and multiple-output (MIMO) system G isassumed linear. Network topology changes are represented by switches indynamic gain matrices K and K0, which may or may not be deliberate.

We classify the response of the system in Fig. 1.1 as one of two types:transient (sometimes termed a ringdown) and ambient. The basic assumptionfor the ambient case is that the system is excited by low-amplitude variations atq and uE and that the variations are typically random or pseudorandom innature. This results in a response at y that is colored by the dynamics G.

1 Signal Processing Methods for Small-Signal Dynamic Properties 3

A transient response is typically larger in amplitude and is caused by a sudden

switch at s or s0, or a sudden step or pulse input at uE. The resulting time-domain

response is a multimodal oscillation superimposed with the underlying ambient

response.The different types of responses are shown in Fig. 1.2, which shows a widely

published plot of the real power flowing on a major transmission line during a

breakup of the wNAPS in 1996. Prior to the transient at the 400 s point, the

system is in an ambient condition. After the ringdown at the the 400 s point, the

system returns to an ambient condition. The next event in the system causes an

unstable oscillation.

Fig. 1.2 Real power flowing on a major transmission line during the western North Americanpower system breakup of 1996

G

K '

K

unknown dynamics

known dynamics

s' = unknown topology changes

s = known topology changes

µ

yi(t)

i(t)

+

+q(t)

uE(t))(ˆ tyi

Fig. 1.1 A structure for information sources in process identification


In developing and applying measurement-based modal analysis algorithms,

it is imperative that one considers the stochastic nature of the system. Power

systems are continually excited by random inputs with high-order indepen-

dence. This is modeled by q(t) in our formulation. Because of this stochastic

nature, no algorithm can exactly estimate the modal properties of the system

from finite-time measurements. There will always be an error associated with

the estimate. When evaluating estimation algorithms, one must address these

error properties. This includes the bias error as well as the variance of the

estimate.In terms of application, we classifymodal frequency and damping estimation

algorithms into two categories: (1) ringdown analyzers and (2) mode meters. A

ringdown analysis tool operates specifically on the ringdown portion of the

response; typically the first several cycles of the oscillation (5–20 s). Alterna-

tively, a mode meter is applied to any portion of the response: ambient,

transient, or combined ambient/transient. Ultimately, a mode meter is an

automated tool that estimates modal properties continuously and without

reference to any exogenous system input.

1.3 Signal Processing Methods for Estimating Modes

Many parametric methods have been applied to estimate power system electro-

mechanical modes. As stated above, we classify these methods into two cate-

gories: ringdown analyzers and mode meters. In this section, we provide an

overview of some of the algorithms that have been used to solve these problems.

1.3.1 Ringdown Algorithms

Ringdown analysis for power system modal analysis is a relatively mature

science. The underlying assumed signal model for these algorithms is a sum of

damped sinusoids. The most widely studied ringdown analysis algorithm is

termed Prony analysis. The pioneering paper by Hauer, Demeure, and Scharf

[8] was the first to establish Prony analysis [9] as a tool for power system

ringdown analysis. Expansion to transfer function applications, multiple out-

puts, and improved numerics were progressively established in [10–17]. Other

ringdown analysis algorithms have been successfully applied to power system

applications. These include the minimal realization algorithm first introduced

in [18], the eigenvalue ralization algorithm (ERA) in [19], the matrix pencil

method [20], and the Hankel total least squares (HTLS) [20]. The conclusions

and discussions in [21] point to the vast similarities between Prony analysis and

the ERA. A comparative analysis between matrix pencil, HTLS, and Prony

analysis in [20] conclude that HTLS and matrix pencil estimate the mode


damping more accurately. These conclusions are certainly subject to the exam-ple case and the parameters chosen for the analysis.

It is beyond the scope of this chapter to provide the equations for all theringdown methods. As an overview, we provide the basic equations for Pronyanalysis. The reader is directed to the above references for more details.

While ignoring noise content and assuming nonrepeated poles, if one appliesan impulse input to the system in (1.1), the response at the ith output can bewritten as

yj tð Þ ¼Xn

i¼1Bie

lit (1:2)

where li is the ith pole (mode). If we let t = kT, where T is the constant sampleperiod, this equation can be converted to discrete-time form as

yj kTð Þ ¼Xn

i¼1Biz

ki ; for k ¼ 0; 1; . . . ;m (1:3)

where zi ¼ eliT is the discrete-time pole. Equation (1.3) is expanded into matrixform as

y 0ð Þy Tð Þ

..

.

y mTð Þ

266664

377775¼

1 1 � � � 1

z1 z2 � � � zn

..

. ... . .

. ...

zm1 zm2 � � � zmn

266664

377775

B1

B2

..

.

Bn

266664

377775

(1:4)

It is relatively easy to show [9] that

y nTð Þy ðnþ 1ÞTð Þ

..

.

y mTð Þ

2

66664

3

77775¼

y ðn� 1ÞTð Þ y ðn� 2ÞTð Þ � � � y 0ð Þy nTð Þ y ðn� 1ÞTð Þ � � � y Tð Þ

..

. ... . .

. ...

y ðm� 1ÞTð Þ y ðm� 1ÞTð Þ � � � y ðm� nÞTð Þ

2

66664

3

77775

a1

a2

..

.

an

2

66664

3

77775(1:5)

where the ai’s are the coefficients of the characteristic equation

zn � a1zn�1 þ a2z

n�2 þ � � � þ an� �

¼ 0 (1:6)

The solution of (1.6) are the zi poles.Prony analysis involves solving (1.5) for the ai characteristic equation coeffi-

cients. Then (1.6) is rooted to obtain the zi discrete-time poles. Lastly, (1.4) issolved for the Bi’s. As described in the above references, these equations can beextended to the multioutput case. Selection of model order n, sample period T,and number of data points mþ1 are also addressed throughout the literature.


1.3.2 Mode-Meter Algorithms

Ambient analysis of power system data estimates the modes when the primary

excitation to the system is random load variations, which results in a low-

amplitude stochastic time series (ambient noise). A good place to begin ambient

analysis is with nonparametric spectral estimation methods, which are very

robust as they make very few assumptions. The most widely used nonpara-

metric method is the Welch periodogram [22, 23] spectrum which provides

an estimate of a signal’s strength as a function of frequency. Thus, usually

the dominate modes are clearly visible as peaks in the spectral estimate. The

estimates of the mode frequencies are identifiable in the locations of the peaks.

The narrower the peaks, the lighter is the damping.Welch spectral estimates are

also used in estimating mode shape as will be discussed in Section 1.7. While

robust and insightful, nonparametric methods do not provide direct numerical

estimates of a mode’s damping ratio and frequency. Therefore, to obtain

further information parametric methods are applied.Ambient-based mode estimation can be conducted in the time domain or

frequency domain. Time-domain algorithms operate directly on the sampled

data while frequency-domain methods require the estimation of the power

spectral density (PSD) function (usually using Welch’s method). The first

available ambient-based mode estimation work [6] used a frequency-domain

strategy. The method described in [6] was applied to actual system measure-

ments. With this approach, Welch periodogram averaging is used to estimate

the PSD of a signal. Frequency-domain identification is then used to estimate

the system modes. A disadvantage of the approach in [6] is that the frequency-

domain identification process used requires an initial estimate of the system

modes prior to analysis which is difficult to automate.There are two basic types of parametric mode estimation algorithms: block

processing and recursive. With block processing algorithms, the modes are

estimated from a window of data. For each new window of data, a new estimate

is calculated. For example, assume one is using a 5min window length. For each

window of data, a single set of modes is calculated. All data in the 5 min block

are equally weighted. A new mode estimate can be calculated as often as

required, but each calculation requires 5 min of the most recent data. The

first application of block processing is contained in [24] where the Yule–Walker

(YW) algorithm is used to estimate modes using an autoregressive (AR) model.

The method is extended to the overdetermined modified YW method [35] to

estimate an autoregressive moving average (ARMA) model in [25]. The

approach is further extended to multiple signals in [26], which can improve

the performance. Block processing methods using subspace methods CVA

(canonical variate algorithm) and N4SID (numerical algorithm for subspace

state–space system identification) were first introduced in [27] and [34], respec-

tively. A variation of the YW approach that estimates the autocorrelation

function using a frequency-domain calculation is introduced in [28]; this


method is termed the Yule–Walker spectrum (YWS) method. Also in [28], theYW, YWS, and N4SID algorithms are compared. Another frequency-domainmethod is the frequency-domain decomposition (FDD) method described in[29], which decomposes the signals’ estimated power spectrum.

For recursive methods, the estimated modes are updated for each newsample of the data. The new estimate is obtained using a combination of thenew data point and the previous mode estimate. A forgetting factor is used todiscount information based on previous data; therefore, new data is weightedmore in each calculation. Similar to the block processing methods, all recursivemethods tested to date require many minutes of data to converge to a steady-state solution. Published results include the least-mean squares (LMS) method[30] and the regularized robust recursive least-squares (R3LS) method [31, 32].

The R3LS method described in [32] offers several advances to previousalgorithms. First, it accommodates an autoregressive moving average exogen-ous (ARMAX) model to account for ambient noise as well as a known input,which can enhance performance during probing. Second, it has a robust objec-tive function to reduce the impact of missing or outlier data, and third, it canincorporate a priori knowledge of the modes. The full impact of these advancesis the subject of current and future research.

An important component of a mode meter is the automated application ofthe algorithm. With all algorithms, several modes are estimated and many ofthem are ‘‘numerical artifacts.’’ Typically, ‘‘modal energy’’ methods are used todetermine which of the modes in the frequency range of the inter-area modeshave the largest energy in the signal [28]. It is then assumed that this is the modeof most interest.

It is beyond the scope of this chapter to provide the equations for all themode-meter methods described above. The reader is directed to the abovereferences for more details and for information on preprocessing the databefore application of the mode-meter algorithms.

1.4 Power System Identification Using Known Probing Signals

It is absolutely imperative to understand that because of the stochastic nature ofthe system, the accuracy of any mode estimation is limited. It is possible tosignificantly improve the estimation by exciting the system with a probingsignal. A signal may be injected into the power system using a number ofdifferent actuators such as resistive brakes, generator excitation, or modulationof DC intertie signals. For example, operators of the wNAPS use both the1,400 MW Chief Joseph dynamic brake and modulation of the Pacific DCintertie (PDCI) to inject known probing signals into the system. The wNAPSis shown in Fig. 1.3 with the PDCI being the DC line flowing from Oregon tosouthern California. The PDCI has been modulated with a number of differentsignals including short duration mid-level probing resulting in transient


PHOENIXMOJAVE

CRANBROOK

SAN FRANCISCO

CORNERSNAVAJO

PINTO

SHASTA

BUCKLEY

SUNDANCE

FT. PECK

KEMANO

PEACE CANYON

MICA

VANCOUVER

SEATTLE

PRINCE RUPERT

AREA

AREA

COLSTRIP

BOISE

PORTLANDAREA

MALIN

TABLE MTN

ROUND MTN

SALT LAKECITY AREA

MEXICO

EL PASOAREA

PALO

LUGO

MIDPOINT

AREA

LOS ANGELESAREA

ALBUQUERQUEAREA

VERDE

DENVERAREA

HOOVER

AREA

LANGDON

HOT SPRINGS

HELLSCANYON

CHIEF JOSEPH

GRAND

BURNS

FOUR

HVDC TERMINAL

COULEE

DEVERS

Fig. 1.3 Major buses and lines in the western North American power system


responses and long-duration low-level probing that result in measured signalsonly slightly above the system ambient noise floor. Low-level probing should becarried out at a level low enough to not be a significant disturbance.

The wNAPS has a long history in the use of probing signals for electromecha-nical mode identification [7, 33]. During the 1980s and 1990s the Chief Josephbrake was frequently used to benchmark system characteristics. In the late 1990s,with synchronized wide-area measurements becoming readily available, modu-lating the PDCI became more common. In 1999, mid-level probing signals wereused to characterize the mode damping. In 2000, low-level pseudorandom noisewas injected into the system. The application of system identification methods tothe input and output data from that test showed great promise for mode estima-tion [34]. In 2005, 2006, and 2008, a number of extensive tests were carried outusing low-level multisine probing signals modulate at the PDCI. The synchro-nized measurements of the system response to those tests proved to be rich ininformation about the system’s dynamic characteristics.

With known input signals, not only can the electromechanical modes beidentified with improved performance, but complete input/output system mod-els, such as transfer functions and state–space models, can be estimated from theinput location to the measured output locations. Many different system identifi-cation methods can be used. This includes extending the R3LS [32] and N4SID[34] methods described previously. There is a tremendous amount of literature onsystem identification given measured inputs and outputs. Some of these algo-rithms work on the time-domain data while other algorithms utilize the fre-quency-domain data. The literature is too extensive to review here; the reader isreferred to one of many textbooks (e.g., see [36]). Classical nonparametric meth-ods such as ETFE (empirical transfer function estimation) and spectral methods[36] may be used to estimate the system magnitude and phase response. Theadvantage of the nonparametric methods is that theymake very few assumptionsabout the underlying systemmodel. Thus, they play an important role in validat-ing parametric system models where one looks for consistency from the fre-quency response identified from a parametric method and the nonparametricmethods. The parametric methods provide much more information about thesystem such as a state–space model or a transfer function equation. It is impor-tant that the parametric algorithm chosen matches well with the underlyingcondition. For example, if an algorithm designed to analyze a transient response(i.e., a sum of damped sinusoids) is applied to ambient data, which is not the sumof damped sinusoids, then poor results are expected.

1.4.1 Probing Signal Selection

In choosing a low-level probing signal to inject into the system, many factorscome into play. The objective in probing signal design is to create an input thatwill result in accurate estimates of the electromechanical inter-area modes andpossibly other system dynamic characteristics while maintaining safe operation


of the power system. The choice of probing signals has a very substantialinfluence on the observed measured data. The protection of the power systemis of the utmost importance. Other important considerations include the shape,amplitude, duration, and repetition of the injected signal. Some identificationtechniques were developed for specific input signals.

A few limitations on the input design are specific to the power systemapplication. It is desirable not to have too many sharp transitions in themodulated signal on a DC intertie. Thus, this rules out many common systemidentification probing signals, which typically transition from rail to rail. Also,the signal should begin and end near a value of zero creating smooth transitionswhen injected into the system. Second, it is desirable to keep the peak probingamplitude small when probing for a long duration. For example, with thePDCI, the maximum input magnitude has been limited to �20 MW for long-duration probing. Another constraint is that the probing input should not looklike a single sinusoidal component as it could be mistaken for a sustainedoscillation. A pseudorandom signal is preferred.

System identification theory gives much guidance for input design. It isimportant to keep in mind that when probing, the measured outputs are acombination of the response to the probing signal and the ambient signal,which is always present in the measured outputs. The ambient signal is stochas-tic (random) in nature. Thus, when the probing signal is present, only a portionof the measured output is the system response to the probing signal, and theother portion is the ambient noise process. When it comes to the quality of theestimated parameters, it is the spectrum of the probing signal which is mostimportant, not the particular time-domain wave shape. The general idea is toplace the content of the probing signal in the frequency band of interest, in thiscase the frequency range of the inter-area electromechanical modes.

The amplitude and time duration of the low-level injected signal are critical.Clearly, the amplitude needs to be small enough not to interfere with the normaloperation of the power system. Yet, there is a well-known trade-off in systemidentification between the observation time and the signal strength. Perfor-mance of system identification algorithms improves with signal-to-noise ratio(SNR) and with observation time. The repeatability of the pseudonoise isimportant to fully take advantage of the repetition of the injected signal.Also, knowing the specific frequency content is critical.

An important quality of a probing signal is its crest factor. The crest factor ofa zero mean waveform u[n] is defined as

Cr¼�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimaxn u2½n�

1=Nð ÞPN

n¼1 u2½n�

s(1:7)

whereN is the number of samples in the waveform and n is the nth time sample.The crest factor is the ratio of the maximummagnitude of the signal to the rootmean square (RMS) value. It is desirable to have a probing signal with as large


an RMS value as possible for a given maximum peak magnitude. Thus, a good

waveform design should have a small crest factor while maintaining the desired

spectrum and power carried by the waveform. The minimum crest factor is

unity and this only occurs in signals which transition from rail to rail. Because

these types of signal are undesirable in this application, the minimum crest

factor cannot be achieved.There are some important advantages to using a periodic input signal.

Output waveforms can be averaged over the periods giving an effective

increase in the SNR by the number of periods averaged. This increase is

known as the processing gain. A similar gain in SNR can be seen in the

frequency domain at the frequency bins of the harmonics of the periodic

input. It is very important to inject an exact integer number of cycles so that

there is no leakage effect in the frequency domain. Also, periodic inputs allow

for methods to estimate the noise signal. The signal period, T, is important as

it determines the frequency resolution as �f=1/T when conducting fre-

quency-domain analysis. Because the inter-area electromechanical modes

are usually in the frequency range from approximately 0.1 to 1.0 Hz, a

frequency resolution in the neighborhood of 0.01 Hz should be adequate.

Note, there is a trade-off between the frequency resolution and the number of

averages. For a given input signal duration, the larger the period, the better

the frequency resolution, but fewer periods of the signal are available for

averaging, so the processing gain is less.For the system tests carried out in the wNAPS in 2005, 2006, and 2008,

a multisine input signal was used because of its favorable characteristics

relative to the above discussion. The bottom line is that the injected signal

should be chosen to not disrupt the normal operation of the power system,

yet to provide an accurate system model given a specific identification

method.

1.5 Mode Estimation Examples

Many papers have been published demonstrating the power system applica-

tion of signal processing methods for estimating modal frequencies and

damping. These papers include ringdown analysis and mode-meter applica-

tions. Many of these papers are referenced in the previous sections. In this

section, we provide a few examples to emphasize some of the significant

challenges. We refer the reader to the references for a complete view of the

application issues.Two types of examples are considered. With the first, a simulated system

with known properties is employed. The advantage of this system is that the

exact solution is known; therefore, algorithm properties can be evaluated. The

second type of examples uses actual system cases from the wNAPS.


1.5.1 Simulation System

The simulation test system is shown in Fig. 1.4. A modified version of the

system was originally developed as a simplified model of the western North

American power grid in [37]; detailed information is presented in the appendix

of [38]. It has been used in many publications as a research demonstration

model for stability-limited issues and mode estimation analysis.The system consists of major generation buses 17 through 24 and 45, and load

buses 31 through 41. Each generator is represented using a detailed two-axis

transient model equipped with a fast-acting voltage regulator, a power system

stabilizer (PSS) unit, and a turbine governor. Two identical generators are

attached to buses 17 through 24. Overall, the system order is 203. Each load is

split into a portion consisting of constant impedance, constant current, constant

power, and random. The random portion of both the real and reactive loads is

obtained by passing independent Gaussian white noise through a 1/f filter. It has

been hypothesized that such a filter is appropriate for load modeling [7].

23

24

1920

21

25

17

18

22

26

500230

230500

28

29

500230

2775, 35

70, 30

82, 40

44, 2644, 26

30, 12

14, 4

54, 20

Real, reactive load (pu)

62, 28

55, 25

3130, 1230

34

33

3235

37

36

38

39

4041

500/110

45

Rectifier

Inverter

42

500 kVHVDC

500/110

4330

Fig. 1.4 Simulation test system


Two primary system operating conditions are used with the simulations thatfollow. With the first condition, termed the 17-machine system, all generators areconnected to the system. Under this condition, the most dominant inter-areamodes are shown in Table 1.1. With the second condition, generator bus 45 isdisconnected from the system; this condition is termed the 16-machine system.Under this condition, the dominant inter-areamodes are shown in Table 1.2. Themodes shown in Tables 1.1 and 1.2 were calculated by conducting an eigenana-lysis of the entire system’s small-signal model under nominal steady-state operat-ing conditions. The eigenanalysis was conducted using the methodology in [4].

For the examples that follow, a typical time-domain simulation consists ofdriving the systemwith independentGaussian load variations tomimic ambientconditions. The system’s response consists of small random variations in thesystem states. As an example, the top plot of Fig. 1.5 shows the resultingrandom variations of bus 22 frequency for a 10 min simulation. The frequencyis calculated using the derivative of the bus phase angle.

To mimic a transient condition, a 0.5 s long load pulse is applied to bus 35.The bottom plot of Fig. 1.5 shows the system’s response to a 700 MW loadpulse. Figure 1.6 shows the response to a 1,400 MW pulse.

1.5.2 Ringdown Analysis Performance

As described in Section 1.3, ringdown analysis is used to estimate the modalproperties from a transient. One important property we wish to emphasize isthat the accuracy of the estimate is strongly related to the SNR. That is, howlarge the ringdown is compared to the ambient noise.

As an example, consider the ringdowns in Figs. 1.5 and 1.6. For the ring-down portion, the SNR in Fig. 1.6 is four times as large as that in Fig. 1.5.Table 1.3 compares the Prony analysis results for these two responses. For eachcase, the Prony analysis was conducted from 31 to 50 s into the simulation. Asseen in Table 1.3, the higher SNR signal provides a more accurate mode

Table 1.1 Inter-area modes of 17-machine system

Frequency (Hz) Damping (%) Buses vs. Buses

0.318 10.74 North half vs. Southern half

0.422 3.63 North half vs. Southern half + bus 45

0.635 3.94 18 vs. Rest of system

0.673 7.63 20,21 vs. 24

Table 1.2 Inter-area modes of 16-machine system

Frequency (Hz) Damping (%) Buses vs. Buses

0.361 6.59 North half vs. Southern half

0.618 3.57 18 vs. Rest of system

0.673 7.66 20,21 vs. 24


estimate of the dominant mode. To exactly quantify the accuracy, a MonteCarlo simulation must be conducted.

1.5.3 Mode-Meter Performance

In this example, the performance of mode-meter algorithms is demonstrated.Specifically, we consider estimation accuracy in the context of the mode damp-ing, the analysis window size, and transient versus ambient conditions. A moreextensive overview of this comparison is contained in [32, 28]. Time-domain

0 2 4 6 8 10

59.98

60

60.02

Time (min.)

Hz

0 0.5 1 1.5 2

59.98

60

60.02

Hz

Ringdown

Zoom

Fig. 1.6 Bus 22 frequency for 16-machine system under a transient simulation response to a1,400 MW 0.5 s load pulse at bus 35. Pulse is applied 30 s into simulation

Ambient

Ringdown

0 2 4 6 8 10

59.98

60

60.02

Hz

0 2 4 6 8 10

59.98

60

60.02

Hz

Ringdown

Ambient

Time (min.)

Fig. 1.5 Bus 22 frequency for 16-machine system. Top plot, ambient condition. Bottom plot,transient simulation response to a 700MW 0.5 s load pulse at bus 35. Pulse is applied 30 s intosimulation


simulation of the Fig. 1.4 system is employed to investigate properties. Because

of the random nature of the data, Monte Carlo simulations are employed to

fully evaluate the properties. In each simulation case, the measurement noise

terms (�i(t) in Fig. 1.1) are represented by passing white noise through a low-

pass first-order filter with a corner at 5 Hz. In each case, �i(t) is scaled such that

the SNR between yiðtÞ and �i(t) is 12 dB. This represents a relatively high

measurement noise condition.Figure 1.7 shows the estimates of the 0.361 Hz mode from the 16-machine

system under varying damping conditions and ambient operation using the YW

Table 1.3 Prony analysis estimates for 16-machine modes

Actual

Estimated Estimated

(700 MW pulse) (1400 MW pulse)

Frequency(Hz)

Damping(%)

Frequency(Hz)

Damping(%)

Frequency(Hz)

Damping(%)

0.361 6.59 0.362 7.2 0.361 6.8

0.618 3.57 0.619 3.9 0.618 3.6

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

. (H

z)

YW Estimated

Actual

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

. (H

z)

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

. (H

z)

1%D 1%D

1%D

20%D20%D

20%D

Actual damping:upper left plot = 6.59% upper right plot = 3% lower plot = 1%

Fig. 1.7 Mode estimates for 16-machine system for estimating the 0.361 Hz mode from 100ambient Monte Carlo simulations using the YW algorithm. 1%D and 20%D are constantdamping lines


algorithm. The figure shows three s-plane plots representing varying damping

conditions. Mode conditions in the system can be varied by modifying system

operating conditions (e.g., steady-state loading) resulting in new small-signal prop-

erties. For each plot, 100 independent simulationswere conductedusing an analysis

window length of 10 min. The relative voltage angles between buses 25–21, 20–22,

19–23, and 26–17 are used as the inputs to the algorithm. For each case, the

estimated mode with the largest pseudoenergy in the region of the s-plane bound

by 0.2, 0.5 Hz, and 20% damping is plotted for each simulation. This tests the

algorithms capability to automatically estimate a mode without human software

interaction. As the mode becomes more lightly damped, it is estimated with con-

siderably more accuracy. This is a property of all mode estimation algorithms.A more challenging situation is that of estimating two closely spaced modes

as with the 17-machine system. Figure 1.8 shows plots for the YW, YWS, and

N4SID algorithms using an analysis window size of 10 min. The two modes

with the largest pseudoenergy terms in the region of the s-plane bound by 0.2,

0.5 Hz, and 20% damping are estimated with a mode-meter algorithm for each

simulation. The YW and YWS provide more accurate estimates than the

N4SID algorithm. Further investigation in [39] shows that the poor perfor-

mance of the N4SID is related to the measurement noise. That is, the

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

Estimated

Actual

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

–0.4 –0.2 00.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

N4SID

YWS

YW

1%D20%D

1%D1%D20%D20%D

Fig. 1.8 Mode estimates for 17-machine system for estimating the 0.318 and 0.422 Hz modesfrom 100 ambient Monte Carlo simulations


performance of the N4SID degrades significantly as measurement noiseincreases while the YW and YWS algorithms continue to perform well withmeasurement noise. Note that the frequency estimates are much more accuratethan the damping estimates, and the accuracy of estimating the less dampedmode is better than the more highly damped mode.

The analysis window size is the amount of historical data required to obtainan estimate. Certainly, one would expect better performance from the mode-meter algorithms as total data used for analysis increases. The length of analysisdata is indicated byN samples in the algorithms or byTtotal¼TN seconds whereT is the sample period.

As before, a series of Monte Carlo simulations are conducted to demonstratethe effect of different window sizes on the performance of the algorithms. In eachof the simulations, the system is operated under the 17-machine condition. Thegoal is to estimate both the 0.318 Hz mode and the 0.422 Hz mode. For eachsimulation, the two modes with the largest pseudoenergy terms in the region ofthe s-plane bound by 0.2, 0.5 Hz, and 20% damping are estimated with a mode-meter algorithm. The results for the YW algorithm are summarized in Fig. 1.9.One can certainly see the improved performance as the window size is increased.

The basic assumption for the mode-meter algorithms is that the system isexcited by random inputs. In real-world automated conditions, data may oftencontain transient ringdowns and nonstationary conditions. A ringdown occurswhen the system is excited by a sudden input such as a fault, generator trip, or loadtrip. Often, the post-transient steady-state condition of the system will change.That is, the mode damping and frequencies will change following the transient. Ingeneral, the mode estimates actually improve if a ringdown is present in the data.

A condition often encountered is a transient which causes the mode tobecome less damped. As an example, a 700 MW, 0.5 s pulse is added to bus35 of the 16-machine system. Prior to the transient, the mode is at 0.361 Hz, 6%damping. During and after the transient, the mode shifts to 0.25 Hz, 1%

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 5 10 15 20 25

Window size (min.)

Fre

q. (

Hz)

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25Window size (min.)

Dam

ping

(%

)

Fig. 1.9 Accuracy of the YW algorithm for 0.318 and 0.422 Hz modes for varying Ttotal.¼mean � standard deviation, ¼median. Ambient condition


damping. Figure 1.10 shows the system’s response. The resulting mode esti-mates using the 10 min data window in the top plot of Fig. 1.10 are shown inFig. 1.11 using three different algorithms. The corresponding estimates usingthe 2 min data window in the bottom plot of Fig. 1.10 are shown in Fig. 1.12.The modes estimates converge to the more lightly damped post-transient con-dition. This is certainly the desired result.

1.5.4 Field Measured Data

As described earlier, operators of the wNAPS periodically conduct extensivedynamic tests. These tests typically involve 0.5 s insertion of the Chief Joseph1,400 MW braking resistor in Washington; and probing of the power referenceof the PDCI (see Fig. 1.3). The resulting system response provides rich data fortesting mode estimation algorithms. This section presents a few of these results.

Figure 1.13 shows the system response from a brake insertion along withseveral minutes of ambient data. The signal shown is the detrended real powerflowing on a major transmission line.

Two recursive mode estimation algorithms are applied to the data: the RLSand RRLS [32] algorithms. The resulting mode estimates are shown inFigs. 1.14 and 1.15. The damping estimates for the 0.39 Hz mode are shownas this is the most lightly damped dominant mode. The results are compared toa Prony analysis of the ringdown. More detailed results are shown in [32]. TheRRLS algorithm provides a more accurate mode-damping estimate and theaccuracy improves after the ringdown.

0 2 4 6 8 10–0.04

–0.02

0

0.02

0.04

Rad

ians

Time (min.)

8 8.5 9 9.5 10–0.04

–0.02

0

0.02

0.04

Time (min.)

Rad

ians

zoom

Fig. 1.10 Angle between buses 25 and 21 voltages for the 16-machine system. At 540 spoint in simulation, a 700 MW, 0.5 s pulse is added to bus 35. Pre- and post-transientmodes differ


1.5.5 Probing Test Results

As discussed in Section 1.4, exciting the system with a probing signal improves

the mode estimates. This is demonstrated in this section using results from

wNAPS probing test results from August 2006.Figure 1.16 summarizes the responses from a typical wNAPS probing test.

The pseudorandom probing shown in the blue region is a multisine signal with

the phases adjusted to optimize the signal crest factor. The frequency content

is 0.1–0.9 Hz; after 1.0 Hz, the signal rolls off at 120 dB/decade and drops to

zero at 2 Hz. Note the probing signal is scaled to�20MWon the DC line. The

resulting response on the AC system is just above the ambient noise.Figure 1.17 shows the mode estimates comparing probing versus ambient

for a mode known to be near 0.38 Hz. In both the ambient case and the

probing case, 20 min of data are analyzed. The ambient and probing occur

at the same hour of the test and the system remained in the same relative

operating point; therefore, the actual systemmode likely remain constant. The

black circles are the mode estimates from a 680 s window sliding over every

60 s in the ambient data. The crosses show the same for the probing data.

Estimated

Pre-transient

Post-transient

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

YW 1%D

20%D

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

YWS 1%D

20%D

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

N4SID 1%D

20%D

Fig. 1.11 Mode estimates for 16-machine system with a transient from 100 Monte Carlosimulations for each algorithm. Pre- and post-transient damping differ. Ttotal¼ 10 min


The green markers show the corresponding estimates using the entire 20 min

of data. As can be seen in the plot, the variance of the estimated damping

during probing is more than twice as small as the variance during the ambient

condition.

Estimated

Pre-transient

Post-transient

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

YWS 1%D

20%D

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

YW1%D

20%D

–0.5 –0.4 –0.3 –0.2 –0.10.2

0.3

0.4

0.5

Real (1/sec)

Imag

./(2p

i) (H

z)

N4SID 1%D

20%D

Fig. 1.12 Mode estimates for 16-machine system with a transient from 100 Monte Carlosimulations for each algorithm. Pre- and post-transient damping differ. Ttotal¼ 2 min

Fig. 1.13 Brake response ofwestern North Americanpower system. Brakeinserted at the 300 s point.Combined ambient andringdown data from fieldmeasurements. Detrendedpower flowing on a majortransmission line


Figure 1.18 shows similar results for varying window sizes. The plot shows

that as the window size is increased, the accuracy of the mode estimate

increases. This is seen in the graph by the decrease in the standard deviation

of the estimate as the window size increases. A bootstrap technique [40] was

used to estimated the standard deviation. Also, in this particular case, estimates

during probing are more than twice as accurate as during ambient conditions

illustrating the improvement in mode estimation performance resulting from

low-level probing.

Fig. 1.14 Frequency estimation of the major modes using the RRLS algorithm

Fig. 1.15 Damping ratio (DR) estimation of the major mode around 0.39 Hz


–20 –15 –10 –5 00.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Real (1/sec)

Imag

. (ra

d/se

c)

YW

AmbientProbingAll AmbientAll Probing

Fig. 1.17 Mode plots for wNAPS probing test series C of the August 2006 probing test.Window¼ 680 s. Window sliding over in one minute sections. Ambient¼ test series C1;Probing¼ test series C2; All ambient¼mode estimates using all of C1; All probing¼modeestimates using all of C2

0 10 20 30 40 50 60500

600

700

800

900

1000

1100

Time from 22-Aug-2006 20:00 GMT (minutes)

Pow

er le

vel (

MW

)

Fig. 1.16 wNAPS response to probing tests. Bottom signal is DC power flow. Top signal is realpower flowing on a major transmission line. 10 min to 17 min region contains brake responses,22 min to 42 min region contains PDCI pseudorandom probing response, 44 min to 47 minregion contains mid-level probing response, and all other regions are ambient responses


1.6 Model Validation and Performance Assessment

As seen in the previous examples, there are limits to the accuracy of any

estimator. To fully solve the problem, estimates need to be validated and the

accuracy of the estimates needs to be assessed. The stochastic nature of the

problem cannot be overemphasized. This stochastic nature of the measured

data results in statistical variability of the estimates. The validity of the esti-

mates needs to be tested with asmany techniques as possible and the accuracy of

the estimates need to be determined. This section discusses numerous methods

to validate the mode estimates and to assess the accuracy of those estimates for

both ambient and probing. The resulting estimates of the modes are just that –

estimates (best guesses). A key question is the following: ‘‘is there adequate

agreement between the observed data and the estimated modes?’’

1.6.1 Model Validation

In real-time operation, there are a limited number of ways to validate mode

estimates from ambient data. In this case, one of the most direct ways to

validate the mode estimates is to compare the measured signal’s nonparametric

spectrum with the parametric spectrum corresponding to the estimated modes.

Fig. 1.18 Mode-damping estimate standard deviation for wNAPS probing test in August2006. Probing versus ambient. One multisine cycle¼ 136 s


For example, if the YW method is being applied to estimate the modes, thecorresponding YW parametric spectrum estimate [35] should be compared forconsistency with the nonparametric Welch periodogram spectrum estimate. Ifthere is not a strong similarity in the spectrums, then one may question thevalidity of the mode estimates. Another approach would be to find comparablemode estimation results using different output channels where the modes arehighly observable. Consistency in results between different mode identificationalgorithms would also be evidence of a valid model. For some algorithms,residual analysis [36] can also be employed. Residual analysis studies the portionof the data which the estimated model cannot reproduce and provides a goodindicator of the validity of the estimated models. Another method of validatingthe ambient mode estimates is by comparing those estimates with the estimatesfrom the analysis of a transient response occurring immediately before or after theambient data. This would assume nomajor system configuration change occurredat the transient event and that the true mode values were nearly the same.

When a known probing signal is used, the same validation methods as in theambient case can be used but additional validation methods are possible. Asdiscussed earlier, injecting known probing signals can improve the accuracy ofthe mode estimates. Comparisons can be made between nonparametric fre-quency response estimates and parametric methods. Moreover, validation datamay be used. Validation data are data not used in the estimation of the modes,but used to validate the estimated model. For example, in the wNAPS probingtests carried out in 2000, 2005, 2006, and 2008, short-duration mid-level probingpulses with a peak amplitude of 125 MWwere applied to the PDCI immediatelyfollowing the long-duration low-level probing. The input and output data fromthe low-level probing is used to estimate system transfer functions. The actualsystem response from the mid-level probing is then compared to the responsefrom applying the same mid-level probing signal to the estimated transfer func-tion. Consistency between the actual response and estimated response is a posi-tive indicator of the validity of the estimated transfer function. There are twosources for differences – errors in the transfer function estimate and system noise.Thus the mid-level probing needs to be performed such that a reasonable SNR isachieved so that the difference in the responses coming from the noise is small.The mid-level probing can be applied multiple times and the responses averaged,which increases the SNR by the number of averages.

1.6.2 Performance Assessment

Because the measured time series of data is stochastic (random) in nature, thereis always some variability in the mode estimates around the true value of themode. Thus, the estimates of the modes are essentially a best guess given theobserved time series of data. It is important to try to quantify the quality ofthe estimate. One way to do this is to not only provide a point estimate of themode frequency and damping but also provide an estimate of the mean square


error in those quantities, or to provide a confidence interval as the estimateinstead of just a point estimate. For example, instead of just stating that themode-damping ratio estimate is 6%, state that the mode-damping ratio esti-mate is 6% with an RMS error of 1%. This gives some indication of theaccuracy of the estimate. Achieving this measure of performance is a difficulttask. In [26], a bootstrapping approach was first applied to electromechanicalmode estimation. The central idea of the bootstrap method is re-sampling andthe method was first introduced in [41]. Performance assessment is certainly anarea of current and future research.

1.7 Estimating Mode Shape

Similar to the modal damping and frequency information, near-real-timeoperational knowledge of a power system’s mode-shape properties may providecritical information for control decisions. For example, modal shape maysomeday be used to optimally determine generator and/or load trippingschemes to improve the damping of a dangerously low damped mode. Theoptimization involves minimizing load shedding and maximizing improveddamping. This section describes how mode shape can be estimated from time-synchronized measurements.

Results published in [42, 43] demonstrate how one can use spectral analysisto estimate the mode shape from synchronized measurements. The followingsummarizes these results.

1.7.1 Defining Mode Shape

The eigenvalues and eigenvectors for (1.1) are defined from the equations

liI� Aj j ¼ 0; Aui ¼ liui; viA ¼ livi (1:8)

where li is the ith eigenvalue (i= 1. . .n), ui (order n � 1) is the ith righteigenvector, and vi (order 1 � n) is the ith left eigenvector, and I is the n � nidentity matrix. As shown in [4] when considering the ambient case, each systemstate can be written as

xðtÞ ¼Xn

i¼1ziðtÞui (1:9)

where

ziðtÞ ¼ vixðtÞ (1:10)

_ziðtÞ ¼ liziðtÞ þ viBLqðtÞ (1:11)

for i= 1. . .n.


The solution of (1.11) results in zi(t), which is the ith mode’s response to q(t).Equation (1.9) provides information on how the modes are combined to createthe system states. Examination of (1.9) reveals that element ui,k (the kth elementof ui) provides the critical information on the ith mode in the kth state. Theamplitude of ui,k provides the information on the magnitude of mode zi in statexk. It is a direct measure of the observability of the mode in the state. The angleof ui,k provides the information on the phasing of zi in state xk. By comparingthe ffui;k for a common generator state (such as the speed), one can determinephasing of the oscillations for the ith mode. As such, ui has been termed the‘‘mode shape’’ vector [4]. Knowledge of ui provides all the required informationto completely determine the pattern of generator swings for the ith mode [4].

1.7.2 Estimating Mode Shape

As described above, the right eigenvector ui completely describes the modeshape ofmode zi. The question here is: How can the properties of ui be estimatedfrom direct power system measurements without the dependence on the lineardifferential model (1.1)? This section summarizes how spectral analysis providesthe required information; see [42] for more details.

Begin by defining two spectral functions.

Skl !ð Þ ¼ limT!1

1

TE Y�k !ð ÞYl !ð Þ� �

(1:12a)

Skk !ð Þ ¼ limT!1

1

TE Y�k !ð ÞYk !ð Þ� �

(1:12b)

where Sk,l(!) is the cross-spectral density (CSD) function between generalsignals yk(t) and yl(t), Sk,k(!) is the PSD of signal yk(t), Yk(!) is the discreteFourier transform of signal yk(t) at frequency !,Y

�k !ð Þ is the complex conjugate

of Yk(!), and E{} is the expectation operator. These definitions are found inmany signal processing textbooks such as [44].

Now assume that li is a lightly damped mode with

li ¼ �i þ j!i (1:13)

where �i << !i. As shown in [42], the following relationships result

ffSkl !ið Þ ffi ffui;l � ffui;k (1:14)

Skk !ið Þ ffi ui;k�� 2Knoise (1:15)

where Knoise is an unknown constant.


Equations (1.14) and (1.15) are used to estimate the mode shape. Assume forthe moment that all generator speed signals are time-synchronized sampled.Also assume that the frequency of the oscillation mode !i is known. The PSD iscalculated for each generator speed signal. From (1.15), the PSD of each signalis scaled by ui;k

�� 2; therefore, the PSD is a direct measure of the observability ofthe mode at that generator.

The phasing of the mode among the generators is directly estimated from theangle of the CSD by (1.14). A reference generator with high mode observabilityis chosen as the reference generator k. The angle of the CSD is calculated for allother generators at mode frequency !i. From (1.14), the angle of the CSD forgenerator l represents the phasing of the oscillation.

1.7.2.1 The Coherency

The squared coherency function is defined as

�2k;l !ð Þ ¼Sk;l !ð Þ�� 2

Sk;k !ð ÞSl;l !ð Þ(1:16)

It represents a measure of the correlation between two signals as a func-tion of frequency [44]. As the two signals become uncorrelated, the coherencyconverges to zero. Similarly, as the signals become totally correlated, thecoherency converges to unity. Basically, it is a measure of percentcorrelation.

As shown in [45], the coherency function can be used to determine if a modeof oscillation is due to one mode or multiple modes at the same frequency. Forexample, given two signals y1(t) and y2(t), if both S1,1(!) and S2,2(!) have peaksat frequency !i, this indicates that the system contains one ormoremodes at thisfrequency. If �21;2 !ið Þ is near unity, then the same mode is contained in both y1and y2. Alternatively, if �21;2 !ið Þ is near zero, then the system contains at leasttwo different modes at frequency !i.

1.7.2.2 Calculating Spectral Terms

Using traditional periodogram averaging methods [44], calculating the PSD,CSD, and coherency requires several minutes of time-synchronized measure-ments. In theory, the measurement should be either the generator angle orspeed; but, in most cases, these signals are rarely time synchronously measured.An excellent approximation to a speed signal is the frequency of the generatorbus voltage (or a nearby bus). Such a signal is obtained from standard synchro-phasor measurements (PMUs, phasor measurement units). More efficientmethods that calculate the required terms from less data are being researchedwith initial results found in [43].


1.7.3 16-Machine Example

This example demonstrates the application of the spectral approach for esti-

mating the mode-shape properties of the 0.36 Hz mode of the 16-machine

system. For comparison, Table 1.4 shows the right eigenvector (ui) terms for

the 0.36 Hz mode and the speed state variable for one generator at each

generation bus. Note that ui has been normalized by the largest term (i.e., the

ui,k for bus 19). Examination of Table 1.4 shows that the mode is primarily a

north–south mode with buses 19 and 20 swinging against buses 18, 22, 17, and

23. Buses 24 and 21 are in the middle of the shape and have very low participa-

tion in the mode. This is indicated by the small value for |ui,k| for these two

buses.The goal is to estimate the mode-shape information in Table 1.4 using only

time-synchronized frequency measurements at the buses. It is assumed that the

mode frequency is known and the system is simulated for 10 min. To compare

with the eigenanalysis results in Table 1.4, bus 19 is selected as the reference bus

for the correlation analysis.Figures 1.19, 1.20, 1.21 show the results for the spectral analysis. The

significant peaks in the PSD and coherency estimates at 0.36 Hz for buses 17,

18, 19, 20, 22, and 23 indicate that the generators connected to these buses all

significantly participate in a single mode at that frequency. The lack of signifi-

cant peaks for buses 21 and 24 indicate that the generators at these buses do not

significantly participate in the mode. The angle of the CSD shown in Fig. 1.21

provides the information for the mode phasing. Bus 20 generators swing in

phase with bus 19 generators, while generators at buses 17, 18, 22, and 23 swing

against bus 19 generators.Using Eqs. (1.14) and (1.15), the spectral analysis results can be directly

compared to the eigenanalysis. The normalized estimated ui;k�� is calculated by

taking the inverse decibel from Fig. 1.19 and normalizing by the value for bus

19. Similarly, the estimated ffui;k is directly taken from Fig. 1.21. The results are

shown in Table 1.5. As seen in the table, the spectral analysis very accurately

estimates the eigenvector solution.

Table 1.4 Eigenvector results for 16-machine, 0.361 Hz mode

Bus Angle (ui;k) (degrees) jui;kj (relative)19 0 1.00

20 1 0.76

18 154 0.64

22 151 0.39

17 151 0.38

23 146 0.34

24 61 0.17

21 –7 0.13


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–65

–60

–55

–50

–45

–40

–35

Frequency (Hz)

PS

D (

dB) Bus 17

Bus 18

Bus 19

Bus 20

PS

D (

dB)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–65

–60

–55

–50

–45

–40

–35

Frequency (Hz)

Bus 21

Bus 22

Bus 23

Bus 24

Fig. 1.19 PSD estimates for 16-machine example. * indicates the 0.361 Hz mode

Coh

eren

cy2

Bus 17

Bus 18

Bus 20

Bus 21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Coh

eren

cy2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Bus 22

Bus 23

Bus 24

Fig. 1.20 Coherency estimates for 16-machine example. Frequency at bus 19 is the referencesignal. * indicates the 0.361 Hz mode

1.7.4 Field Measured Data

Now consider the actual system data from the wNAPS in Fig. 1.3. The modal

properties of the system are routinely investigated through system probing

testing that employs a synchronized measurement system. The system contains

several inter-area modes including significant ones near 0.25 and 0.37 Hz.

The mode’s frequency and damping are estimated using previously described

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–200

–100

0

100

200

Frequency (Hz)

CS

D a

ngle

(de

gree

s)

Bus 17

Bus 18Bus 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–200

–100

0

100

200

Frequency (Hz)

CS

D a

ngle

(de

gree

s)

Bus 22

Bus 23

Fig. 1.21 CSD angle estimates for 16-machine example. Frequency at bus 19 is the referencesignal. * indicates the 0.361 Hz mode

Table 1.5 Comparison of eigenvector and spectral analysis for the 0.362Hz mode of the16-machine example

Bus

Eigenvector Spectral analysis

Angle (ui;k)(degrees)

jui;kj(relative)

Angle (ui;k)(degrees)

jui;kj(relative)

19 0 1.00 0 1.00

20 1 0.76 3 0.74

18 154 0.64 161 0.66

22 151 0.39 157 0.41

17 151 0.38 158 0.40

23 146 0.34 157 0.38


mode-meter methods. The goal of this section is to demonstrate the application

of spectral analysis to actual systemmeasurement using the 0.25 Hz mode as anexample.

For demonstration purposes, six buses near major generation sitesspread across the system are selected. The actual locations of the sitesare not revealed in order to protect data confidentiality. The buses are

termed bus A through bus F. Nearly 20 min of phasor data was collectedduring a recent probing test from a PMU at each bus. The frequency errorat each bus is estimated from the phasor angle using a forward differen-

cing calculation [33].Results from the spectral analysis are shown in Figs. 1.22–1.24. The

relatively large peaks at 0.25 Hz for buses B through F in the PSD andcoherency indicates that these buses participate in a single mode at thisfrequency. The relatively small peak for bus A indicates that this bus does

not significantly observe this mode. Bus D is selected as the reference busfor the analysis as it has a large peak in the PSD. It should be noted thatany signal with a large peak may be selected as the reference bus in theanalysis.

The CSD in Fig. 1.24 shows the mode phasing. The plot indicates that buses

D and C swing together against buses B, E, and F.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–65

–60

–55

–50

–45

Frequency (Hz)

PS

D (

dB)

A

BC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–60

–55

–50

–45

–40

Frequency (Hz)

PS

D (

dB)

D

EF

Fig. 1.22 Estimated PSD for wNAPS mode-shape example


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–200

–100

0

100

200

Frequency (Hz)

CS

D a

ngle

(de

gree

s)

B

C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8–200

–100

0

100

200

Frequency (Hz)

CS

D a

ngle

(de

gree

s)

E

F

Fig. 1.24 Estimated CSD angle for wNAPS example. Bus D is the reference signal. *indicates the 0.25 Hz mode

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Coh

eren

cy2

Coh

eren

cy2

A

BC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Frequency (Hz)

E

F

Fig. 1.23 Estimated coherency for wNAPS example. BusD is the reference signal.* indicatesthe 0.25 Hz mode

1.8 Conclusion

This chapter has presented an overview of many of the more successful analysistechniques and challenges to electromechanical mode estimation from time-synchronized data. The theoretical basis for these methods is described as wellas application and performance properties. Several examples are used todemonstrate some of the more challenging issues. A long list of references isprovided for further details.

Acknowledgments The authors wish to acknowledge the contribution of the many graduatestudents over the years. Also, the technical leadership of Dr. JohnHauer of Pacific NorthwestNational Laboratory (retired) and Mr. Bill Mittelstadt of the Bonneville Power Administra-tion (retired) are acknowledged. Much of this work was supported by the US Department ofEnergy; the authors wish to thank Mr. Phil Overholt for his support.

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39. D. J. Trudnowski, J. W. Pierre, and N. Zhou, ‘‘Performance and properties of ambient-data swing-mode estimation algorithms, version 1.0,’’ Report no. ENGR 2006-1, Engi-neering Dept., Montana Tech of the University of Montana, Butte, MT, USA, 2006.

40. F.K. Tuffner, ‘‘Computationally efficient weighted updating of statistical parameter esti-mates for time varying signals with application to power system identification,’’ Ph.D.dissertation, Department of Electrical andComputer Engineering,University ofWyoming,Laramie, WY, USA, 2008.

41. B. Efron, R. Tibshirani, ‘‘Bootstrap methods: another look at the jackknife,’’ The Annalsof Statistics, vol. 7, no. 1, pp. 1–26, 1979.

42. D. Trudnowski, ‘‘Estimating electromechanical mode shape from synchrophasor mea-surements,’’ IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 1188–1195, Aug.2008.

43. L. Dosiek, D. Trudnowski, J. Pierre, ‘‘New algorithms for mode shape estimation usingmeasured data,’’ IEEEPower & Energy Society GeneralMeeting, paper no. PESGM2008-001014, July 2008.

44. J. S. Bendat, A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis,2nd Ed., John Wiley & Sons, New York, 1993.

45. D. Trudnowski, J. Hauer, J. Pierre, W. Litzenberger, D. Maratukulam, ‘‘Using thecoherency function to detect large-scale dynamic system modal observability,’’ Proceed-ings of the 1999 American Control Conference, pp. 2886–2890, June 1999.


Chapter 2

Enhancements to the Hilbert–Huang Transform

for Application to Power System Oscillations

Nilanjan Senroy

Abstract TheHilbert–Huang transform is introduced for time–frequency analysisof oscillatory signals representing power system dynamic behavior. Fundamen-tal assumptions of the Hilbert–Huang transform are revisited, particularly theability of empirical mode decomposition to yield monocomponent intrinsicmode functions. In the context of the specific application, some enhancementsto the original algorithms are discussed. A wide variety of application examplesare employed to demonstrate the efficacy of the improved Hilbert–Huangtransform.

2.1 Introduction

Modern interconnected power system dynamics is characterized by oscillatorybehavior. These oscillations are produced as a result of a variety of disturbancessuch as changes in loads, tripping of lines, faults, and other discrete events. Theoscillations manifest themselves as variations in line flows and generator angleexcursions. A detailed study of these oscillations is necessary to gain a thoroughunderstanding of the system dynamics.

Modern computer processing speeds and memory have allowed engineers toapply analysis techniques that were previously considered impractical due totheir computational burden. Increasing usage of strategically located sensorsand measuring devices is leading to vast amounts of data, whose analysiscan lead to a deeper insight into the underlying processes. In this context, it isworthwhile to mention the advent of wide-area measurement systems(WAMSs) in modern power systems. Using such data, utilities are increasinglyrelying on distributed measurement and control of power systems. An effectiveway to process such real-time measurement data is to analyze the constituent

N. Senroy (*)Department of Electrical Engineering, Indian Institute of Technology, New Delhi, Indiae-mail: [email protected]


37

modes. The frequency and damping of these modes indicate the nature andseverity of a disturbance, reveal the health of the system, and predict the out-come of an evolving contingency. Such information can be further used to plancontrol actions, instill situational awareness, and enable health monitoring andpreventive maintenance procedures.

Spectral analysis of power system oscillatory signal data is a challenging task.As power systems are inherently nonlinear, the signals representing their behaviorcontain possible time-varying waveform distortions. Further, modal interactionbetween closely spaced frequency components may lead to nonlinearities in thesignal. Thus, the spectral analysis of these signals is not a trivial task.

A popular tool for spectral analysis is the fast Fourier transform (FFT). TheFFT basically transforms a distorted signal from time domain to frequencydomain by resolving it into stationary equivalents, with frequency and ampli-tude information. Therefore, in the event of nonstationary components in theoriginal distorted signal, FFT results are suspect. The challenge of estimatingnonstationary distortion components is met by computing the FFT in a finitetime window. The signal components are assumed to be stationary within thistime window. The complete signal is analyzed by sliding the FFT window alongthe entire duration of the signal. The size of the analysis window must be largeenough to allow all the constituent modes to manifest completely, yet smallenough to satisfy assumptions of linearity and stationarity. This is a dauntingtask particularly in the case of power system oscillatory signals which arefrequently damped or ‘ringdown’ signals, containing possible nonlinearinteractions.

An alternative approach to the spectral analysis of power system oscillatorysignals is to repose the problem as an instantaneous frequency/amplitudetracking problem. Such an approach implies retaining the time-domain infor-mation of the signal without compromising on its frequency-domain analysis.The Hilbert–Huang technique is one such technique, recently proposed, totrack the temporal variations in the frequency and amplitude of the variouscomponents within a time-varying distorted signal. It was initially proposed forgeophysics applications, but has been widely applied with success in problemsin biomedical engineering, image processing, and structural safety. It has alsobeen applied in power systems in the area of power quality, subsynchronousresonance, and to analyze inter-area oscillations. The original Hilbert–Huangtechnique consists of the empirical mode decomposition (EMD) followed bythe Hilbert transform. While the Hilbert transform is a well-known mathema-tical technique widely applied in signal processing and communications, theEMDwas developed byNorden Huang [1]. As the name suggests, the EMD is adata-driven technique, empirical in nature with limited analytical justificationas of date.

There are some shortcomings of the original Hilbert–Huang technique, parti-cularly in the EMD algorithm, that limit its application to signals of the typeencountered in power system stability studies. The intention of this chapter is tohighlight these shortcomings and further resolve them using come enhancements

38 N. Senroy

to the EMD technique. The application of the enhanced Hilbert–Huang techni-que is demonstrated for a wide variety of problems in power system monitoringand control.

2.2 Hilbert–Huang Transform

Before going into the modified Hilbert–Huang transform, a brief exposition ofthe fundamental Hilbert–Huang transform is in order. TheHilbert–Huang trans-form consists of the EMD followed by the Hilbert transform. The original EMDalgorithm as provided in [1] is referred to as the standard EMD in this chapter.

2.2.1 Empirical Mode Decomposition

The underlying philosophy behind EMD is the concept of instantaneous fre-quency. From a Fourier analysis point of view, the frequency of a signal wouldbe derived from its time period, which is the time taken to complete onestationary time period. Therefore, for a nonstationary waveform whereinthere exist incomplete time periods, the frequency would be hard to define.Essentially, a full wave is required to be present in the signal, to be recognized asa legitimate frequency. However, the frequency is also equivalent to the angularvelocity which can be defined as the rate of change of phase. Hence, if a uniquephase can be defined for a real-valued signal, it would be easy to compute itsrate of change of phase and thereby its frequency. This point will be furtherelaborated in the next subsection.

The frequency obtained in this manner is unique at any instant in time, andhence termed as instantaneous frequency. In other words, it is possible to defineonly one instantaneous frequency for a signal at any point in time. This poses aproblem for multicomponent signals which have more than one frequencycomponents existing at a given time. The instantaneous frequency obtainedfor such a signal would be meaningless, unless the individual components areisolated before applying Hilbert transform on them. EMD is a method pro-posed by Norden Huang to decompose a multicomponent waveform intointrinsic mode functions (IMFs) that have well-defined Hilbert transforms.

The EMD technique essentially involves identifying a baseline signal existingin a signal. Such a baseline signal, when subtracted from the original signal,leaves behind a monocomponent signal, i.e., whose Hilbert transform is welldefined. This is called an IMF, which is characterized by the followingproperties:

� local uniformity around zero,� all the maxima (minima) are greater (less) than zero,� the numberof extremaand thenumberof zeros are equalordiffer byatmost one.

2 Enhanced Hilbert–Huang Technique for Power System Oscillations 39

The baseline signal is identified as the local mean of an upper signal envelopeand lower signal envelope. The upper signal envelope arises from the interpola-tion between the local maxima points, while the lower envelope is similarlyobtained from the local minima points. The steps to obtain an IMF from adistorted signal are as follows:

A1. identify local extrema of the signal, s(t),A2. interpolate between maxima (minima) to obtain the upper (lower) envel-

ope, eM(t) (em(t)),A3. compute the baseline signal, m(t) = (eM(t) + em(t))/2A4. extract c(t) = s(t) – m(t),A5. if c(t) is not an IMF then improve it by applying steps A1–A4 replacing s(t)

by c(t), until the new c(t) obtained satisfies the conditions of being an IMF,A6. if c(t) is an IMF, then compute the residue, r(t) = s(t) – c(t),A7. if r(t) is not monotonic, then repeat steps A1–A6 replacing s(t) with r(t), to

obtain the next IMF.

In practice, an appropriate stopping criterion must be used in step A5 toavoid overimproving the IMF, as that can potentially lead to loss of modalinformation. Once all the IMFs have been extracted, the final residue is mono-tonic in nature, i.e., lack of extrema points. The IMFs ‘sifted’ in this mannerfrom a distorted signal are orthogonal in nature, and appear in decreasing orderof frequency. In other words, the first IMF extracted contains the highestfrequency component, while the last IMF extracted contains the lowest fre-quency component. If n IMFs are extracted in this manner, the original dis-torted signal can be recovered from the IMFs as follows:

sðtÞ ¼X

n

ciðtÞ þ rðtÞ (2:1)

2.2.2 Hilbert Transform

In 1946, Gabor developed the concept of an analytic signal to provide anunambiguous definition of the phase of a signal [2]. If the instantaneousfrequency is defined as the rate of change of phase, the instantaneous phase ofthe signal is required. For a real-valued signal, its instantaneous phase cannotbe computed without defining its imaginary counterpart. While there are infi-nite ways of arriving at an imaginary counterpart of a given real signal, theappropriate method, as defined by Gabor, is as follows. In the Fourier trans-form of the original real-valued signal, the negative frequencies are suppressed,and an inverse Fourier transform applied to the resultant spectrum. The signalthus obtained is complex in nature and is referred to as the analytic form of theoriginal signal. The real part of this analytic signal is the original real-valuedsignal, while its imaginary part equals the Hilbert transform of the original

40 N. Senroy

signal. Hence, the first step toward the calculation of the instantaneous fre-quency (and amplitude) of a signal, is the application of the Hilbert transform.

The Hilbert transform of a real-valued signal, s(t), is defined as

sH tð Þ ¼ H x tð Þf g ¼ 1

p}

Z1

�1

x tð Þt� y

dt (2:2)

where } indicates the Cauchy’s principal value of the integral. Accordingly, theintegral is evaluated in the range [–1, y– e] and [y+ e,1]. The analytic form ofs(t) can be defined from its Hilbert transform as sA(t) = s(t) + jsH(t) , such that

= sA tð Þf g ¼ Re = s tð Þf gf g (2:3)

where = �f g refers to the Fourier transform operator. The instantaneous phaseangle of sA(t) is accordingly defined as

� tð Þ ¼ tan�1Im sA tð Þf gRe sA tð Þf g ¼ tan�1

sH tð Þs tð Þ (2:4)

The phase, thus obtained, must be unwrapped to be meaningful.

2.3 Modified Hilbert–Huang Transform

In this section, the limitations of the standard EMD are highlighted, and theenhancements to the EMD are presented. The modified Hilbert–Huang trans-form was initially suggested as an analysis technique for signals encountered inpower quality [3]. These signals are characterized by the presence of a strongfundamental frequency (50 or 60 Hz), distorted by the presence of higherharmonic and inter-harmonic frequency components that are relatively weakerin magnitude. Additionally, the frequencies of these components lie within anoctave posing a challenge in separation. Encouraged by the results, theenhanced algorithms were applied to signals representing power system oscilla-tory behavior. While there are significant differences between signals encoun-tered in power quality studies and those that are used in this chapter, thephilosophy behind the enhanced Hilbert–Huang transform remains the samefor both kind of signals. The nature of the signals that are the focus of thischapter is (a) signals are nonstationary and possibly nonlinear, (b) signals are oflow frequency in the range of 0–2 Hz, and (c) individual component frequencieslie within an octave.

Before proceeding further, an explanation of a frequency octave is provided.An octave is the frequency range between one frequency and its double or half-frequency. Examples of octaves are 7.5–15 kHz and 0.5–1.0 Hz. Two


frequencies are said to share an octave if their ratio lies between 0.5 and 2.Hence, 0.45 and 0.6 Hz share an octave, while 0.35 and 0.8 Hz lie in differentoctaves.

2.3.1 Limitations of EMD

It is interesting to note here the similarity between an IMF and a real-zero (RZ)signal as defined by [4]. RZ signals are bandpass signals whose zeros are all realand distinct. A subclass of bandpass signals that share only real simple zeroswith their Hilbert transform, is completely described by zero crossings only ifthe bandwidth is less than an octave. An IMF would fall in this subclass ofsignals. Extensive experiments have already established the ability of EMD todecompose white noise into IMFs whose frequency spectrum comprises anoctave. The mean frequencies of the extracted IMFs show a doubling phenom-enon. Thus, the implication is that if a signal contains two modes whosefrequencies lie within an octave, standard application of EMD is unable toseparate the modes.

Figure 2.1 shows the result of application of EMD to separate the constitu-ent frequencies in a distorted signal. When the signal is of the form sin pt +0.7 sin 0.3pt, the standard application of EMD yields two monocomponentIMFs corresponding to the two frequency components present in the signal.These two frequencies are 0.5 and 0.15 Hz, both of which clearly lie in differentoctaves. However, when the signal is of the form sin pt + 0.7 sin 0.75pt, stan-dard application of EMD fails to yield the two components. Rather, the firstIMF comes out to be identical to the original signal as shown in Fig. 2.1. This isbecause the two frequencies, 0.5 and 0.375 Hz, lie within the same octave.

0 10 20–2

0

2 s1 = sinπt + 0.7sin0.3πt

time (s)

0 10 20–2

0

2 IMFs extracted from s1

time (s)

0 10 20–2

0

2 s2 = sinπt+0.7sin0.75πt

time (s)

0 10 20–2

0

2 IMFs extracted from s2

time (s)

Fig. 2.1 Standardapplication of EMD toextract IMFs from amulticomponent signal.When the signalcomponents havefrequencies 1 and 0.3 Hz, thetwo IMFs obtained aremonocomponent. When thesignal constituentfrequencies are 1 and0.75 Hz, the first IMF isidentical to the signal

42 N. Senroy

Based on the above example, a 30 s synthetic signal sampled at 100 Hz,

s(t)=10 sin 2pt + m sin 2pft where 5 � m�40, 0 � f� 1, was analyzed using

the standard EMD. The signal consists of two components. The first compo-

nent was held steady at 1 Hz, with amplitude of 10. The second component’s

frequency and amplitude varied from 0 to 1 Hz and 5 to 40, respectively. The

standard EMD was applied to the signal, and the spectral content of the first

IMF was analyzed using FFT. Note, the signal is a stationary signal, and FFT

is expected to yield the spectral content accurately. Normally, the first IMF is

expected to contain the highest frequency component, which is this case is

10 sin 2pt. The multicomponent character of the first IMF arises from the

inclusion of the second component (whose frequency and magnitude are

being varied), by EMD. Hence, the percentage of the second component

included in the first IMF was analyzed, and plotted in Fig. 2.2. In Fig. 2.2,

the x-axis shows the variation of the frequency of the second component, while

the y-axis shows the variation of the ratio of the amplitudes of the two

components.

01

2

3

4

prop

ortio

n of

IInd

com

pone

ntex

trac

ted

in Is

t IM

F

0.5

1

1.5

0.2

frequency of IInd component (Hz)

amplitude ratio of IInd

component toIst component

0.40.6

0.8

Fig. 2.2 Percentage of lower frequency component included in the first IMF by the standardapplication of EMD. The second component frequency was varied from 0 to 1 Hz (x-axis),while its amplitude was varied from half to four times the amplitude of the first component(y-axis)


FromFig. 2.2, some observationsmay bemade. As the frequency of the second

component approaches the frequency of the first component, the likelihood of it

being included in the first IMF increases. For relatively small amplitudes of the

second component, a frequency of more than 0.5 Hz means that the standard

EMDwill not be able to separate the two components. Additionally, as the second

component amplitude increases, the ‘cutoff’ frequency for the first IMFdecreases.

In Fig. 2.2, if the second component amplitude is four times the first component

(i.e., the 1 Hz component), it will be included in the first IMF even if its frequency

is less than 0.4 Hz. It may thus be concluded that the first IMF obtained by

applying the standard EMD to a distorted signal, may be multicomponent underthe following conditions:

1. if the highest two frequencies in the distorted signal fall in an octave,2. if the distorted signal consists of a weak high-frequency component along

with a dominating lower frequency component.

2.3.2 Masking Signal-Based EMD [5]

The use of masking signals has been proposed to enhance the discriminatingcapability of EMD [5]. A masking signal of frequency higher than the highest

frequency component present in the original distorted signal is artificially added

and subtracted from the signal to obtain two new signals. EMD is performed on

these two new signals to obtain the first IMF only. The average of the two IMFs

is then computed to yield the correct IMF for the original signal. This ‘improved’

IMF would be monocomponent only if the frequency of the masking signal is

chosen such that only the highest frequency component in the original signal falls

in the same octave as the masking signal [3]. A systematic method to construct

masking signals to apply on a distorted signal is given as follows:

B1. Perform FFT on the distorted signal, s(t) to estimate frequency compo-nents f1, f2, . . . , fn, where f15f25 � � �5fn. In the case of a complicatedsignal, f1, f2, . . ., fn are stationary equivalents of the possibly time-varyingfrequency components.

B2. Construct masking signals, mask2, mask3, . . ., maskn, where maskk =Mk sin(2p( fk+fk – 1)t). In the case of power quality signals [3], the valueofMk is suggested to be 5.5 times the magnitude of fk obtained in the FFTspectrum. In experiments with power system oscillation signals, this valueis found to work satisfactorily. However, it is stressed here that the value ofMk is empirical, and individual experiences about appropriate maskingsignal amplitudes may vary.

B3. Compute two signals, s(t) + maskn and s(t) – maskn Perform EMD (stepsA1–A6) on both the signals to obtain their first IMFs only, IMF+ andIMF_. Then c1(t) = (IMF+ + IMF_)/2. This is the correct IMF of thedistorted signal.

44 N. Senroy

B4. Obtain residue, r1(t) = s(t) – c1(t).B5. Perform steps B3 and B4, replacing s(t) with the residue obtained in step B4,

iteratively until n – 1 IMFs containing frequency components f2, f3, . . ., fn havebeen extracted. The final residue rn(t) will contain the remaining component f1.

As a demonstration of the masking signal-based EMD, consider the second

signal of Fig. 2.1, i.e., s(t) = sinpt + 0.7 sin 0.75pt. Using a masking signal,

5.5 sin 3.5pt, two IMFs were obtained, as shown in Fig. 2.3. The first IMF was

monocomponent in nature, as revealed by its FFT. However, there is a significant

loss of amplitude information of the 1 Hz component in the first IMF. The second

IMF contains traces of the 1 Hz component along with the 0.75 Hz component.

2.3.3 Frequency Heterodyne Technique [6]

Frequency shifting has also been proposed as an alternative to the masking

signal-based EMD [6]. As demonstrated in the previous section, the masking

0 5 10 15 20–2

0

2 sinπt + 0.7sin0.75πt

time (s)

0 5 10 15 20–2

0

2 IMFs extracted

time (s)

0 1 2 3 4 50

0.2

0.4

0.6FFT of first IMF

frequency (Hz)

III

Fig. 2.3 Masking signal based EMD on a signal, s(t) = sinpt+ 0.7 sin 0.75pt. The first IMFobtained is monocomponent as evident from its FFT. However, there is significant loss ofamplitude information in its retrieval


signal-based EMD sometimes leads to loss of amplitude information while

extracting the highest frequency component. While the philosophy behind

masking signals is to include the highest frequency component in a different

octave, the basic idea behind frequency heterodyne technique is to alter the

spectral distance between two adjacent frequency components. This is achieved

by nonlinear mixing of the distorted signal with a pure tone of frequency greater

than the highest frequency present in the distorted signal. This is possible only if

the frequencies of the distorting components are known in advance. The prin-

ciple is similar to the heterodyne detection commonly used in communication

theory. The signal of interest is amplitude modulated by mixing it with an

appropriate ‘carrier signal.’ Such a signal is referred to as double sideband

modulated (DSB) with suppressed carrier as it contains two frequency shifted

copies of the original signal on either side of the carrier frequency. Further, a

single sideband modulated (SSB) signal may be obtained by removing one of

the sidebands using an appropriate filter or a Hilbert transformer. This also

effectively reduces the bandwidth of the modulated signal.Consider a signal, s(t), with its Hilbert transform sH(t). The frequency content

of s(t) is shifted around a new carrier frequency, F, bymultiplying it by the analytic

representation of the carrier signal. The DSB signal thus obtained is as follows:

fDSB tð Þ ¼ s tð Þe j2pFt (2:5)

The SSB signal is obtained by using the analytic form of s(t). The lower

sideband signal is obtained as follows

fSSB tð Þ ¼ Re s tð Þ þ jsH tð Þð Þe�j2pFt� �

(2:6)

In the context of EMD, suppose the distorted signal contains two frequencies

f1 and f2 ( f1< f2), both of which lie in an octave. The heterodyne frequency, F, is

selected as f2 < F < 2f2 – f1. The length of the original distorted signal will

determine how close F can be to f2. After heterodyning, the SSB frequencies

obtained are F– f1 and F– f2. If the value of F is chosen correctly, the two shifted

frequencies will lie in different octaves, and subsequent application of the

standard EMD results in separation of the components as individual IMFs of

frequencies F– f1 and F– f2. The IMFs are then translated back to the original

frequencies using the same heterodyne technique.The same signal of Fig. 2.3 was heterodyned with a signal of frequency

1.2 Hz. The result was the shifting of the 1 Hz component to 0.2 Hz, and the

0.75 Hz component to 0.45 Hz. Figure 2.4 shows the heterodyne technique as

applied to the signal. The new heterodyned signal contains 0.2 and 0.45 Hz,

which do not fall in the same octave. Hence, regular application of EMD is able

to separate the two components in different IMFs, which are subsequently

heterodyned back to their original frequencies. Figure 2.5 shows the IMFs

46 N. Senroy

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

X: 0.9995Y: 0.9984

X: 0.7496Y: 0.7015

X: 0.4498Y: 0.6991

X: 0.1999Y: 1

X=1.2Y=1

frequency (Hz)

heterodynefrequency

originalfrequencies

heterodynedfrequencies

Fig. 2.4 Frequency shifting using the heterodyne technique. The original signal contained twofrequency components at 1 and 0.75 Hz. After heterodyning with 1.2 Hz signal, thefrequencies were shifted to 0.2 and 0.45 Hz, respectively

0 5 10 15 20–2

–1

0

1

2 sinπt+0.7sin0.75πt

time (s)

0 5 10 15 20–1.5

–1

–0.5

0

0.5

1

1.5 IMFs obtained by heterodyne technique

time (s)

Fig. 2.5 IMFs obtained from applying standard EMD after heterodyning the distorted signal.Both the IMFs are monocomponent in nature with negligible loss of amplitude information


obtained finally, from the distorted signal of Figure 2.3. Both the IMFs aremonocomponent and suffer negligible loss of amplitude information.

The error in extraction from the frequency heterodyne technique is signifi-cant at the edges of the window of observation and near switching instants only.This is in contrast to the masking signal-based EMD, wherein the error inextraction persists throughout the observation window. However, one of thesignificant drawbacks of the frequency heterodyne technique is the difficulty inselection of an appropriate heterodyne frequency. Accurate knowledge of thespectral content of the distorted signal is required before making a decision onthe heterodyne frequency. This knowledge could be obtained from an FFT onthe distorted signal; however, FFT-based spectral analysis of time-varyingdistorted signals yields approximate and possibly inaccurate information.

2.4 Case Studies

In this section several case studies are presented that demonstrate the applica-tion of the modifications to the Hilbert–Huang technique. All the signals to beanalyzed are derived from power system applications. However, the exactnature of the signal varies according to the particular application. The firstcase is the study of power flow oscillations in a critical transmission line in alarge interconnected power system, in response to a contingency. The objectiveis to characterize the amplitude and frequency of the undamped oscillationsusing the frequency heterodyne-based Hilbert–Huang technique. The secondcase is the application of the masking signal-based Hilbert–Huang techniquefor time–frequency–magnitude characterization of the oscillations in load tor-que and field current, in a specialized ship propulsion motor. The third case isthe application of the masking signal-based EMD to understand slow coher-ency in multimachine power systems. The fourth and final case study involvesactual wide-area measurement signals of frequency, recorded in the westernUnited States in 2005.

2.4.1 Power Flow Oscillations in Large Power Systems

This section presents the application of the Hilbert–Huang transform toanalyze the power flow oscillations in a six-area, 377-machine detailedmodel of the Mexican interconnected system [7]. Time-domain simulation ofthe outage of one of the units in the Laguna Verde nuclear power plant, wascarried out. The contingency involved disruption of 650 MW without anyfault in the southeastern part of the system. No supplementary dampingcontrollers like power system stabilizer were considered in the study.Figure 2.6 shows the resultant oscillations in the real power flow on the230 kV transmission line fromMalpaso Dos (MPD) substation toMacuspanaDos (MCD) substation, due to the tripping of the Laguna Verde unit. The

48 N. Senroy

contingency is a critical contingency as it results in undamped oscillations.

The FFT (Fig. 2.6) reveals prominent modal frequencies at 1.5 and 0.8 Hz.The 230 kV MPD–MCD transmission line is located within the southeastern

area, and the oscillations in its line flow represent the localized interaction

between machines within the area. The objective of the study is to estimate the

damping and frequency of the oscillations. Due to the close spacing of the

frequency components, the standard EMD was unable to discriminate between

the two principal modes. A masking signal-based EMD was able to accurately

extract the frequencies of the two modes; however, some amplitude information

was lost in the process. Hence, the MPD–MCD signal was heterodyned with

a frequency of 2.20 Hz. Using the procedure outline in Section 2.3.3, the

two principal modes were separated. Further application of Hilbert transform

yielded their instantaneous frequency and amplitude. Figure 2.7 shows the almost

monotonic rise in amplitude of both the frequency components (1.5 and 0.8 Hz).

This is a typical undamped oscillatory behavior, wherein the damping may be

accurately computed from the variation of the amplitude shown in Fig. 2.7.

0 5 10 15 20 25 30–50

0

50

time (s)

[MW

]

line flow

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25 FFT of signal

frequency (Hz)

Fig. 2.6 Oscillations in thereal power flow on the230 kV MPD–MCDtransmission line in responseto the tripping of the LagunaVerde 650 MW unit. TheFFT of the oscillationsreveal prominent modalfrequencies at 1.5 and 0.8 Hz

0 5 10 15 20 25 300

20

40

60 instantaneous magnitude

time (s)

[MW

]

0 5 10 15 20 25 300

1

2

3

4 instantaneous frequency

time (s)

[Hz]

Fig. 2.7 Instantaneousfrequency and amplitudes ofthe two most prominentIMFs extracted from theoscillations in the line flowof the 230 kV MPD–MCDtransmission line. Note, themonotonic rise in amplitudeover time signifyingundamped oscillations


2.4.2 Torque and Field Current Variations in HTS PropulsionMotors [8]

A slightly different kind of application of the modified Hilbert–Huang trans-

form is presented in this section. Moving away from traditional interconnected

power systems, this section devotes itself to shipboard power systems. Typi-

cally, an integrated power system on an all-electric ship implies that electricity is

used to fulfill all the energy needs of the ship including propulsion. Thus,

specialized electric machines are employed as propulsion motors that are not

only energy efficient but also contribute to the robustness of the integrated

power system.A critical aspect of machine design is the understanding of electromagnetic

stator–rotor interactions during varying load conditions. Recently, there has been

renewed interest in the application of high-temperature superconducting (HTS)

motors for the propulsions needs of future generations of warships [8]. HTS

motors are characterized by a significantly high-power density and efficiency

when compared to conventional motors. They are typically synchronousmachines

with theHTS technology for the rotor field winding only, carrying theDC current.

The stator armature containing the AC windings follow conventional technology.

While in operation at sea, a propulsion motor is subject to the application of

varying sea states, which lead to slow variations in load torque. Such low-fre-

quency oscillations in the torque result in variations in the field current. Joule

heating due to these current variations in the cryogenically cooled field windings

has significant implications on the refrigeration aspects of the machine.A transfer function between the load torque variations and the field current

oscillations is very difficult to obtain analytically. Even with finite element-

based models, computing the relationship is a challenging task, especially in the

presence of varying rotor speed. Appropriate measurements of the relevant

quantities can enable transfer function estimation, provided suitable empirical

methods are applied. Traditionally, empirical methods for transfer function

estimation are primarily FFT based, wherein the FFT of an artificially intro-

duced input signal, torque, is compared with the FFT of the output signal, field

current. Such an exercise entails systematic analysis, wherein the machine may

have to be taken offline. During sea-state conditions, the transfer function

estimation is possible only if an appropriate technique is available to study

time-varying waveform distortions. The modified Hilbert–Huang technique is

applied to accurately estimate the relationship between the motor torque and

field current variations. Figure 2.8 shows an example of the oscillations in load

torque and field current due to the application of a sea state.FFT analysis of the torque variations revealed possible modal frequencies at

2.3, 1.74, 1.26, 0.87, and 0.535Hz. Accordingly, themasking signal-based EMD

was applied, with the masking frequencies fixed at 4.04, 3.00, 2.13, 1,405, and

0.815 Hz. Similarly, the principal modes identified from an FFT analysis of the

field current variations were 1.8, 1.4, 1.07, 0.87, 0.535, and 0.28 Hz.

50 N. Senroy

Accordingly, the masking frequencies were fixed at 3.2, 2.47, 1.94, 1.405, 0.815,

and 0.38 Hz. Figure 2.9 shows the instantaneous frequencies of the principal

IMFs extracted using the masking signal-based EMD. It can observed that the

frequency tracking improves for lower frequencies. The reason for this is that

the amplitudes at these frequencies are greater. This is an inherent property of

the Hilbert–Huang technique. Another significant observation can be made

from Fig. 2.9. When the instantaneous frequencies of the modes present in the

15 20 25 30–0.1

–0.05

0

0.05

0.1

15 20 25 30–0.02

–0.01

0

0.01

0.02

time (s)

field current variationstorque variations

Fig. 2.8 Typical variations of load torque and field current of a HTS propulsion motor due tothe application of sea states. The axis for the field current variations is to the right, while thesame for the torque is to the left

15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (s)

freq

uenc

y (H

z)

Fig. 2.9 Instantaneous frequencies extracted from torque and field current variations in anotional HTS motor due to the application of sea states. The dashed lines represent theinstantaneous frequencies of field current variations, while the solid lines represent the same forthe torque variations. Note, both the field current and torque contain identical frequency modes


load torque are compared with that of the modes present in the field current,there is a clear correspondence. This indicates that all the significant modespresent in the load torque oscillations also appear in the field current variations.

Because the application of the sea states result in the excitation of a fewmodes in the load torque and field current, the relationship between the two canbe gauged for those modal frequencies only. The instantaneous amplitude gainwas calculated directly from the instantaneous amplitudes of the most domi-nant IMFs, whose frequencies are shown in Fig. 2.9. A scatter plot was used torepresent this gain, because for every sampling instant, there is a calculatedamplitude gain. In order to validate the computed amplitude gains, a traditionalFFT-based empirical transfer function estimation was also carried out for awide range of frequencies. This technique involves calculating the ratio of theamplitude of the output (oscillations in the field current) to a single frequencyinput (load torque variation). The system was found to respond linearly overthe entire frequency range, except for oscillation frequencies close to the naturalmodes of the system [8]. Figure 2.10 shows a reasonably good agreementbetween the trend of the scatter plots obtained from online sea-state analysisand the frequency response curve obtained independently using off-line FFT-based techniques. The spread evident in the scatter plots is an inherent artifactof the Hilbert–Huang technique, which defined instantaneous frequency as therate of change of the instantaneous phase of the analytic form of a real-valuedsignal. The spread at any frequency may reduce considerably if the amplitudesof the respective oscillations are higher.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

frequency (Hz)

tran

sfer

func

tion

gain

Fig. 2.10 Transfer function gain between variations in the field current and load torque of aHTS propulsion motor. The scatter points represent results from the Hilbert–Huangtransform using measurements taken during simulated sea-state conditions. The solid linerepresents FFT-based off-line empirical transfer function estimation. Note, the increase inscatter as the frequency increases, because of which the amplitude of oscillations due to the seastates die out

52 N. Senroy

2.4.3 Analyzing Slow Coherency [9]

The aim of this section is to confirm some aspects of the linearized analysis of a

power system using generator swing curves obtained using nonlinear time-

domain simulations. Coherency of generators is used as an aggregation criter-

ion to simplify a complex power system network. It has been observed in a

multimachine system that in the aftermath of a sudden disturbance, some

machines tend to ‘swing together.’ This means that there is relatively little

difference between their swing curves. Such coherent machines may be grouped

together and represented as equivalent machines in further computations. In a

practical dynamic system, perfectly coherent machines are rarely encountered.

Hence, near-coherent machines are grouped together by examining the rows of

an eigenbasis matrix. This eigenbasis matrix is formed by combining the eigen-

vectors corresponding to some modes of interest. Coherency of these machines

is always defined with respect to a set of modes of the system. Coherent states

are insensitive to the system perturbation because only those modes are selected

that are excited by the initial perturbation and subsequently the states with the

same content of perturbed modes are identified.In a power system with n natural modes of oscillation, there are some modes

that are of high frequency and well damped. These modes do not have much

impact on the slow transients in the system. Hence, the coherency between the

machines is always defined for r modes of the system which are usually ‘slow’

and have a tangible impact on the system behavior. This kind of coherency is

referred to as slow coherency. In slow coherency theory, generators are coherent

with each other over large timescales. Coherent groups may exhibit very fast

oscillations within themselves and a disturbance will propagate within a group

very rapidly. Therefore, over very short timescales, each of these tightly bonded

groups may be studied in isolation from the rest of the system. However, over

larger timescales, the weak interactions between various tightly bonded groups

become significant. Over large timescales, these tightly bonded slow coherent

machines are represented as equivalent models that are insensitive to the level of

detail used in modeling the generators.A six-machine system was linearized around its base case operating point [9],

and the natural modes were calculated to be 1.29, 0.99, 0.72, 0.49, 0.38, and

0.0 Hz. Further eigenanalysis revealed that generator 1 and 6 were coherent for

themodes – 0.0, 0.38, and 0.49Hz. Because this coherency was established using

the linearized model of the system, the modified Hilbert–Huang method was

employed for confirmation using the nonlinear model of the system. Generator

swing curves were obtained for a line tripping without a fault. The 10 s long

swing curves are shown in Fig. 2.11. While a rough idea of the coherency

between the machines can be had from visual inspection of the swing curves,

a detailed analysis is readily provided by the masking signal-based EMD.Masking signals were used with EMD to extract monocomponent IMFs.

For both swing curves, the masking signal frequencies used were 1.6 and 1.1 Hz.


Figure 2.11 shows the three important IMFs extracted from the swing curves.

The highest frequency IMFs, i.e., IMF1 for both the machines are anti-phase toeach other. The lower frequency IMFs are in phase with each other. In termsof magnitude, the third IMF was the most significant for both the generator

swing curves. Hilbert transform was applied to all the three sets of IMFs, andthe instantaneous frequency obtained is shown in Fig. 2.12. It is clear fromFig. 2.12, that the instantaneous frequency of the corresponding IMFs,

extracted from the two generators’ swing curves, show good agreement.The selection of IMFs for comparing the instantaneous phase difference must

be done on the basis of their instantaneous frequencies which must match.Hence, an instantaneous phase difference is calculated between those IMFswhose instantaneous frequencies match.

It is also interesting to relate the instantaneous frequencies obtained for the

IMFs with the natural frequencies of the system as revealed by the eigenvalueanalysis. The coherency between the machines was computed for the slow

0 2 4 6 8 10

0

5

10

15

time (s)

swing curves

0 2 4 6 8 10–0.5

0

0.5

time (s)

IMF1

0 2 4 6 8 10–1

0

1

time (s)

IMF2

0 2 4 6 8 10

–1

0

1

time(s)

IMF3

Fig. 2.11 Analysis of slow coherency between two generators using the masking signal-basedEMD. The top plot shows the original swing curves obtained for a line tripping. The lowerthree plots show the extracted IMFs from the two swing curves in decreasing order offrequency. The generators are coherent for lower frequencies, and out of phase for the highestfrequency

54 N. Senroy

modes 0.0, 0.39, and 0.49Hz. Furthermore, it was found that generators 1 and 6participate primarily in the 1.29 Hz mode (participation factors: 0.305 and0.191, respectively), and the 0.49 Hz mode (participation factors: 0.115 and0.199, respectively), with minor participation in the 0.72 Hz mode (participa-tion factors: 0.006 and 0.026, respectively). Thus, slow coherency is verified byrelating the instantaneous frequencies of the coherent IMFs and the naturalmodes of the system.

2.4.4 Wide-Area Measurement Signals [9]

This section presents the application of the modified EMD to analyze actualwide-area measurement data. An Internet-based WAMS is presented in [10]. Itconsists of frequency deviation recorders (FDRs) monitoring system frequencyat strategic locations in the USA. The FDRs measure frequency at the distribu-tion level at a sampling rate of 10 Hz, by plugging into 110 V sockets. Themeasurement data from the FDR is consequently noisy, and preprocessing isrequired before the data is ready to be analyzed. The frequency measurementsare time stamped using Global Positioning System synchronizing signals.

0 1 2 3 4 5 6 7 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time [s]

freq

uenc

y [H

z]

Fig. 2.12 Instantaneous frequency of the three set of IMFs extracted from the swing curvesshown in Fig. 2.11. The dashed line corresponds to the IMFs extracted from the swing curvesof one machine while the solid lines correspond to the IMFs extracted from the other machineswing curve


The Pacific DC Intertie is a �500 V, 3,100 MW HVDC line connecting the

Cellilo substation in Oregon to the Sylmar substation near Los Angeles, south-

ern California. This 1,354-km-long line is critical in improving the reliability of

the electricity grid located in the western United States, by supplementing the

thermal and nuclear generation in the southwest with hydropower from the

northwest. OnAugust 25, 2005 at 3:53 PM (PDT, Pacific Daylight Time) due to

the loss of a converter, the HVDC line was taken out of service, resulting in the

disruption of 1,750 MW. Figure 2.13 shows the frequency deviations resulting

out of the loss of the HVDC tie line. They were recorded by FDR14, FDR16,

and FDR21 located at Arizona State University (ASU), Tempe AZ, USA, Los

Angeles, CA, USA, and Washington State University (WSU), WA, USA,

respectively. It is clear from the oscillations that the FDR21 signal located in

the northwest is out of phase with the other two signals both located in the

southwest. However, due to the fact that all the signals aremulticomponent, it is

difficult to arrive at a reliable phase comparison without the application of

EMD.The standard EMDof [1] was applied to extract the overall trend from all the

three signals. This manifests as the last IMF extracted. Figure 2.14 shows the

extracted trends from the three WAMS signals. The system frequency was

initially at 59.8 Hz, when the HVDC line tripped. Following the tripping, the

frequency dipped to less than 59.75 Hz, after which it settles to 59.8 Hz. The

oscillations extracted from the frequency signals are shown in Fig. 2.15.

03:52:58 PM 03:53:08 PM 03:53:18 PM59.7

59.75

59.8

59.85

59.9

59.95

Pacific Daylight Time

freq

uenc

y (H

z)

FDR14

FDR16

FDR21

Fig. 2.13 Frequency oscillations recorded by FDR14 at ASU, Tempe, AZ; FDR16 at LosAngeles, CA; FDR21 at WSU, Washington, WA. The event is the tripping of the Pacific DCIntertie on August 25, 2005

56 N. Senroy

03:52:58 PM 03:53:08 PM 03:53:18 PM59.7

59.75

59.8

59.85

59.9

59.95


freq

uenc

y (H

z)

FDR16

FDR14

FDR21

Fig. 2.14 Trend extracted from the WAMS signals shown in Fig. 2.13. The signals wererecorded by FDR14, FDR16, and FDR21 located in Arizona State University, Tempe, AZ,Los Angeles, CA, and Washington State University, WA, respectively

03:52:58 PM 03:53:08 PM 03:53:18 PM–0.08

–0.06

–0.04

–0.02

0

0.02

0.04

0.06


freq

uenc

y (H

z)

FDR14

FDR16

FDR21

Fig. 2.15 Oscillations in frequency extracted using the standard EMD from the WAMSsignals shown in Fig. 2.13


The birth of frequency oscillations are clearly observable at 03:52:58 PM,

arising out of the HVDC line tripping. These oscillations die out within 18 s,

when the frequency settles to around 59.8 Hz.The FFT spectrum of the frequency variations, reveals the multicomponent

character of the oscillations. Thus, the straightforward application of Hilbert

transform would not yield meaningful phase information. Further examination

reveals that there are possible low-amplitude frequency components in all the

three signals around 0.6–0.7 Hz. Straightforward application of the EMDwould

include these components along with the dominant 0.31 Hz component in the

first IMF.Hence, a masking signal is required to extract the weak components of

frequency ranging from 0.6 to 0.7 Hz. Amasking signal of frequency 0.88Hz and

appropriate amplitude was used for both FDR14 and FDR16 signals. Similarly,

a 0.99 Hz masking signal was used for the FDR21 signal.The dominant IMFs extracted from the three WAMS signals are shown in

Fig. 2.16(a). Hilbert transform on the IMFs revealed their instantaneous

phases. These were further used to compute the instantaneous phase differences

as shown in Fig. 2.16(b). It can be observed from Fig. 2.16(b) that the instanta-

neous phase difference between the dominant IMFs of FDR14 and FDR16

hovers around 08. Similarly, the phase difference between the dominant IMFs

of FDR21 and FDR14/FDR16 hovers around 1808. This indicates that FDR14

and FDR16 are in phase and coherent, while FDR21 is out of phase with both

FDR14 and FDR16.

03:52:58 PM 03:53:08 PM 03:53:18 PM–90

0

90

180

250

Inst

anta

neou

sph

ase

diffe

renc

e(d

eg)

(b)

FDR14 and FDR16FDR14 and FDR21FDR16 and FDR21

03:52:58 PM 03:53:08 PM 03:53:18 PM–0.05

0

0.05


freq

uenc

y de

viat

ion

(Hz)

(a)FDR21

FDR14 FDR16

Fig. 2.16 Instantaneous phase difference of the dominant IMFs extracted from the WAMSsignals of Fig. 2.3: (a) dominant IMFs of FDR14, FDR16, and FDR21 and (b) instantaneousphase differences computed using Hilbert transform. Masking signal frequencies used withEMD are 0.88 Hz for FDR14 and FDR16 and 0.99 Hz for FDR21

58 N. Senroy

2.5 Discussion

In this chapter the Hilbert–Huang method has been presented along with

improvements to enhance its application to signals representing power sys-

tem dynamic behavior. All the modifications to the original Hilbert–Huang

technique presented have been aimed at improving the capability of the

EMD to discriminate between closely spaced as well as weak amplitude

frequency components in a time-varying distorted signal. Case studies of the

application of the technique to various kinds of signals have also been

presented. While the case studies clearly demonstrate the efficacy of the

suggested improvements, it is worthwhile to note the limitations of the

modified Hilbert–Huang technique.The technique of constructing masking signals depends heavily on the

approximate spectral information obtained using the FFT. When applied to

nonstationary and nonlinear waveforms, the FFT results in a stationary or

periodic interpretation. Caution must be exercised while deriving masking

signals from the FFT spectrum, as heavily damped signals result in spurious

peaks in the FFT spectrum.The frequency chosen to heterodyne the distorted signal must be such that it

shifts the frequency spectrum to a lower frequency scale, such that adjacent

frequency components lie in different octaves. While maximum value of the

heterodyne frequency is well defined, the minimum value is harder to define. A

heterodyne frequency selected very close to the highest frequency component,

results in a waveform wherein the highest frequency component is translated to

a very low frequency. Accordingly, the total length of the observed signal must

be large enough to accommodate the dynamics of this low-frequency compo-

nent. If the length of the signal is not adequate, the heterodyne frequency must

be selected as a larger value. Hence, the minimum value of the heterodyne

frequency depends on the dynamics of the highest frequency component present

in the distorted signal.A key component of the EMD technique is the interpolation between

extrema points that is achieved using spline fitting. Spline fitting is associated

with serious errors at the end points of the data, which affect lower frequency

IMFs more. In fact, spline fitting-induced errors accumulate as more and more

IMFs are extracted. Thus, the instantaneous frequency and amplitudes for

these IMFs show considerable error at the ends of the window, as can be

observed in Fig. 2.12. Additionally, one problem with the Hilbert transform is

that at very small magnitudes, it is difficult to track the instantaneous fre-

quency. One possible direction of future research is to focus on the minimum

threshold amplitude an IMF must have, to facilitate satisfactory frequency

tracking.The Hilbert–Huang transform is noncausal by nature; however, it may be

possible to implement it as an online application. Several aspects of the techni-

que must be resolved before that. One critical aspect is that the signal extrema


points must be clearly and unambiguously defined for accurate sifting of the

IMFs. While this indicates that the signal must be significantly oversampled, it

also makes the algorithm sensitive to sensor-induced jitters. These jitters appear

as high-frequency components riding over the original signal, and appropriate

threshold criteria are needed to recognize such phenomena. Further, the dura-

tion of observation of the signal must be large enough to include adequate

number of extrema points to facilitate the spline algorithm. In the context of

online measurement data analysis, a sliding window of appropriate length may

be slid along the incoming data.

2.6 Conclusion

In this chapter, the time–frequency and time–magnitude localization capabil-

ities of the Hilbert–Huang transform have been significantly improved by the

suggested modifications. These modifications have been proposed keeping in

mind the specific nature of the signals to be analyzed, namely those that

represent power system dynamic behavior. The wide variety of case studies

demonstrate that it is possible to extract instantaneous signal parameters with

minimal knowledge of the system from which the signal is obtained. For

instance, the analysis of wide-area measurements implies that it is possible to

extract accurate coherency information using distributed measurements. Thus,

dynamic reduction of the system can be achieved empirically, with limited

knowledge of system parameters leading to model validation. Further coher-

ency can be monitored in real time, which will be useful in designing

special protection schemes relying on generator aggregation. The modified

Hilbert–Huang technique is also a reliable post-processing tool to analyze

sequence of events leading to a better understanding of evolving system

dynamics as well as to validate existing system models. Similarly, the adaptive

transfer function estimation technique described in this chapter may lead to

improvements in online health and condition monitoring of critical power

system equipment. Further research should focus on adapting the modified

Hilbert–Huang technique to real-time applications involving online data pro-

cessing, as well as to improve its capability for accurate instantaneous frequency

and magnitude estimation.

Acknowledgments The author acknowledges the contribution of Siddharth Suryanarayananof Colorado School of Mines, Golden, Colorado, USA in the development of the algorithmspresented in this chapter. The following other people are also acknowledged for their technicalcontributions: PauloM.Ribeiro of Calvin College,Michigan, USA;Michael ‘Mischa’ Steurerof Center for Advanced Power Systems, Florida State University, Tallahassee, Florida, USA;Stephen Woodruff of NASA Dryden Flight Research Center, California, USA; and ArturoMessina of CINVESTAV, Guadalajara, Mexico. Financial support from the Office of NavalResearch, USA, the Department of Energy, USA and the Industrial Research and Develop-ment Unit, IIT-Delhi, India, is also gratefully acknowledged.

60 N. Senroy

References

1. Huang N E, et al., ‘‘The empirical mode decomposition and the Hilbert spectrum fornonlinear and nonstationary time series analysis,’’ Proc. R. Soc. Lond. A., vol. 454, 1998,pp. 903–995

2. Gabor D., ‘‘Theory of communication,’’ IEE J. Comm. Eng., vol. 93, 1946, pp. 429–457.3. Senroy N, Suryanarayanan S, Ribeiro P F, ‘‘An improved Hilbert-Huang method for

analysis of time-varying waveforms in power quality,’’ IEEE Trans. Power Sys., vol. 22,No. 4, Nov. 2007, pp. 1843–1850.

4. RequichaAG, ‘‘The zeros of entire functions: theory and engineering applications,’’Proc.IEEE, vol. 68, no. 3, Mar. 1980, pp. 308–328.

5. Deering R, Kaiser J F, ‘‘The use of masking signal to improve empirical mode decom-position,’’ Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing (ICASSP ’05), vol.454, 2005, pp. 485–488.

6. Senroy N, Suryanarayanan S, ‘‘Two techniques to enhance empirical mode decomposi-tion for power quality applications,’’ IEEE PES General Meeting, June 2007, pp. 1–6.

7. Messina A R, Vittal V, ‘‘Nonlinear, Non-stationary analysis of interarea oscillations viaHilbert spectral analysis,’’ IEEETrans. Power Sys., vol. 21, No. 3, Aug. 2006, pp. 1234–1241.

8. Senroy N, Suryanarayanan S, Steurer M, ‘‘Adaptive transfer function estimation of anotional high-temperature superconducting propulsion motor,’’ Accepted for publica-tion, IEEE Trans. Ind. Appl., Feb. 2008.

9. Senroy N, ‘‘Generator coherency using the Hilbert-Huang transform,’’ IEEE Trans.Power Sys., vol. 23, No. 4, Nov. 2008, pp. 1701–1708.

10. Wang J K, et al., ‘‘Analysis of system oscillations using wide-area measurements,’’ IEEEPES General Meeting, June 2006, pp. 1–6.


Chapter 3

Variants of Hilbert–Huang Transform

with Applications to Power Systems’

Oscillatory Dynamics

Dina Shona Laila, Arturo Roman Messina, and Bikash Chandra Pal

Abstract Power system dynamic processes may exhibit highly complex spatial

and temporal dynamics and take place over a great range of timescales. When

frequency analysis requires the separation of a signal into its essential compo-

nents, resolution becomes an important issue. The Hilbert–Huang transform

(HHT) introduced by Huang is a powerful data-driven, adaptive technique for

analyzing data from nonlinear and nonstationary processes. The core to this

development is the empirical mode decomposition (EMD) that separates a

signal into a series of amplitude- and frequency-modulated signal components

from which temporal modal properties can be derived. Previous analytical

works have shown that several problems may prevent the effective use of

EMD on various types of signals especially those exhibiting closely spaced

frequency components and mode mixing. The method allows a precise char-

acterization of temporal modal frequency and damping behavior and enables a

better interpretation of nonlinear and nonstationary phenomena in physical

terms.This chapter investigates several extension to the HHT. A critical review of

existing approaches to HHT is first presented. Then, a refined masking signal

EMD method is introduced that overcomes some of the limitations of the

existing approaches to isolate and extract modal components. Techniques to

compute a local Hilbert transformation are discussed and a number of numer-

ical issues are discussed.As case studies, the applications of the various EDM algorithms in power

system’ signal analysis are presented. The focus of the case studies is to

accurately characterize composite system oscillation in a wide-area power

network.

D.S. Laila (*)Department of Electrical and Electronic Engineering, Imperial College London,Exhibition Road, London SW7 2AZ, UKe-mail: [email protected]


63

3.1 Introduction

Slow electromechanical oscillations are inherent in stressed interconnected

power system operation [1]. In many occasions [2], they appear to be precursor

to severe system upset. Obviously, it is important to detect and mitigate their

influence for secure operation.The existing technology is to run series of dynamic security analysis which

essentially utilizes eigenanalysis on the linearized model of the system [3, 4].

However, the oscillatory behavior of power system is nonlinear, which causes

the eigenanalysis less accurate in estimating the damping and frequency asso-

ciated with the oscillations apart from the uncertainty in the model. Techniques

using measured data such as Prony analysis, spectral analysis, and MIMO

(multiple-input multiple-output) state–space identification were applied in [5, 6,

7, 8]. These techniques, however, rely on the linear and periodicity assumption of

the system response. In [9, 10] Fourier analysis has been used for off-line studies

of power system dynamics. However, Fourier spectral analysis requires the data

to be strictly periodic or stationary. The Fourier spectrum defines uniform

harmonic components globally and therefore needs many additional harmonic

components to simulate nonstationary data. As a result, it spreads the energy

over a wide frequency range making little physical sense. Although linear and

stationary approximation may not be as crude, the information retrieved from

this approach is often insufficient. Recently, nonlinear and nonstationary analy-

sis have been used to characterize the time–frequency attributes of system

response [11, 12], applying Hilbert–Huang transform (HHT).The HHT is an empirically based signal analysis method developed by

Huang et al. [13], which is a potentially powerful tool to analyze nonlinear and

nonstationary signals. This method has been applied to various fields such as

medical [14], geology and geodesic [15], and power systems [16]. TheHHTmethod

consists of twomain processes, the empiricalmode decomposition (EMD) and the

Hilbert transform (HT). The EMD process aims at decomposing the signal into

its frequency components called intrinsic mode functions (IMFs). The main

procedure in EMD is called sifting. The EMD process targets each IMF to be a

monofrequency signal, namely signal with single frequency component. For the

EMD to give a good analysis, resolution, namely the ability of the method to

separate each frequency component during the sifting process is very important.

It is further established that the EMD behaves as a dyadic filter bank [17]. It

decomposes white noise into IMFs whose frequency spectrum comprises an

octave. The mean frequencies of extracted IMFs show period doubling phenom-

enon. Although the original sifting procedure can perform this task quite well for

certain frequency range and composition of a signal, unfortunately, it often fails to

work for signalswith low frequency, lowmagnitude, or narrow frequency, i.e., two

consecutive frequency components within an octave. This is seen as a potential

limitation of the standard EMD when analyzing power system signals which are

dominated by some inter-area modes with frequency close to each other.

64 D.S. Laila et al.

There has been considerable recent research interest to improve the EMD

process, in particular, the sifting procedure to overcome the difficult situations

mentioned earlier. In [18], a masking technique is first introduced, and in [19,

20], the masking technique is improved, giving a constructive algorithm on how

to choose the masking signals. Other works in this directions are, for instance

[21, 22]. As the limitation of the EMD process somehow stems from the loose

definition of an IMF, another approach to improve the method is by refining

the definition of this function [23]. In this work, it is suggested to impose

additional condition to a function to be called an IMF.EMD with masking algorithm from [19] are particularly interesting.

Although it has been shown that the masking technique improves the sifting

process in the power quality analysis as shown in [19], the result is not generic.

The algorithm has worked well for signals in high-frequency range, but it has

not been able to accurately capture the close frequencies associated with inter-

area mode oscillations. Moreover, the algorithm contains some limitations as it

depends on the use of Fourier analysis.In this chapter, several extensions are to the EMDwith masking is presented,

based on the use of masking signal in the sifting procedure. A systematic way to

construct the masks for signals with various frequency ranges and compositions

is also provided, extending both the results from [18, 19]. A local HT based on

convolution is also presented that circumvents some of the limitations of dis-

crete Fourier transform (DFT)-based approaches.The rest of the chapter is as follows. In Section 3.2, we present some

preliminaries of EMD and HT. In Section 3.3, we present several algorithms

of refinement to standard EMD and HT algorithm and demonstrates its effec-

tiveness on a synthetic signal. Some applications to synthetic signals and also

real measurement signals are provided in Section 3.4 and the chapter concludes

by some remarks in Section 3.5.

3.2 Preliminaries

In an effort to make this chapter reasonably self-contained, some preliminaries

that are needed to support the techniques presented in this chapter are presented.

The standard algorithm of the HHT and its components, the EMD technique,

and HT are also briefly reviewed. Our development follows the development of

Huang [13], and we refer the readers to this source for more details.

3.2.1 Fourier Analysis

Fourier analysis is one of the most important tools in signal spectrum analysis.

Its frequencial description can be the basis of a better comprehension of the

3 Variants of Hilbert–Huang Transform with Applications 65

underlying phenomena as it complements the temporal description [24]. Given asignal xðtÞ, the Fourier transform is defined as follows:

XðfÞ ¼Z þ1

�1xðtÞe�i2pft dt : (3:1)

The computation of one frequency value XðfÞ requires knowledge of the com-plete history of the signal ranging from �1 to þ1. On the other hand, theinverse Fourier transform is defined as

xðtÞ ¼Z þ1

�1XðfÞei2pft df : (3:2)

Any value xðtÞ at one instant t can be regarded as an infinite superposition ofcomplex exponentials or everlasting and completely nonfocal waves.

However, there is the limitation of this tool. From the definition of Fouriertransform that involves the integration over the range from �1 to 1, it isobvious that the length of the signal matters. It may give quite an accuratespectrum for long or steady-state signal, but it may distort the physical reality ofshort or transient signals.

3.2.2 The Empirical Mode Decomposition Method

The EMDmethod provides an analytical basis for the decomposition of a signalxðtÞ into a set of basis functions, called IMFs. An IMF is defined as a signal thatsatisfies the following criteria.

1. Over the entire time series the number of extrema and the number of zero-crossings differ by, at most, one, i.e., an essentially oscillatory process.

2. At any point the mean value of the envelope defined by the local maxima andthe envelope defined by the local minima is zero.

The basic EMD method adopted to extract the IMFs essentially consists of athree-step procedure called sifting [12]. The goal is to subtract away the large-scalefeatures of the signal repeatedly until only the fine-scale features remain. A signalxðtÞ is thus divided into the fine-scale details cðtÞ and the residue rðtÞ, hencexðtÞ ¼ cðtÞ þ rðtÞ. The components contained in the fine-scale details are theIMFs.

The standard EMD process can be summarized as follows.

S1. Given the original signal xðtÞ; set roðtÞ ¼ xðtÞ, j ¼ 1.S2. Extract the jth IMF using the sifting procedure:


a. Set i ¼ 1 and hi�1ðtÞ ¼ rj�1ðtÞ.b. Identify the successive local minima and the local maxima for hi�1ðtÞ.

The time spacing between successive maxima is defined to be the time-scale of these successive maxima.

c. Interpolate the local minima and the local maxima with a cubic spline toform an upper emaxi�1ðtÞ and lower emini�1ðtÞ envelope for the whole dataspan.

d. Compute the instantaneous mean of the envelopes,

mi�1ðtÞ ¼emini�1ðtÞ þ emaxi�1ðtÞ

2; (3:3)

and determine a new estimate hiðtÞ ¼ hi�1ðtÞ �mi�1ðtÞ, such thatemini�1ðtÞ � hiðtÞ � emaxi�1ðtÞ for all t. Set i ¼ iþ 1.

e. Repeat steps S2b–S2d until hiðtÞ satisfies a set of predeterminedstopping criteria (follows the criteria 1 and 2 of an IMF). Then setcjðtÞ ¼ hiðtÞ.

S3. Obtain an improved residue rjðtÞ ¼ rj�1ðtÞ � cjðtÞ. Set j ¼ jþ 1. Repeat stepS2 until the number of extrema in rjðtÞ is less than 2.

This approach allows elimination of low-amplitude riding waves in the timeseries and eliminates asymmetries with respect to the local mean, i.e., it makesthe wave profile more symmetric. At the end of this process, the EMD yields thefollowing decomposition of the signal xðtÞ,

xðtÞ ¼Xn

j¼1cjðtÞ þ rnðtÞ ¼

Xp

k¼1ckðtÞ þ

Xn

l¼pþ1clðtÞ þ rnðtÞ; (3:4)

where ckðtÞ; k ¼ 1; . . . ; p contain the physical behavior of interest and theremaining terms clðtÞ; l ¼ pþ 1; . . . ; n and rnðtÞ contain less relevant, nonsinu-soidal characteristics. We emphasize that the refined masking signal methodenables a superior analysis of dynamic behavior than Fourier-based techniquesand provides essential information that may be used to determine modalbehavior, i.e., instantaneous damping and phase as discussed below.

3.2.3 Hilbert Transform

Given a real signal xðtÞ. Its complex representation is

zðtÞ ¼ xðtÞ þ ixHðtÞ ; (3:5)

where xHðtÞ is the HT of xðtÞ, given by

xHðtÞ ¼1

pP

Z þ1

�1

xðsÞt� s

ds ; (3:6)


with P the Cauchy principal value of the integral. Equation (3.5) can be rewrittenin an exponential form as

zðtÞ ¼ AðtÞei ðtÞ ; (3:7)

where AðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðtÞ2 þ xHðtÞ2

qand ðtÞ ¼ arctan xHðtÞ=xðtÞ.

The time derivative of (3.7) is

_zðtÞ ¼ AðtÞei ðtÞði!ðtÞÞ þ ei ðtÞ _AðtÞ ; (3:8)

where !ðtÞ is the instantaneous angular frequency, which, by definition, is thetime derivative of the instantaneous angle

!ðtÞ ¼ _ ðtÞ ¼ d

dtarctan

xHðtÞxðtÞ : (3:9)

Hence, the instantaneous frequency can be defined as

fðtÞ ¼ !ðtÞ2p

; (3:10)

and using (3.7) and (3.8), it can be computed as

fðtÞ ¼ 1

2pIm

_zðtÞzðtÞ

� �¼ 1

2pxðtÞ _xHðtÞ � xHðtÞ _xðtÞ

x2ðtÞ þ x2HðtÞ: (3:11)

Remark 3.1 In HHT technique, HT is applied to each IMF to compute itsinstantaneous frequency, as well as instantaneous magnitude. As instantaneousfrequency is best defined for monofrequency signals, i.e., signals that containonly one (dominant) frequency, it makes sense to expect each IMF to bemonofrequency. However, as pointed out earlier, the IMFs may contain amixture of frequencies (frequency modulation) and are difficult to interpret interms of conventional modal analysis. This has motivated the need for demo-dulation techniques that extract from each IMF the dominant interactingfrequencies.

3.2.4 Instantaneous Damping

3.2.4.1 Computation Based on the Exponential Decay

The knowledge about the instantaneous magnitude and instantaneous fre-quency of a signal allows us to further compute the instantaneous damping ofthe signal. Damping characterization is another useful alternative to the


analysis of local behavior of the oscillation. Consider the signal (3.7). We canrewrite the signal as [25]

zðtÞ ¼ AðtÞei ðtÞ ¼ �ðtÞe��ðtÞþi ðtÞ : (3:12)

Then the time-dependent decay function can be modeled as

�ðtÞ ¼ �Z t

0

�ðtÞ dt : (3:13)

Moreover, using (3.7) and (3.8), we obtain

_zðtÞzðtÞ ¼ ��ðtÞ þ

_�ðtÞ�ðtÞ

� �þ i!ðtÞ

� �: (3:14)

Noting that

Re_zðtÞzðtÞ

� �¼

_AðtÞAðtÞ ; (3:15)

we have the instantaneous damping coefficient � as

�ðtÞ ¼ � d�ðtÞdt¼ �

_AðtÞAðtÞ �

_�ðtÞ�ðtÞ

� �: (3:16)

Moreover, if �ðtÞ is constant, which means the signal is purely exponential,_�ðtÞ ¼ 0, and hence (3.16) is simplified into

�ðtÞ ¼ � d�ðtÞdt¼ �

_AðtÞAðtÞ : (3:17)

3.2.4.2 Computation Based on the Second-Order System Approach

Another way of computing the instantaneous damping is through the second-order system emulation approach [26, 27]. Given a signal

xðtÞ ¼ AðtÞ cosð ðtÞÞ : (3:18)

The complex representation of the signal is

zðtÞ ¼ xðtÞ þ jxHðtÞ (3:19)

with xHðtÞ the HT of xðtÞ that follows the form


xHðtÞ ¼ AðtÞ sinð ðtÞÞ : (3:20)

A general form of an initial differential equation of motion in the analyticalsignal form can be written in a frequency-dependent (viscous) damping form as

€zþ 2�ðAÞ _zþ !2oðAÞz ¼ 0 ; (3:21)

or also in a frequency-independent (structural) damping form as

€zþ !2oðAÞ 1þ j

�ðAÞp

� �¼ 0 ; (3:22)

with z the system solution in the analytical signal form, � the instantaneousdamping coefficient, and !o the instantaneous undamped natural frequency,� ¼ 2p� the logarithmic decrement and � the damping ratio.

Using the analytical signal form (3.19) together with its two first derivatives

_z ¼ zðtÞ_AðtÞAðtÞ þ j!ðtÞ� �

€z ¼ zðtÞ€AðtÞAðtÞ � !

2ðtÞ þ j 2!ðtÞ_AðtÞAðtÞ þ _!ðtÞ

� �� (3:23)

to solve the dynamic equation (3.21), we get the equation for free vibrationanalysis

z€A

A� !2 þ !2

o þ 2�_AðtÞAðtÞ þ j 2

_AðtÞAðtÞ!þ _!þ 2�!

� �� ¼ 0 (3:24)

where A and ! ¼ �1 are, respectively, the envelope and instantaneous fre-quency of the vibratory system solution. Solving two equations for real andimaginary parts of (3.24), we can write the expression for the instantaneousmodal parameters as functions of the first and the second derivatives of thesignal envelope and the instantaneous frequency, i.e.,

!2oðtÞ ¼ !2 �

€A

Aþ 2

_A2

A2þ

_A _!

A!(3:25)

�ðtÞ ¼ �_A

A� _!

2!; (3:26)

where !oðtÞ is the instantaneous undamped natural frequency and �ðtÞ is theinstantaneous damping coefficient of the system. The parameters ! and A arethe instantaneous frequency and the envelope (amplitude) of the oscillation,respectively. Moreover, if the signal is monofrequency, which is the case for


IMFs when the EMD works well to decompose a composite signal, we have_!ðtÞ ¼ 0, and hence (3.26) simplifies into

�ðtÞ ¼ �_AðtÞAðtÞ : (3:27)

The instantaneous damping ratio �ðtÞ is computed following the relation

�ðtÞ ¼ �ðtÞ2!oðtÞ

: (3:28)

Remark 3.2 Observing the formulae given by Eqs. (3.16) and (3.26) it can beseen that the first approach takes into account the changing of the ‘‘unexpo-nential’’ instantaneous magnitude to compute the instantaneous damping,whereas the second approach takes into account the frequency variation ofthe signal. When the signal is purely exponential and monofrequency, the twoformulae yield an equivalent instantaneous computation as given by Eqs. (3.17)and (3.27).

3.2.5 Completeness, Orthogonality, and Orthogonality Index

The decomposition through EMD results in IMFs that possess completenessand orthogonality properties. The completeness is given by virtue of the decom-position as observed from (3.4), the sum of all decomposed elements yields theoriginal composite signal. In Section 3.4, we present some numerical examplesto demonstrate the completeness property of the decomposition (see also [13Section 6]). The orthogonality property is satisfied in all practical sense, but isnot guaranteed theoretically [13]. By virtue of the decomposition, the elementshould all be locally orthogonal to each other, because each element is obtainedfrom the difference between the composite signal and its local mean through themaximal and minimal envelopes, i.e.,

ðxðtÞ � xðtÞÞ � xðtÞ ¼ 0 : (3:29)

Ideally, if the mean is computed analytically, (3.29) is true and the IMFs areorthogonal. However, as the mean is computed through geometrical identifica-tion of the envelopes, hence it is not the exact mean. Moreover, each successiveIMF component is only part of the signal constituting xðtÞ. Because of theseapproximation, some errors, however small, cannot be avoided.

While we can assume the IMFs to be ‘‘approximately’’ orthogonal, the moreaccurate orthogonality of these EMD components should be checked a poster-iori numerically as follows. Rewrite (3.4) as


xðtÞ ¼Xnþ1

j¼1cjðtÞ; (3:30)

where the residue is regarded as cnþ1. The orthogonality of the IMFs is thenchecked based on a parameter called the index of orthogonality. Squaring thesignal (3.30), we obtain

xðtÞ ¼Xnþ1

j¼1c2j ðtÞ þ 2

Xnþ1

j¼1

Xnþ1

k¼1cjðtÞckðtÞ: (3:31)

If the decomposition is orthogonal, the cross terms on the right-hand side of(3.31) are zero. Hence, the index of orthogonality is defined as

IO ¼XT

t¼1

Xnþ1

j¼1

Xnþ1

k¼1cjðtÞckðtÞ=x2ðtÞ

!; (3:32)

where T40 is the number of the samples. The closer the value of IO to zero, thecloser the decomposition to orthogonality.

Theoretically, the EMD guarantees orthogonality only on the strength ofEq (3.29), and orthogonality also depends on the decomposition algorithm.Moreover, orthogonality is only important for linear decomposition system,and it would not make any physical sense for nonlinear decomposition. Hence,orthogonality of EMD for nonlinear signal is not guaranteed, but neverthelessthe orthogonality index is usually quite small which means the nonlineardecomposition is still close to orthogonal in some sense. More discussion andcomputational example for the orthogonality of EMD are presented in [13].

3.3 Masking Techniques to Improve Empirical

Mode Decomposition

This section discusses extensions to conventional EMD analysis to study oscil-latory dynamics. First, a synthetic example is introduced to examine conditionsunder which the conventional masking technique may fail. Then, various algo-rithms to refine the existing HHT are proposed and tested.

3.3.1 The Standard EMD Method and Its Limitation

Consider a two-component signal, xðtÞ,

xðtÞ ¼ 8 sinð1:6ptÞ þ 20 sinðptÞ : (3:33)

The time evolution of this signal is shown in Fig. 3.1. The clear feature of signal(3.33) is that it consists of low-frequency components and the magnitude of


the higher frequency component is significantly lower than that of the lower

frequency component.The standard EMD [28] that has been described in Section 3.2.2 is applied to

the testing signal (3.33). Figure 3.2 shows the decomposition result, with the

dashed background of the first two plots show the individual components of the

composite signal, which IMF1 and IMF2 should imitate. However it is seen that

0 5 10 15 20 25 30−30

0

30

Time (s)

Fig. 3.1 The synthetic signal (3.33) with two components

0 5 10 15 20 25 30−2

0

2

0 5 10 15 20 25 30–1

0

1

Res

idue

Time (s)

0 5 10 15 20 25 30–30

0

30

IMF 1

IMF 2

IMF 3

0 5 10 15 20 25 30–20

0

20

Fig. 3.2 IMFs of the signal (3.33) obtained using the standard EMD (components of thecomposite signal are plotted as dashed line background)


the two IMFs do not match the background. IMF1 is not a monofrequencysignal, but instead it exhibits mode mixing, making little sense to expect usefulphysical interpretation through the application ofHilbert analysis. The error onIMF1 is transmitted to the next IMFs. This shows that the standard EMD doesnot perform the decomposition well to this signal.

This drawback of standard EMD has been recognized in many cases. In anattempt to improve the performance and effectiveness of EMD, in [18,29] amasking technique is introduced. The technique aims at solving the problem ofmode mixing due to intermittency that occurs during transient events and theproblem of ambiguity that happens when two or more frequencies are not wellseparated. This technique has helped solving parts of the problems. More in-depth discussion about the background and technicalities of this technique ispresented in [29]. Further development of EMDwithmasking is proposed in [19].

Although larger classes of composite signals can be decomposed using theexisting EMD with masking techniques, there are still signals where the techni-ques do not work. To indicate the existing problem, we also applied themaskingtechnique in [19] to signal (3.33). This masking technique also fails to workas the obtained IMFs are almost indistinguishable to those obtained using thestandard EMD. Figure 3.3 compares the spectra of the IMFs obtained usingconventional EMD with that of the approach in [19]. As discussed in

0 0.2 0.5 0.8 1.2 1.6 20

5

10

15

20(a)

0 0.2 0.5 0.8 1.2 1.6 20

5

10

15

20

Frequency (Hz)

(b)

Fig. 3.3 Fourier spectrum of the first IMFs of signal (3.33) obtained using (a) the standardEMD and (b) the EMD with masking [19]


Section 3.1, the EMD results in mode mixing in which the 0.5 and 0.8Hzmodes are seen to interact and create a lower frequency mode at 0.2Hz whichin fact is not the component of the original signal. Techniques to identify andisolate the individual frequency components are discussed in the followingsubsection.

3.3.2 EMD Method with Fourier-Based Masking Technique

Simulation results in Section 3.3.1 show that the ability of EMD to deal withmode mixing is affected by three main factors:

� The signal consists of low frequency components;1

� The magnitude of the highest frequency component is much lower thanothers, particularly the second component, which is directly next to it inthe Fourier spectrum;

� The frequency components are high enough, but they are relatively close toeach other.

Based on the above considerations, the use of masking signal to improveempirical mode decomposition was investigated that extends the previouswork in [19]. The proposed approach permits the analysis of more generalsignals that exhibit mode mixing and results in improved characterization offrequency behavior. The algorithm called the Refined EMD (R-EMD), can besummarized as follows:

R1. Perform FFT on the original signal xðtÞ to estimate the frequency compo-nents f1; f2; . . . ; fn, with f14f24 � � �4fn. These captured frequencies arethe stationary equivalence of the possibly time varying frequency compo-nents of the signal xðtÞ.

R2. Construct the masking signals mask1;mask2; . . . ;maskn�1 using the fol-lowing sinusoidal signal:

maskkðtÞ ¼ Mk sinð2pðfk þ fkþ1ÞtÞ : (3:34)

The value of Mk is empirical and following [19] it is chosen to beMk ¼ 5:5 �Mk, with Mk40 the magnitude of the spectrum of the kthfrequency component.

R3. Identify two cases depending on the physical values of the highest fre-quency components f1 and f2, and their associated amplitudesM1 andM2:Case 1: If one of the following conditions hold:

(a) f1 � 1 and M15R21M2,(b) f141 and f1 � R1f2,

1 We consider the frequency 1Hz as the boundary between the low- and high-frequencysignals, and this is seen from the highest frequency component of a signal.


(c) f141 and R1f25f15R2f2 and M15R22M2,(d) f141 and f1 � R2f2 and M15R23M2,

where R21 ¼ 1:1, R1 ¼ 1:5, R2 ¼ 2, R22 ¼ 2, and R23 ¼ 0:5, then

1.1. Use only the first masking signal

mask1ðtÞ ¼ M1 sinð2pðf1 þ f2ÞtÞ (3:35)

for the whole process.1.2. Construct two signalsxþðtÞ ¼ xðtÞ þmask1ðtÞ andx�ðtÞ ¼ xðtÞ �mask1ðtÞ.

PerformEMDon each signal following steps S1 to S3 from the standardEMDto obtain all IMFs from each of them, i.e., cþi ðtÞ and c�i ðtÞ; i ¼ 1; 2; . . . ; n andalso the residue rþn ðtÞ and r�n ðtÞ.

1.3. The IMFs and the residue of the signal xðtÞ are

ciðtÞ ¼ðcþi ðtÞ þ c�i ðtÞÞ

2; i ¼ 1; 2; . . . ; n; (3:36)

rnðtÞ ¼ðrþn ðtÞ þ r�n ðtÞÞ

2: (3:37)

1.4. The total reconstructed signal ~xðtÞ is

~xðtÞ ¼Xn

i¼1ciðtÞ þ rnðtÞ : (3:38)

Case 2: If other than conditions (a) – (d) hold, then

2.1. Use all the constructed masking signals (3.34).2.2. Construct two signalsxþðtÞ ¼ xðtÞ þmask1ðtÞ andx�ðtÞ ¼ xðtÞ �mask1ðtÞ.

Perform EMD to each signal to obtain the first IMF only from each one, i.e.,cþ1 ðtÞ and c�1 ðtÞ. The first IMF of xðtÞ is

c1ðtÞ ¼ðcþ1 ðtÞ þ c�1 ðtÞÞ

2: (3:39)

2.3. Obtain the residue r1ðtÞ ¼ xðtÞ � c1ðtÞ.2.4. Use the next masking signal, perform steps 2.2 and 2.3 iteratively using

each masking signal while replacing xðtÞ with the residue obtained ateach iteration, until n� 1 IMFs containing the frequency componentsf2; f3; . . . ; fn are extracted. The final residue rnðtÞ will contain theremainder.

2.5. Compute the final residue, rnðtÞ ¼ xðtÞ � cnðtÞ.2.6. If the residue rnðtÞ is above the threshold value of error tolerance, then

repeat step S2 of the sifting process presented in Section 3.2.2 on rnðtÞ toobtain the next IMF and new residue.


2.7. The total reconstructed signal ~xðtÞ is

~xðtÞ ¼Xn

i¼1ciðtÞ þ rnðtÞ : (3:40)

Remark 3.3 In the complete R-EMD algorithm, we combine the proposedmasking algorithm, referred to as Case 1, and the masking algorithm from

[19], referred to as Case 2. As can be seen from the required conditions stated in

the algorithm, Case 1 is active when f1 � 1, it is clear that this algorithm takes

care of low-frequency signals and on the other hand Case 1 takes care of the

decomposition of high-frequency signals, so that the whole process can handle

the decomposition for a large sets of signals both with high-and low-frequency

components.Moreover, the values of the parameters R1, R2, R21, R22, and R23 in Case 1

are chosen based on the relation between the frequency as well as the amplitude

of the first two highest frequency components of the composite signals. In this

paper, the values are chosen to suit the application for signals that contain inter-

area oscillation. The choice helps classifying signals that satisfies the three

reasons given at the beginning of this section. Although they are not optimal,

the chosen combination yields effective decomposition for a large set of signals.

In general, seeing the EMD algorithm as a filtering process, we can think of the

parameters as filter gains that are possible to tune if necessary.The R-EMD algorithm gives different procedures for dealing with high-

and low-frequency signals. The main difference is in the way the masking

signals are utilized. For Case 2, we use as many masking signals as the number

of frequencies (or ideally the number of frequencies minus one) we want to

extract from the signal, and we subtract the effect of each masking signal at

every sifting stage, after each IMF is obtained. On the other hand, for Case 1,

we use only the first masking signal, constructed from the first two highest

frequency components peaking on the Fourier spectrum and let the masking

signal stay until the end of the decomposition process. The effect of this

masking signal is then automatically removed from the signal through the

use of formula (3.36).

Remark 3.4 The use of only one masking signal constructed using the twohighest frequency components of the spectrum in Case 1 is justified, as it

satisfies the condition of a masking frequency to be higher than the frequency

to be masked. The significant advantage of this algorithm is that it preserves

well the magnitude of the signal components, which is not the case for other

algorithms as the decomposition often fails. Hence, not only that the instanta-

neous frequency of the IMFs obtained using the R-EMD algorithm is more

meaningful but also we can obtain a quite good estimation of the instantaneous

magnitude of the IMFs.


3.3.3 EMD Method with Energy-Based Masking Technique

In the previous subsection we use FFT to construct the masking signals, which

implies that to some extent we rely on FFT to separate the frequency compo-

nents of the composite signals. Due to the limitation of FFT, particularly in

dealing with nonlinear and nonstationary signals, this step may deteriorate the

decomposing power of the proposed algorithm as FFT may give wrong infor-

mation in selecting the frequency of the masking signals.Drawing on Case 1 in Section 3.3.2 and the notion of instantaneous mean

frequency in [18], an alternative approach to determining an appropriate mask-

ing signal is suggested, relaxing the dependence to Fourier spectrum for detect-

ing the frequency components of the signal. The algorithm, called A-EMD, is

summarized as follows:

A1. Perform the standard EMD algorithm on the original signal xðtÞ to obtainthe IMFs. Use only the first IMF, c1ðtÞ, which is expected to contain thehighest frequency component of the signal, fmax, but may also containmode mixing with other lower frequency components. Perform HT onc1ðtÞ to obtain its instantaneous frequency f1ðtÞ and instantaneous magni-tude A1ðtÞ.

A2. In the spirit of Hilbert analysis, compute the energy weighted mean of f1ðtÞover L samples, i.e.,

�f ¼PL

i¼1 A1ðiÞf 21ðiÞPLi¼1 A1ðiÞf1ðiÞ

: (3:41)

A3. Observe Case 1 from R3, then replace step 1.1 with the following.

1.1. Construct the masking signal

mask1ðtÞ ¼ M1 sinð2pðm �f ÞtÞ ; (3:42)

whereM1 ¼ maxi¼1;...;L A1ðiÞ and m41.

The rest follow the steps given in the R-EMD algorithm.

Remark 3.5 Note that if the maximum frequency of the composite signal, fmax,is lower than 1Hz, it is common to choose m ¼ 2 as choosing a higher value of

mmay cause the masking signal to be ineffective as its frequency, m�f, would be

much higher than fmax. Comparing with [18], where the masking signal is

computed as mask1ðtÞ ¼ a0 sinð2pð�f=fsÞtÞ, the parameter m replaces the para-

meter fs, the sampling rate. Moreover, we introduced M1 ¼ maxi¼1;...;L A1ðiÞfor analytical choice of a0 in [18]. To complete the formulation of the method,

an efficient algorithm to extract instantaneous attributes is now explored based

on the use of a local HT.


3.3.4 Local Hilbert Transform

Existing approaches to the calculation of the complex trace (3.5) are based onthe computation of the analytic signal through the Fourier transform. Thistransform, however, has a global character and suffers from problems such asend effects and leakage. In this section, an alternative approach based on filterbanks is proposed that circumvents some of these effects.

Given a signal

xðtÞ ¼X

!

að!Þ cosð!tÞ þ bð!Þ sinð!tÞ ; (3:43)

where a and b are the Fourier coefficients

að!Þ ¼ 1

T

Z T

0

xðtÞ cosð!tÞ dt ;

bð!Þ ¼ 1

T

Z T

0

xðtÞ sinð!tÞ dt :(3:44)

The transformation to a complex time series is

zðtÞ ¼X

!

að!Þ cosð!tÞ þ bð!Þ sinð!tÞ

þ i bð!Þ cosð!tÞ � að!Þ sinð!tÞ½ �

¼ xðtÞ þ ixðtÞ ;

(3:45)

where xðtÞ ¼ xHðtÞ is the quadrature function, or the HT in (3.5).The HT used in this construction is obtained directly by operating the real

component with a convolution filter

xðtÞ ¼ xHðtÞ ¼XM

l¼�Mxðt� lÞhðlÞ ; (3:46)

where hð�Þ is the convolution filter with unit amplitude response and 90 phaseshift. A simple filter that provides an adequate amplitude response and p=2phase response is given by [30] as

hðlÞ ¼2lp sin

2ðpl=2Þ; l 6¼ 0;

0; l ¼ 0;

((3:47)

where �M515M. As M!1 the filter (3.47) yields an exact HT. For Mfinite, the filter introduces ripple effects. To limit these effects, a local HT has


been developed based on filter banks. As suggested in [31,32], the filter banksare developed such that the flatness of the frequency response is maximal for thelength of the filter. Defining z ¼ ej!, a maxflat filter can be defined by

hðzÞ ¼ 1þ z�1

2

� �2p

Q2p�2ðzÞ (3:48)

where p is the number that determine the zeros at ! ¼ p, and Q is chosen suchthat hðzÞ is halfband. The filter hðzÞ is shifted in frequency by p=2.

3.4 Applications

To test the accuracy of the method, we consider both synthetic data and datafrom transient stability simulations. For comparison, the system response isanalyzed using various algorithms described in previous subsections.

3.4.1 Application to a Synthetic Signal

To verify the accuracy of the method we examine again the synthetic signal(3.33) that we have used in Section 3.3.1. It has been noted earlier, conventionalanalysis fails to identify the individual modal components. In this subsectionwe perform two different major tests of our proposed algorithms. First, wetest the decomposing capability and second, we test the reliability to deal withnonlinear/nonstationary signals.

3.4.1.1 Decomposing Capability Test

Figure 3.4 shows the first three IMFs obtained using the R-EMD algorithm,whilst Fig. 3.5 shows the spectra of the first and the second IMFs. For erroranalysis, IMF1 and IMF2 are also compared with the corresponding compo-nents of the composite signal (3.33), which are also plotted in the backgroundwith dashed lines. It can be seen that the two IMFs match quite well thecorresponding components of frequency 0.8 and 0.5Hz, respectively. More-over, Fig. 3.6 shows the correctness of the whole decomposition results.

Figure 3.7 (a) and (b) shows the instantaneous frequency of IMF1 and IMF2,respectively, which show the frequency components of the composite signal.This figure also compares the instantaneous frequency obtained utilizingthe command hilbert in Matlab, with the convolution approach proposed inSection 3.3.4, where the latter is seen to reduce end effects.

The overall observations have shown two things. First, that R-EMDachieves a higher temporal resolution than the standard methods. Second, the


0 5 10 15 20 25 30−3

0

3

0 5 10 15 20 25 30

0

0.2

0.4

Time (s)

Res

idue

0 5 10 15 20 25 30−10

0

10

IMF 1

IMF 2

IMF 3

0 5 10 15 20 25 30−20

0

20

Fig. 3.4 IMFs of signal (3.33) obtained using the R-EMD (components of the compositesignal are plotted as dashed line background)

0 0.4 0.8 1.2 1.6 20

2

4

6(a)

0 0.2 0.5 0.8 1.2 1.6 20

5

10

15

20

Frequency (Hz)

(b)

IMF2

IMF1

Fig. 3.5 Fourier spectra of the first and second IMFs of signal (3.33) with R-EMD


convolution-based Hilbert transformer provides smoother transformation of

the signal by reducing end effects. These yield a more accurate physical char-

acterization of temporal behavior of the signal.We have also tested the energy-based A-EMD algorithm on signal (3.33).

However, we do not include the simulation plots in this paper as they are very

similar to the results from the R-EMD algorithm. We will show the application

of the A-EMD in the next example.

0 5 10 15 20 25 30−30

0

30

Time (s)

Reconstructed

Original

Fig. 3.6 Reconstruction of signal (3.33) from IMFs obtained using R-EMD

0 5 10 15 20 25 300

0.8

1.6

2.4(a) Frequency IMF1

0 5 10 15 20 25 300

0.5

1

1.5(b) Frequency IMF2

0 5 10 15 20 25 300

0.2

0.4

0.6(c) Frequency IMF3

Time (s)

StandardConvolution

Fig. 3.7 Instantaneous frequency of the IMFs of the synthetic signal (3.33).


3.4.1.2 Reliability to Handle Nonlinear/Nonstationary Signals

The reliability of R-EMD to cope with nonlinearity/nonstationarity is also

tested using clipped signal, where we distort signal (3.33) by clipping its com-

ponents at some ranges. The decomposition performance can be observed in

Fig. 3.8 where we can see that IMF1 and IMF2 preserve the frequency of each

corresponding component, while at the same time trying to capture the shape of

the distorted signals.Although we only show two IMFs in Fig. 3.8, this decomposition actually

yields another three insignificant IMFs plus a residue (as shown partly in

Fig. 3.4). However, feeding the distorted signal using a standard Prony analysis

tool (we have used the BPA/PNNL Ringdown Analysis Tool) for comparison,

we obtain more elements of the signals. Moreover,applying the instantaneous

damping computation formula (3.16), we obtain the comparison between HHT

with R-EMD and Prony analysis as provided in Table 3.1.From the quantities shown in Table 3.1, we can observe that HHT with

R-EMD is more reliable than Prony in dealing with nonlinearities, in this case

clipped signals. While the higher order IMFs and the residue from HHT with

R-EMD are insignificant in terms of magnitude with respect to IMF1 and IMF2,

which represent the components of the signal, some of the extra components

yielded by Prony have higher relative energy than the 0.8Hz component, that

0 5 10 15 20 25 30

0

5

10

IMF 1

IMF 2

0 5 10 15 20 25 30–20

–10

–10

–5

0

10

20

Time (s)

Fig. 3.8 The first two IMFs of the distorted signal (3.33) obtained using the R-EMD(distorted components are plotted as dashed line background)


may cause misleading in the analysis. Moreover, the damping computationshows that for the 0.8Hz component HHT with R-EMD is twice more accu-rate, while for the 0.5Hz component it is eight times more accurate than Prony.While in HHT with R-EMD the positive damping comes more due to the signaldistorsion and IMF1 and IMF2 do not show any increment in their magnitude,in Prony the 0.8006 and 0.4994Hz components really show the increment intheir amplitude, whereas they should not.

3.4.2 Instantaneous Damping Computation

3.4.2.1 Test I

Given a monofrequency sinusoidal signal

yðtÞ ¼ sinð2ptÞ : (3:49)

as plotted in Fig. 3.9. It is obvious that the signal contains a single frequencyf ¼ p Hz with unit amplitude. Using the information obtained from applyingthe Hilbert transformation to signal (3.49), the instantaneous damping compu-tation gives the result as depicted in Fig. 3.10.

Table 3.1 Comparison of R-EMD results and Prony analysis results

Distorted signal HHT (mean values) Prony

Frequency(Hz) z

Frequency(Hz) z

Frequency(Hz) z Relative Energy

0.8 0.0000 0.8016 0.0013 0.8006 0.0030 0.1496

0.5 0.0000 0.5083 0.0001 0.4994 0.0048 1.0000

0.2151 –0.0238 0.3390 0.0409 0.0375

0.1369 0.0106 0.1866 0.1980 0.0301

0.0833 0.0055 0.0921 0.9536 0.3269

6.3298 0.0615 0.3180

0 5 10 15 20 25 30−1

0.5

0

0.5

1

Time (s)

Fig. 3.9 Test signal (3.49)


Note that as the signal is monofrequency the decomposition process is notrequired. It can be observed clearly that the instantaneous magnitude of thesignal AðtÞ ¼ 1, the instantaneous frequency fðtÞ ¼ 1 and the instantaneousdamping �ðtÞ ¼ 0.

From this example we have been able to see that the instantaneous dampingcomputation is valid for a monofrequency signal. In the next example, we willtest the case when the signal contains several frequency components.

3.4.2.2 Test II

Given the following sinusoidal signal:

yðtÞ ¼ 6 sinð1:6ptÞ þ 2t sinðptÞ : (3:50)

It is obvious that the signal contains two frequency components 0.8 and 0.5Hz,respectively, with the first component has constant amplitude and the secondone with increasing amplitude. As the signal is a composite signal, we will firstneed to run an EMD process to decompose the components of the signal, andcompute the instantaneous damping of each component.

0 5 10 15 20 25 300.5

1

1.5

0 5 10 15 20 25 300.5

1

1.5

0 5 10 15 20 25 30−1

0

1

Time (s)

(a) Magnitude

(b) Frequency

(c) Damping

Average 2−28 s = −0.000000378≈0

Fig. 3.10 Instantaneous attributes of signal (3.49)


Up to this point, we have verified that our proposed algorithms provide a better

alternative implementation of HHT in certain applications, and it has shown

to be particularly more powerful when dealing with nonlinear/nonstationary

signals.

0 5 10 15 20 25 30

−60

−30

0

30

60

Time (s)

Fig. 3.11 Test signal (3.50)

0 5 10 15 20 25 300

5

10

Mag

nitu

de

0 5 10 15 20 25 300

0.8

1.6

Freq

uenc

y

0 5 10 15 20 25 30–0.2

0

0.2

Time (s)

Dam

ping

0 5 10 15 20 25 30–6–3036

IMF 1

FilteredOriginal

FilteredOriginal

Average = −0.0048519

Fig. 3.12 Instantaneous attributes of IMF1 from signal (3.50)


3.4.3 Application to Simulated Data

To verify the proposed method further, we consider simulation data from

transient stability simulations of a complex system. Figure 3.14 depicts a

simplified diagram of the test system showing the study area and major inter-

faces selected for study [11].Several simulation studies have been conducted to assess the applicability of

the proposed technique to analyze composite oscillations resulting from major

system disturbances. In these studies, the southeastern–central interface was

chosen for analysis because this corridor has a dominant participation in three

major inter-area modes. Figure 3.15 shows the power flow response of a key

transmission line interconnection, to the loss of Laguna Verde (LGV) unit #1 at

the Southeastern system. This particular contingency results in undamped

oscillations involving three major inter-area modes at 0.25, 0.50, and 0.78Hz.Using the R-EMDmethod, we decompose the signal into four nonstationary

temporal IMFs and a trend. Figure 3.16 shows the decomposition over the

entire simulation window.It is interesting to compare our result with the one presented in [11, Fig. 5]

which shows the IMFs of the same signal obtained from the standard EMD

algorithm. While in [11, Fig. 5] (the first three IMFs are re-plotted in Fig. 3.17)

0 5 10 15 20 25 300

30

60

Mag

nitu

de

0 5 10 15 20 25 300

0.5

1

Freq

uenc

y

0 5 10 15 20 25 30−1

−0.5

0

Time (s)

Dam

ping

0 5 10 15 20 25 30

−50−25

02550

IMF 2

Filtered

Original

Filtered

Original

Average = −0.12643

Fig. 3.13 Instantaneous attributes of IMF2 from signal (3.50)


all frequency components appear in IMF1, or in other words the shifting

process does not decompose the signal properly, in Fig. 3.16 we can see that

particularly the first three IMFs are pretty monofrequency. Moreover, it can be

seen fromFig. 3.18 that the proposed R-EMD algorithm accurately extracts the

Fig. 3.14 Geographical scheme of the Mexican interconnected power system

0 5 10 15 20 25 30

600

650

700

750

Time (s)

Rea

l pow

er f

low

(M

W)

Fig. 3.15 Tie-line oscillations following the loss of Laguna Verde unit #1


0 5 10 15 20 25 30–20

020

IMF 1

0 5 10 15 20 25 30–80

080

IMF 2

0 5 10 15 20 25 30–20

020

IMF 3

0 5 10 15 20 25 30–1

01

IMF 4

0 5 10 15 20 25 30–2

–1

0

Time (s)

Res

idue

Fig. 3.16 The IMFs obtained using R-EMD algorithm

0 5 10 15 20 25 30–80

0

80

IMF 1

0 5 10 15 20 25 30–25

0

25

IMF 2

0 5 10 15 20 25 30–7

0

7

Time (s)

IMF 3

Fig. 3.17 The first three IMFs obtained using standard EMD algorithm


three dominant frequencies as we can see the instantaneous frequency value of

each IMF is quite constant throughout the time. This has shown that the

decomposition works well.The frequency component of the inter-area modes obtained from the power

signal in this study are 0.7625, 0.4888, and 0.2542Hz; these modes coincide very

well with detailed eigenvalue analysis of the system [11].Another advantage of this approach over other existing methods is that

modal damping can be determined more accurately as the individual (modal)

components are isolated and extracted. This issue will be addressed in our

future research.In order to demonstrate that Hilbert analysis correctly identifies system beha-

vior, we also show that the damping ratio listed in [11, Table III] for the frequency

components 0.7625, 0.4888, and 0.2247Hz, which are respectively 0.0173,

–0.0209, and –0.0351, matches the trend of magnitude of each frequency compo-

nent. As we can observe from Figs. 3.16 and 3.19, the 0.7625Hz component is

decreasing, the 0.4888Hz is increasing, and the 0.2542Hz is also increasing.Figures 3.20 and 3.21 are the corresponding IMFs and instantaneous frequency

computed using the A-EMD method. Comparisons between Figs. 3.20 with 3.16

and Figs. 3.21 with 3.18 show that the two methods give results that show good

agreement. In both cases, the local HT is found to reduce the end effects.

0 5 10 15 20 25 30

0.4

0.8

1.2

(a) Frequency IMF1

0 5 10 15 20 25 30

0

0.5

1

(b) Frequency IMF2

0 5 10 15 20 25 30

0

0.3

0.6

(c) Frequency IMF3

Time (s)

StandardConvolution

Fig. 3.18 Instantaneous frequency of the IMFs showing the frequency of the inter areaoscillation


0 5 10 15 20 25 300

10

20

(a) Magnitude IMF1

0 5 10 15 20 25 300

40

80

(b) Magnitude IMF2

0 5 10 15 20 25 30

10

20

(c) Magnitude IMF3

Time (s)

StandardConvolution

Fig. 3.19 Instantaneous amplitude of the IMFs showing the growth of each component

0 5 10 15 20 25 30–20

020

IMF 1

0 5 10 15 20 25 30–80

080

IMF 2

0 5 10 15 20 25 30–20

020

IMF 3

0 5 10 15 20 25 30–2

02

IMF 4

0 5 10 15 20 25 30

–2–1

0

Time (s)

Res

idue

Fig. 3.20 The IMFs obtained using A-EMD algorithm


The use of a modified masking signal in our numerical formulation deservessome comment. It may be tempting to question why we are using (3.42) with theterm m�f instead of using the maximum value of the instantaneous frequencyf1ðtÞ of the first IMF that is logically the maximum frequency component of thesignal and replacem�fwithmf1;max where 15m52. Figure 3.22 shows the reason

0 5 10 15 20 25 30

0.4

0.8

1.2

(a) Frequency IMF1

0 5 10 15 20 25 30

0

0.5

1

(b) Frequency IMF2

0 5 10 15 20 25 30

0

0.3

0.6

(c) Frequency IMF3

Time (s)

StandardConvolution

Fig. 3.21 Instantaneous frequency of the IMFs obtained using A-EMD algorithm

0 5 10 15 20 25 300

2

4

6

Time (s)

Fig. 3.22 Instantaneous frequency of the first IMF obtained using the standard EMD algorithm


for this. The odd peak appears on the plot will give a wrong information of thevalue of the maximum frequency component that lead to the frequency of theconstructed masking signal too high. Clearly, the use of �f in (3.42) helps infiltering fictitious variations which in turn results in improved systemcharacterization.

To complete our study, we also make a comparison between HHT withA-EMD and Prony analysis tool. The result is shown in Table 3.2. It can beobserved that the results obtained using Prony involve some ambiguities as canbe seen for the components 0.4915Hz and 0.5276Hz as well as the components0.2494 and 0.2758 as they are coming as pairs. Although the relative energy ofthe pairing components are significantly different, it tells us that the dampinginformation does not show the real damping ratio of the true component 0.5and 0.25Hz, respectively. If the components of the monitored signal are notknown, this creates confusion in interpreting the results. On the contrary, HHTwith A-EMD gives more reliable and consistent results for the decompositionand the damping computation.

These findings are very useful for monitoring and analysis of the inter-areaoscillation for power system. It has simplified the analysis, as in this way theinstantaneous frequency and instantaneous damping of the inter-area oscilla-tion can be seen clearly and directly from visual observation, which is veryuseful when engineers have to make quick decision to take action in urgentsituations.

3.4.4 Application to Measured Data

To further test the ability of themethod to treat complex data, we analyze in thissection data from time-synchronized phasor measurements and the results arecompared to those of previous investigations based on standard EMD analysis.The data set chosen for investigation is the real power flow between two majorsubstations obtained from a recording of an oscillatory event in the Mexicansystem [12] which is depicted in Fig. 3.14. The two substations are TTH, which

Table 3.2 Comparison between HHT with A-EMD and Prony for inter-area modes analysis

Modes HHT (mean values) Prony

Frequency Frequencyz

Frequencyz Relative Energy(Hz) (Hz) (Hz)

0.78 0.7625 0.010 0.7678 0.0119 0.1032

0.5 0.4888 �0.010 0.4915 �0.0271 1.0000

– – 0.5276 0.0250 0.0489

0.25 0.2542 �0.007 0.2494 �0.0257 0.2542

– – 0.2758 0.0257 0.0622

0.0978 �0.030 1.1983 0.1548

0.8635 0.0420


is located at the Central system, and TTE, which is located at the Southeastern

system. The measurements were obtained over a 4 s period at a sampling rate of

2ms.Figure 3.23 shows the time evolution of the selected signal and Fig. 3.24

shows its corresponding spectrum. The power spectrum discloses the presence

of three major modes at 0.48, 0.44, 0.39, and 0.25Hz. Detailed examination of

the record suggests the presence of nonstationary characteristics. Also, the

signal exhibits a strong nonlinear trend associated with the frequency recovery

characteristics of the system.Based on the recorded data, the refined EMD method was used to decom-

pose the signal into several components, each with single frequency. The

objective is to assess the ability of the method to capture transient behavior in

real-world observational data.Figure 3.25 shows the three dominant components decomposed from the

data by the A-EMD. The other components have small amplitudes compared

with the first two IMFs and are not studied here. Because each component

0 100 200 300 400350

360

370

380

390

400

Time (s)

Rea

l pow

er f

low

(M

W)

Fig. 3.23 Measured real power flow between TTH and TTE of the Mexican system

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Frequency (Hz)

X:0.4472Y:0.9899

X:0.2566Y:0.4291

X:0.3935Y:0.4718

X:0.4863Y:0.7534

Fig. 3.24 Fourier spectrum of the measured signal


captures a particular timescale, the EMD technique effectively decomposes theoriginal signal into a set of IMF components, which can be linear or nonlinearand time-varying. Visual inspection of the IMFs in Fig. 3.25 shows that therefined method produces nearly sinusoidal oscillations from which meaningfulinstantaneous parameters can be extracted.

A key advantage of the method is that the exact time at which each modebecomes dominant can be determined. Useful insight into the physical inter-pretation of the IMFs can be obtained from the spectra of the first two IMFs inFig. 3.26. These results are consistent with the spectra of the original signalshown in Fig. 3.24, hence confirm the accuracy of themethod. The refinedHHTmethod, however, captures, additionally, the evolution of the frequencies pre-sent at every time instant of the records.

0 50 100 150 200 250 300 350 400–50

0

50IM

F 1

0 50 100 150 200 250 300 350 400–10

0

10

IMF 2

0 50 100 150 200 250 300 350 400–5

0

5

Time (s)

IMF 3

Fig. 3.25 Extracted IMFs from the R-EMD algorithm

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Frequency (Hz)

(a)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

Frequency (Hz)

(b)

x:0.4863y:0.6995

x:0.4448y:0.928

x:0.391y:0.3572

x:0.2542y:0.4002

Fig. 3.26 Spectra of the IMF1 (a) and IMF2 (b)


Having determined the individual IMFs, efforts were directed toward asses-

sing the ability of the method to capture abrupt changes in modal behavior.

Figures 3.27 and 3.28 show themodal attributes for modes 1 and 2, respectively,

obtained using the theory in Section 3.3. In these plots, the upper panel and

middle panel show the instantaneous amplitude and frequency computed using

the conventional approach in [18]. The lower panel shows the instantaneous

damping obtained from (3.16).As it may observed from these plots, in all cases, the modal frequency is very

accurately by the refined method. In addition, careful inspection of the middle

panel in Fig. 3.4 shows that the A-EMDmethod suppresses anomalous peaks in

the instantaneous frequency computations. This, in turn, results in improved

characterization of system damping as discussed below.As noted above, each IMF emphasizes a different oscillation mode with

different amplitude and frequency content. The first IMF captures the highest

frequency mode at 0.44 and 0.48Hz, while IMF 2 captures the time evolution of

the 0.25Hz mode.Also of interest, the analysis of modal damping in the lower panels in Figs. 3.27

and 3.28 show that the method is able to characterize local temporal behavior.

This information might be used to trigger control corrective or preventive control

actions. This is, however, not discussed here.

125 130 135 140 145 150 155 160 165 170 1750

10

20

Mag

nitu

de

125 130 135 140 145 150 155 160 165 170 1750

0.5

1

Freq

uenc

y

125 130 135 140 145 150 155 160 165 170 175–0.4

–0.2

0

Time (s)

Dam

ping

FilteredUnfiltered

Average = –0.00124

0.48 Hz0.44 Hz

IMF1

Fig. 3.27 Instantaneous attributes of IMF1


To verify the accuracy of the proposed method, detailed studies were con-

ducted using Prony analysis. Prior to Prony analysis, the slow trend was

removed by subtracting the nonlinear trend from the original signal; this

makes the analysis more accurate and enable closer comparison between both

approaches. Table 3.3 shows Prony analysis results for the recorded bus fre-

quency signal in Fig. 3.23.Overall, the results are found to be in good agreement although it should also

be stressed that Prony cannot capture the modes simultaneously as HHT does.

Consistent Prony results are found only when the time intervals match those of

Hilbert analysis. While several techniques can be used to improve damping

estimates this is not the main focus of this chapter.As discussed in Section 3.2.5, the IMFs form a set of complete basis for

modal decomposition of the oscillatory processes, which is derived from the

data themselves. Also, for linear signals, the resulted IMFs are numerically

100 110 120 130 140 150 160 170 180 190 2000

5

10M

agni

tude

100 110 120 130 140 150 160 170 180 190 2000

0.25

0.5

Freq

uenc

y

100 110 120 130 140 150 160 170 180 190 200–0.5

0

0.5

Dam

ping

Time (s)

Average = –0.00169

0.25 Hz

IMF2

Fig. 3.28 Instantaneous attributes of IMF2

Table 3.3 Prony analysis results

Time intervals Dominant modes Frequency Damping Energy

125–175 2 0.440 0.0025 1.0000

125–140 2 0.414 0.0192 1.0000

3 0.229 0.0436 0.7930

140–160 1 0.487 0.0228 0.0000


orthogonal. Following [13], in Section 3.2.5 the orthogonality index is alsointroduced for the case of nonlinear signals.

To verify the completeness of the decomposition, the full data was recon-structed using the decomposed IMFs. Figure 3.29 compares the original timeseries with the reconstructed signal obtained using the EMD. As shown in thisplot, the results are practically indistinguishable over the entire length of thesignal, thus giving confidence to the results.

Moreover, Table 3.4 compares the IO for the conventional approach and therefined EMD method. As may be seen from this table, the IO is very smallshowing that the proposed method produces physically and numerically realis-tic IMFs. It is also interesting to note that the refined algorithm reduces slightlythe orthogonality condition of the decomposition.

3.5 Conclusion

In this chapter various refinement of EMD algorithms and a local Hilberttransformation have been presented. These algorithms are well suited forextracting and characterizing temporal behavior and can be applied to typicalsignals found in power oscillatory processes. Applications to synthetic signalsas well as measurement signals show that the proposed algorithms provideimproved visualization and characterization of complex oscillations involvingmulti-timescale behavior. Several extensions to the proposed analytical toolsare possible including improvements in the masking signal method, the compu-tation of local Hilbert transformations and the estimation of modal properties.These issues warrant further investigation.

0 100 200 300 400–30

–20

–10

0

10

20

Time (s)

Reconstructed signalOriginal signal

Fig. 3.29 Reconstruction of signal from IMFs obtained using A-EMD

Table 3.4 Index of orthogonality of the measurement signal

Conventional approach A-EMD algorithm

IO 7.128e� 06 6.977e� 06


Acknowledgments The authors are thankful to Jegatheeswaran Thambirajah and NinaThornhill from the Chemical Engineering Department, Imperial College London, for usefuldiscussion and good teamwork in pursuing the research in this topic that make it possible forthe authors to contribute this chapter. The authors also thank ABB, Switzerland, andNational Grid, UK, for the research collaboration done within this research project.

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18. R. Deering and J. F. Kaiser. The use of a masking signal to improve empirical modedecomposition. In Proc. IEEE Int. Conf. on Acoustic, Speech and Signal Proc. (ICASSP’05), 4:485–488, 2005.


19. N. Senroy and S. Suryanarayanan. Two techniques to enhance empirical mode decom-position for power quality applications. In Proc. IEEE Power Eng. Soc. Gen. Meet.,Tampa, Florida, 1–6, 2007.

20. N. Senroy, S. Suryanarayanan, and P. F. Ribeiro. An improved Hilbert-Huang methodfor analysis of time-varying waveforms in power quality. IEEE Trans. on Power Syst.,22(4):1843–1850, 2007.

21. E. Del’echelle, J. Lemoine, and O. Niang. Empirical Mode Decomposition: An analyticalapproach for sifting process. IEEE Signal Process. Lett., 12:764–767, 2005.

22. R. Srinivasan, R. Rengaswamy, and R. Miller. A modified empirical mode decomposi-tion (EMD) process for oscillation characterization in control loops. Control Eng. Pract.,15:1135–1148, 2007.

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Chapter 4

Practical Application of Hilbert Transform

Techniques in Identifying Inter-area Oscillations

T.J. Browne, V. Vittal, G.T. Heydt, and Arturo Roman Messina

Abstract Disturbances in large power systems can exhibit nonlinear, time-

varying behavior. Traditional modal identification from field data is via tech-

niques, such as Prony analysis, which assume data stationarity. The Hilbert

transform and analytic function can be used to analyze inter-area oscillatory

behavior of power systems with the stationarity assumption relaxed. However,

reducing the data to simple numerical results can be achieved more effectively

when stationarity is assumed. The application process is not straightforward

and subtle changes can yield considerable variation in the results observed. An

example is the effect of discrete time calculations of the Hilbert transform over a

window of finite length. Application of the newer modal identification techni-

que, Hilbert analysis, is examined relative to the more established Prony ana-

lysis, with particular reference to the considerable structural differences which

exist between the two methods. Prony analysis yields modes which are directly

expressed as exponentially modulated sinusoids, whereas the Hilbert method

provides amore general solution. Synthetic andmeasured signals are used in the

comparison. Some general conclusions are drawn from the analysis of several

signals, including sets of measured field data.

4.1 Inter-area Oscillations in Power Systems

Inter-area oscillations in power systems [1, 2] occur when geographically iso-

lated generator groups swing against each other. When inter-area oscillations

are unstable, the interconnected network can be broken up, potentially leading

to islanding. Real-time analysis of power system measurements is intended to

assist in assessing the existence and stability of inter-area oscillations.

T.J. Browne (*)Ira A. Fulton School of Engineering, Department of Electrical Engineering, ArizonaState University, Tempe, AZ, USAe-mail: [email protected]


101

Present analysis techniques are yet to be applied to real-time power systemmeasurements for consistent stability identification. One modal parameteridentification method, Prony analysis [3], provides estimates of dampingwhen the power system can be satisfactorily modeled with linear approxima-tions. Prony analysis can be applied successfully to some sections of the mea-sured bus frequency signal given in Fig. 4.1; however, application of the Pronymethod to the entire signal does not yield meaningful results.

The Hilbert transform H of a signal x(t) is obtained by convolution with aninverse time function:

H½xðtÞ� ¼ 1

pt� xðtÞ: (4:1)

Unlike the Fourier transform (FT), the Hilbert transform of a time-domainsignal is itself a time-domain signal. When applied to a single mode, the Hilberttransform offers some useful properties in modal parameter identification. Arelatively new technique, empirical mode decomposition (EMD) [5], isemployed to isolate individual modes within the signal prior to application ofthe Hilbert transform.

This chapter will demonstrate practical considerations associated with theapplication of the Hilbert transform and EMD in identifying modal parametersunder inter-area oscillatory conditions [4], which are typically in the range0.1–2 Hz [6].

4.2 Present Identification Techniques

4.2.1 Prony Analysis

Prony analysis [3, 7, 8] is a curve-fitting technique, which decomposes a signalinto sinusoids modulated by exponential growth or decay – that is, into modesof the form

Fig. 4.1 Measured bus frequency signal during a disturbance in the Mexican InterconnectedSystem. Reproduced, with permission, from [4]. # 2008 IEEE

102 T.J. Browne et al.

xðtÞ ¼Me�t sinð!dtþ ’Þ: (4:2)

Although the underlying mathematics, which is based on linear predic-tion, is eighteenth-century technology, numerical ill-conditioning makes thetechnique computationally intensive [3], delaying widespread use untilaround the 1980s.

Being an eigenvalue-based technique, Prony analysis delivers the real andimaginary parts of the mode, corresponding respectively to the attenuationfactor � and the modal damped frequency !d, fromwhich the natural frequency!n and damping ratio � can be calculated using well-known linear systemtheory. In addition, Prony analysis is able to identify the two additionalmodal parameters, namely amplitude and phase shift, which allow a mode tobe described uniquely in the time domain, making reconstruction of the signalfrom its Prony-identified components feasible.

The key assumption involved in Prony analysis is that the signal is station-ary: where the assumption is not met, the linear system theory underpinning theProny algorithm is inapplicable and so the decomposition is not necessarilymeaningful. Factors such as window length and sampling rate can affect theaccuracy with which a nonstationary signal can be approximated by a sum ofstationary modes.

Prony software [9] has been used to assist in verifying results of Hilbertanalysis and to provide a point of reference. Adjustments to options in thesoftware can have significant impact on the results obtained.

4.2.2 Fourier Methods

Certainly, the oldest and perhaps the most widely used signal analysismethods are based on Fourier technology. This is the resolution of a signalx(t) into its component sine waves, that is, a frequency spectrum. Themethod is based on the Fourier series for x(t). If x(t) is periodic with periodT, then 2p=T rad/s is the fundamental frequency, !o; and one finds that theFourier series is

xðtÞ ¼ ao2þX1

n¼1an þ a�n� �

cosðn!otÞ þ j an � a�n� �

sinðn!otÞ (4:3)

where ai are the Fourier coefficients

ai ¼2

T

Zt1þT

t1

xðtÞe�ji!ot dt: (4:4)

4 Practical Application of Hilbert Transform Techniques 103

In engineering, the series is often rewritten as

xðtÞ ¼ ao2þX1

n¼1bn cosðn!otÞ þ cn sinðn!otÞ (4:5)

where

bn ¼2

T

Zt1þT

t1

xðtÞ cosðn!otÞ dt (4:6)

cn ¼2

T

Zt1þT

t1

xðtÞ sinðn!otÞ dt: (4:7)

Note again that x(t) must be periodic, and also note that there are no ‘‘inter-frequency components’’ (i.e., no ‘‘fractional harmonics’’).

In the present application, it is convenient to use a numerical analysismethod, the discrete Fourier transform (DFT), which is

FDðxðk�TÞÞ ¼ Xðm��Þ ¼ 1ffiffiffiffiNp

XN

m¼1xði�TÞe�jim��T (4:8)

where FD(.) denotes the DFT, �T is the sampling time,N is the number of pointsin the sample, and N must span an integer number of periods of the signalx(t).The frequency domain resolution of the DFT is �� which is 2p/N�T r/s.In the 1960s, several researchers found a clever way to find the DFT of a signalusing certain symmetry properties of the complex exponential. The method iscalled the fast Fourier transform (FFT), and it is an exact method of evaluatingthe DFT of a signal. The advantage of the FFT is that not every term in the DFTneed be explicitly evaluated, and therefore the computational burden is reduced.It is well known that the computational burden in the FFT becomes N log(N)rather than N2 as implied by the definition of the DFT.

As the sampling interval becomes smaller and smaller, and as N goes toinfinity, it is a simple matter to show that the DFT becomes the FT,

FffðtÞg ¼ Fðj!Þ ¼Zþ1

�1

fðtÞe�j!t dt: (4:9)

The inverse FT is

F�1fFðj!Þg ¼ fðtÞ ¼ 1

2p

Zþ1

�1

Fðj!Þeþj!t d!: (4:10)


Note that:

� ‘‘Almost all’’ periodic functions x(t) possess a Fourier series.� A Fourier series is a representation of x(t) as a sum of sines and cosines that

have frequencies that are integer multiples of 1/T Hz (T = the period of thesignal f(t)).

� Each term in the series is called a harmonic.� The Fourier series, the DFT, and the FFT are the minimum least-squares fit

of a sine–cosine series to a signal x(t).That is, there is no better calculation ofthe series coefficients that is possible. However, if the series or the discretetransforms are calculated using faulty assumptions, for example, x(t) isassumed periodic but it is not actually periodic, then one may not obtain asuitable representation of x(t) in the frequency domain.

As an historical note, Fourier methods are credited to Baron Jean BaptisteJoseph Fourier (1768–1830) who worked at the Ecole Normale in Paris. Theliterature of Fourier methods is huge. A sampling of useful references for thesubjects touched upon is [10–14].

Proceeding to examine the use of Fourier technology to find signal damping,use the DFT of the sampled time signal x(k�T),

Xðm��Þ ¼ 1ffiffiffiffiNp

XN�1

i¼0xði�TÞe�jim�T�� (4:11)

where the signal x(k�T) is assumed to be periodic with periodicity N and�� =2p/N�T. Of course, all DFTs have a discrete spectrum. Note that theassumption of periodicity of x(k�T) results in discrete spectrum X(m��) withno ‘‘leakage.’’ Leakage refers to omission of terms in theDFT sum due to failureto span an integer number of periods of x(t). If the DFT (and hence the FFT) dospan an integer number of periods of x(t), and, indeed, if x(t) is truly periodic,X(m��) is ‘‘exact.’’

Using X(m��), the parameter � is calculated,

� ¼ 2pBW

(4:12)

where BW is the (3 dB) bandwidth of the spectrum |X(m��)|. This parameter �is the damping parameter in the case of a single-mode signal.

Because actual measurements give x(k�T) that is not generally periodic, therequired time window for the calculation of X(m��) is infinite and finitesampling is undersampling. Undersampling results in low-pass filtering ofX(m��). This problem is not important if the signal x(t) is nearly periodic,but a damped sine wave, the expected signal, is clearly aperiodic.

Most engineers are familiar with minimum phase systems, and therefore maybe unfamiliar with the trivial property that the pole locations in the right (i.e.,


nonminimum phase systems) or left half-planes may be readily determined fromthe phase spectrum of the DFT (and FFT) near resonance. All methods basedon the assumption of signal behavior as a second-order system suffer from thedifficulty that if the signal is not a simple second-order system, erroneous resultsmay occur – and the DFT approach shown here has that property. Actualpower signal are results of nonlinear, time-varying, high-order systems, andtherefore all methods of identifying modes may give inaccurate results whendissimilar modes are nearly collocated or time variation of the signal (and itssource system) come into play. The short-time Fourier transform (STFT) mayaddress this situation. Parameter identification techniques have been devel-oped, which exploit sliding windows [15] and spectral decay [16].

4.3 The Hilbert Transform and Analytic Function

4.3.1 Hilbert Transform Properties

The Hilbert transform XH(t) of a signal x(t) is defined [17] as the convolution

XHðtÞ ¼ H½xðtÞ� ¼ 1

pt� xðtÞ: (4:13)

An alternative definition [18] reverses the sign. In contrast to the FT, theHilbert transform does not introduce a frequency variable. Rather, the trans-form of a time-domain signal remains in the time domain.

Properties of the convolution mean that the inverse Hilbert transformH–1 ofa signal YH(t) is given by

yðtÞ ¼ H�1 YHðtÞ½ � ¼ �1pt� yðtÞ ¼ �H YHðtÞ½ � (4:14)

and so a double application of the Hilbert transform to a signal returns thenegative of the original signal.

Because the signals of principal concern are damped sinusoids, Hilberttransforms of trigonometric and exponential functions are of interest. It canbe shown that the Hilbert transform of a cosine function is a sine function.Similarly, (4.14) implies that the Hilbert transform of a sine function is anegative cosine function. Hahn [17] gives the Hilbert transform of a rotatingphasor as

H½ej!t� ¼ �sgnð!Þej!t (4:15)

where sgn is the signum (sign) function.Many other Hilbert transform pairs areprovided in [17]. Damped sinusoids will be addressed by the analytic function inSection 4.3.2.


The linearity of the convolution operation confers useful properties upon theHilbert transform. The Hilbert transform of the sum of two functions is the sumof the Hilbert transforms of the individual functions. Similarly, if k is a constantthen for any function x(t),

H½kxðtÞ� ¼ kH½xðtÞ�: (4:16)

Products of functions are more complicated. Bedrosian [19] showed that theproduct of two analytic functions x1(t) and x2(t) is given by

H½x1ðtÞx2ðtÞ� ¼ x1ðtÞH½x2ðtÞ� (4:17)

when x1(t) and x2(t) are spectrally disjoint. However, this spectral condition istoo restrictive to be useful in modal identification: exponential functions arewideband signals, and so an exponential decay is unsuitable for use as eitherx1(t) or x2(t). Cain [20] developed an unrestricted expression for a Hilberttransform product of two complex (or, by implication, real) functions,

H½x1ðtÞx2ðtÞ� ¼ x1ðtÞH½x2ðtÞ� þ x2ðtÞH½x1ðtÞ� þH½H½x1ðtÞ�H½x2ðtÞ��: (4:18)

Because the final termmeans that evaluation of this expression still requires aHilbert transform of a product, (4.18) does not appear to have found wide-spread use, at least in the power engineering literature. More heuristic techni-ques tend to be utilized when Hilbert transform products are required.

4.3.2 Modal Parameters in Terms of the Analytic Function

The analytic function [17, 21] is a complex signal derived from both the originalsignal and its Hilbert transform as

XAðtÞ ¼ xðtÞ þ jH½xðtÞ� (4:19)

For a single-mode signal of the form of (4.2) for any real t, the analytic signalcan be expressed as

XAðtÞ ¼Me�t cosð!dtþ ’Þ þ jMe�t sinð!dtþ ’Þ

¼Me�tejð!dtþ’Þ:(4:20)

Therefore, the magnitude of the analytic function is related to the attenua-tion factor � of the mode via

ln XAðtÞj j ¼ �tþ ln M (4:21)


while the phase angle of the analytic function can be related to the modaldamped frequency !d as

ffXAðtÞ ¼ !dtþ ’ (4:22)

For a signal of the form (4.2), (4.21), and (4.22) imply that the attenuationfactor can be found from the slope of a plot of ln |XA(t)| versus time, whereas themodal damped frequency can be established from the slope of a plot of ffXA(t)versus time. By implication, the two parameters could be found instead using aleast-squares fit for online applications.

As an example of the technique, consider the signal

fðtÞ ¼ e�t=10 sinð2p0:27tÞ; �15t51: (4:23)

The Hilbert transform of f(t) is

FHðtÞ ¼ �e�t=10 cosð2p0:27tÞ; �15t51: (4:24)

giving the analytic signal

FAðtÞ ¼ �je�t=10 ej2p0:27t; �15t51: (4:25)

It can be seen that ln |FA(t)| ¼ –0.1t and so the attenuation factor � ¼ –0.1.Similarly, ffFA(t) ¼ 0.27t, yielding !dt ¼ 0.27 rad/s.

The problem remains to decompose a measured signal xm(t) into individualcomponents of the form (4.2). Two techniques which have been applied to thisend are Prony analysis [3, 7, 8], discussed in Section 4.2, and EMD [5, 22]. AsProny analysis delivers the modal properties without further calculation, onlythe EMD is applied in conjunction with the Hilbert transform.

It should be noted here that it is not the entire set of modal parameters whichis critical to the intended application. The first priority, stability assessment,implies that the sign of the attenuation factor � is the most important parameterto be identified for each mode: a negative value indicates a stable mode whereasa positive value denotes instability. Errors in the other parameters can betolerated acceptably should stability assessment be of sole interest. However,the following implementation of the technique for stability assessment and forthe purposes of control implementation become of increasing interest. If themodal frequency is assessed incorrectly then any control action will be misdir-ected and may be, at best, ineffective.

4.3.3 Hilbert Transform Implementation

The theory presented in Section 4.3.2 was concerned purely with the calculationof the Hilbert transform in continuous time. For the modal identification


problem, utilizing sampled power systemmeasurements, continuous time beha-vior can only be approximated. Some work has been carried out on a specificdiscrete Hilbert transform [17].

It can be shown [23] that the Fourier spectra of the original and analytic signalsare identical at positive frequencies, whereas at negative frequencies, the FT of theanalytic signal is zero. Software [24] employed for calculation of the Hilberttransform exploits this property, using the discrete approximation to the FT togenerate the analytic signal directly, without first determining a Hilbert transform.The Hilbert transform is extracted as the imaginary part of the analytic signal.

A key advantage of calculation via the DFT is the speed and simplicity of theFFT, compared with the operations required for the direct convolution. How-ever, disadvantages exist also, primarily aliasing problems inherent in applica-tion of the DFT to a signal which is unlikely to be band-limited. A highsampling frequency can mitigate some, but not all, of these aliasing problems.

4.3.4 Instantaneous Frequency

The concept of ‘‘instantaneous frequency,’’ arguably meaningless for a single-frequency sinusoid, has been examined for nonlinear signals and is used [5] as ajustification for the decomposition to be introduced in Section 4.5. Only oscilla-tions which lie within the frequency range of 0.1–2.0 Hz characteristic of inter-area modes [25] are of interest. It can be argued that when the sampling rate issufficiently high, a meaningful instantaneous frequency of a signal can becalculated using only two or three samples in the vicinity of every samplinginstant under consideration. Boashash [26] describes several possible techni-ques, and notes that the instantaneous frequency is not necessarily reconcilablewith frequencies obtained from amore global Fourier analysis. The logic of thisstatement can be verified by considering an infinite sine wave: the instantaneousfrequency around the zero crossings is necessarily greater than that at the peaksand troughs, and so a range of frequencies, which would not be identified bydirect application of Fourier techniques, can be conceived.

Instantaneous damping is less obviously definable, despite an assertion [27]in a civil engineering context that the concept makes physical sense. Thedamping in an oscillatory signal has no obvious meaning if only a singleoscillatory cycle is to be examined.

4.4 Application to Single-Mode Signal

The general expression

H cos t ¼ sin t (4:26)

holds over the domain –1< t <1.


However, when applying the theory to data gathered from field measure-ments, only a finite signal window is available. Some examination of theproperties of time-limited signals and their Hilbert transforms is, therefore,warranted.

Consider the signal given in (4.23). Over an infinite window the Hilberttransform is well-defined and offers a closed-form expression, specifically(4.24). Because an infinite window is impractical to measure, suppose that a50 s interval of this signal is recorded, as given in Fig. 4.2.

The Hilbert transform calculated for the signal in Fig. 4.2 will dependheavily on the boundaries over which the transform is to be calculated. Ifonly the 0–50 s window is used, the result in Fig. 4.3 develops. The oscillationseen in the Hilbert transform appears well damped until around 45 s into therecord, when significant change occurs. The effect on the damping calculationis evidenced in Fig. 4.3. The mathematics developed earlier suggests thatFig. 4.3 should be a straight line of slope –1 pu/s. Whilst this is a reasonablerepresentation in the middle part of the record, distortion toward the endcould yield differing interpretations.

Fig. 4.2 Signal given in (4.23), truncated to the 0–50 s interval. Reproduced, with permission,from [4]. # 2008 IEEE

Fig. 4.3 Hilbert transform, calculated over 0–50 s window only, of signal in Fig. 4.2


To alleviate the situation, the original signal can be zero padded prior to

finding the Hilbert transform and the analytic signal. Figures 4.5 and 4.6

demonstrate the improvement made to Figs. 4.3 and 4.4, respectively when

this action is taken.Note that Figs. 4.5 and 4.6 show significant variation before t=0, when the

input to the transform is zero. That is, the Hilbert transform operator is not

causal. This is apparent from the definition via the convolution, and is con-

firmed here by the calculation.

Fig. 4.4 Logarithm of analytic signal magnitude, calculated fromHilbert transform in Fig. 4.3

Fig. 4.5 Hilbert transform of zero-padded record

Fig. 4.6 Logarithm of analytic signal of zero-padded record


The reason for the discrepancy can be traced to the Hilbert transform of arectangular pulse. A time-limited signal xm(t), as recovered frommeasurements,can be constructed as the product of an infinite signal xn(t) and a rectangularpulse ((t – a)/T) having duration T and midpoint t = a.

An alternative perspective can be obtained through consideration of thecalculation method. It has been established that the Hilbert transform can becalculated from the inverse FT of the positive frequency spectrum of the signal.Fourier methods assume periodicity, which is not a property held by the signalunder examination. Aliasing is therefore unavoidable. However, by zero pad-ding the signal on either side, the extent of the aliasing is reduced by making thefalse repetition less significant relative to the actual signal.

The domain on which the Hilbert transform of a signal is calculated can havea strong impact on the result. Distortion around the endpoints of the consideredwindow is most severe when only a short time is used for the calculation. Toreduce the impact of distortion around the endpoints of the signal section underexamination, the first 10% and last 10% of the time interval are discarded whenassessing modal parameters using least squares.

4.5 Multiple Mode Signals: Empirical Mode Decomposition

EMD [5], is a relatively new method of extracting components of a signal.Unlike linear transform techniques, EMD does not assume a constant fre-quency for each component. Instead, components exhibiting ‘‘fast’’ (in somesense) variations are isolated from components varying more slowly. Being atime–frequency analysis, the technique bears closer resemblance to wavelettransforms than to Fourier analysis. However, whereas wavelets offer closed-form expressions, EMD is defined by an algorithm and not a formal mathema-tical expression.

Traditional linear analysis methods, such as Fourier and Prony decomposi-tion, find constant parameters to fit a signal to a specific expression; in the caseof modal analysis, the expression takes the form

xðtÞ ¼X

i

Mie�it sinð!ditþ ’iÞ: (4:27)

By contrast, the components identified by EMD do not take a form which hasbeen previously assigned. EMD is described as a data-driven technique [5]:whereas Prony analysis assumes that a signal can be satisfactorily decomposedinto damped sinusoids, EMD makes no such assumption.

Huang [5] denoted the signal components identified by EMD as ‘‘intrinsicmode functions,’’ or IMFs, and proposed two defining properties of an IMF.The first requires that the difference between the number of zero crossings andthe number of maxima andminima must be –1, 0, or 1. The second requires that‘‘at any point, the mean value of the envelope defined by the local maxima and


the envelope defined by the local minima is zero’’ [5]. An oscillatory mode of theform (4.2) meets – and exceeds – the definition of an IMF. In practice, a signalwith any significant DC component cannot be an IMF.

Huang [5] provides comprehensive instructions on application of the EMD.Many shortcomings in the implementation are examined. Only an overview isgiven here; for further details [5] is recommended. Available EMD software [28]provides additional mechanics.

EMD ‘‘sifts’’ out fast local variations from the signal content. To find thefirst, and fastest, IMF, the mean of the maximum and minimum envelopes isfound at each point and subtracted from the signal. Envelopes are presentlyfitted to the data points using a cubic spline algorithm; problems with the cubicspline fitting have been identified as a key deficiency of EMD [5]. The process offinding envelopes and subtracting their mean from the signal continues until theresulting signal meets the definition of an IMF. This IMF is subtracted from theoriginal signal and the process begins again, continuing until a residue whichnot possessing an envelope is found.

As an example of EMD application, Fig. 4.7 gives a bimodal test signaldefined as

gðtÞ ¼ e�t=10 sinð2p0:27tÞ þ e�t=8 sinð2p0:60tÞ; �15t51 (4:28)

and shows the decomposition into the three most significant IMFs. A secondmode has been added to (4.23) to derive g(t). As expected, the IMFs showing the

Fig. 4.7 Empirical mode decomposition of bimodal test signal given in (4.28). Residue issuppressed


0.60 Hz (IMF1) and 0.27 Hz (IMF2) detail account for the bulk of the energyassociated with the signal. Several spurious IMFs are falsely identified, themostsignificant being IMF3 (displayed) which still has negligible energy relative toIMF1 and IMF2. The most likely cause of these spurious components is endeffects associated with the decomposition. The residue from the EMD is negli-gible and so not shown.

In this instance the additional mode does not present significant cause forconcern as the oscillations are well damped and of negligible energy. However,with other signals the potential exists for spurious unstable components toappear. In such a situation, control logic could incorrectly trigger separation ofthe network into islands. Alternate methods are therefore required to confirm theoutput from the EMD when used in a potential supervisory control scheme.

EMD suffers from a paradox: it analyzes time and frequency behavior of asignal simultaneously, but can only resolve one of these two characteristics bymaking assumptions on the other. In practical terms, EMD can only be used toassess the modal content of a signal numerically by assuming that the signal is infact composed of linear modes; that is, that (4.27) holds. If modal analysis is tobe undertaken, any online modal assessment must therefore operate on a slidingwindow.

Much of the discrepancy arises from a lack of clarity in the meaning of theword ‘‘mode.’’ In linear signal analysis, ‘‘mode’’ implies one of the components of(4.27). By contrast, in the EMD sense ‘‘mode’’ refers to a single componentisolated by the heuristic EMD process, which may bear no resemblance to amode in the linear sense [22, 28]. Neither interpretation of ‘‘mode’’ is necessarilywrong; rather, the intended meaning depends entirely on the context of thediscussion. Because the intended application is online numerical assessment ofmodal content, with a view toward determining the stability or otherwise of theunderlying system, the assumption of a linear system is implicit in the analysis.

In the field of power quality, EMD encounters difficulty in isolating sinu-soids closely spaced in frequency [29]. A solution which was employed involvesthe derivation of masking signals from the FFT of the signal. This solution,whilst having merit for sinusoids, is not appropriate for damped sinusoids:derivation of a masking signal is arguably more complicated than identificationof the mode itself and therefore provides no benefit.

4.6 Factors Affecting Performance of the Technique

4.6.1 Modal Separation

Section 4.5 has demonstrated the application of EMD in order to prepare foridentification of modal parameters in amultimodal signal. The test signal whichwas used in Section 4.5 contained two modes, separated in damped frequencyby a factor of more than two (0.27 and 0.60 Hz) and sharing the same


attenuation factor of 0.1. It was noted that the EMD algorithm was able tosuccessfully separate the two components.

By contrast, in a signal containing twomodes closely spaced in frequency butwidely separated in damping, identification of the two components is notreadily achieved. As an example, the upper trace of Fig. 4.8 gives a signalderived from (4.23) with the addition of a second mode [4]:

hðtÞ ¼ e�t=10 sinð2p0:27tÞ þ e�t=1 sinð2p0:29tÞ (4:29)

The remaining traces in Fig. 4.8 give the IMFs and residue identified fromthe EMD. Application of the Hilbert analysis yields only the 0.27 Hz compo-nent from IMF1; the 0.29 Hz mode is not apparent. Altering the analysis tosuccessive windows of 10 s duration, or to the 0–50 s window examined earlier,provides no improvement. Application of Prony analysis to the same signal,however, successfully identifies both components of the signal, despite themuchlarger energy associated with the 0.27 Hz mode than the 0.29 Hz mode.

These results provide reason for caution in implementation of the EMD andHilbert analysis for any future real-time control application. In this instance, themissedmode is stable and is damped out quickly. However, it is quite conceivablethat the method could fail to identify a slightly unstable – but initially small –mode until after catastrophic system consequences have occurred.

For a further examination of EMD and Hilbert analysis applied to multi-modal signals, Fig. 4.9 gives a tie-line power flow signal from the classic two-areasystem [25, 30]. Over the full 50 s window, both Prony and Hilbert analysis

Fig. 4.8 Empirical mode decomposition of signal described by (4.29) over the 0–10 s interval


identify a 0.21 Hz mode of damping ratio 0.044 and 0.043, respectively [4],comparing well with a damping ratio of 0.04 identified previously [30]. BothProny analysis and Sanchez-Gasca et al. [30] have identified a second harmonic ofthe 0.21 Hz as being present during the first half of the window in Fig. 4.9. TheHilbert analysis does not identify this mode the same way. Rather, a nonlinearityis identified.

4.6.2 Noise Tolerance

In order for the Hilbert analysis to be applicable for the intended stabilityassessment or correction applications, it must be capable of identifying modalparameters even in the presence of noise. In order to examine this capability,sample test signals are corrupted with noise to test the capabilities of theapproach. Figure 4.10 gives three signals corrupted by white Gaussian noise.

Fig. 4.9 Tie-line flow signal from two-area system [25, 30]. Reproduced, with permission,from [4]. # 2008 IEEE

Fig. 4.10 Three noise-corrupted test signals. The first two signals are based on the single-mode signal (0.27 Hz) of (4.29); the third is based on bimodal signal (0.27 and 0.60 Hz)described by (4.28). Reproduced in part, with permission, from [4]. # 2008 IEEE


The upper andmiddle [4] traces add noise, with signal-to-noise ratios (SNRs) of10 and 2, respectively, to themonomodal signal of Fig. 4.2. The lower trace addsnoise, in an SNR of 5, to the bimodal signal of Fig. 4.7.

Considering the monomodal, SNR¼ 10, signal first, the Hilbert analysis is ableto identify the 0.27Hzmode, albeit as 0.269Hz, over both thewindow shown inFig.4.10 and the first 10 s intervalwithin thatwindow.However, the decay time constantidentified is 6.4 s for the 50 s interval and 9.8 s for the 10 s interval, comparedwith theexpected 10.0 s. Prony analysis provides superior estimates of 10.0 and 10.1 s,respectively [4]. The poor time constant estimated by the Hilbert technique overthe longer window is not overly surprising: toward the end of the window themodehas decayed substantially, enabling the noise to mask the signal more effectively.

The center trace in Fig. 4.10 is essentially a substantially noisier version of theupper trace. As such, it is to be expected that modal analysis is more difficultand results are less likely to be satisfactory. Such an expectation is verified byresults [4]: the Hilbert technique cannot identify the 0.27Hz mode in either thefull 50 s interval or in shorter 10 s intervals. Instead, a spurious 0.10Hz compo-nent dominates, and is coupled with different additional modes which varyaccording to the analysis window. Over the first 10 s interval, the dominantadditional mode is at 0.17Hz, whereas over the full 50 s interval a 0.38Hzcomponent is seen. Whilst the sum and the difference respectively of the twokey frequencies in the windows investigated approximate the actual 0.27Hzmode in the original signal, there is no apparent reason to assume that the0.27Hz signal can be derived from the two identified components ex post. Bycontrast, Prony analysis identifies the 0.27Hz mode directly [4], albeit with apoor time constant estimate (7.2 s) from the 0 s through 10 s interval and animproved assessment (9.3 s) over the 0–50 s interval. Components identified byboth techniques to represent the noise are not of interest for stability assessmentor control purposes; results are therefore not provided for these components.

The bimodal corrupted signal in the lower trace of Fig. 4.10 is repeated forthe 0–10 s interval in Fig. 4.11, which includes the three slowest IMFs identifiedby the EMD applied to this signal. Because the EMD isolates the fastestcomponents first, and noise can be thought of as a high-frequency process, itis logical that the modes of interest are the final IMFs developed. It should benoted that the scales on the vertical axes are not consistent throughout Fig. 4.11;therefore, IMFs 6 and 7 are of more interest than IMF 5, which appears toaccount only for the initial primary oscillation in the signal. Additionally, thecontribution of IMF to a signal reconstruction appears to distort IMF 6 fromits expected shape as a regular decaying sinusoid. By inspection, IMFs 6 and 7correspond, respectively, to the 0.60 and 0.27Hz signal components.

The distorting effect of IMF5 on IMF6 is made more apparent when modalparameter identification is undertaken. Figure 4.12 gives the natural logarithmof the analytic signal magnitudes for IMFs 6 and 7. Based on the theory given inSection 4.3.2, it would be expected that these two plots would approximatestraight lines. Whilst this is a fair description of the plot for IMF7, the same isnot true of the plot for IMF6. The residual from the least-squares estimate of


attenuation factor for IMF6 will be significant; the potential exists for high

residuals to be used to indicate to any control algorithm the considerable

uncertainty in such an estimate.

Fig. 4.12 Corrupted test signal and logarithms of analytic signal magnitudes for two keyIMFs identified. Reproduced in part, with permission, from [4]. # 2008 IEEE

Fig. 4.11 Corrupted bimodal test signal and three most significant IMFs identified.Reproduced in part, with permission, from [4]. # 2008 IEEE


Furthermore, although the Hilbert technique is able to correctly identify themodal frequencies from the IMF6 and IMF7 analytic function phase angles, theattenuation factors identified from Fig. 4.12 are reversed: a time constant of 8 s,and not 10 s, is attributed to the 0.27Hz mode, whereas a time constant of 10 s,and not 8 s, is attributed to the 0.60Hz mode. No explanation is readilyapparent for this phenomenon. However, the Hilbert technique is in thisinstance superior to the Prony analysis. Over the 10 s interval Prony analysiscannot identify the 0.27Hz mode and identifies the 0.60Hz mode of 8 s timeconstant as a 0.57Hz, 2.4 s component. Prony is able to identify both frequen-cies over the longer 50 s interval, but without a realistic damping assessment.

The investigation of Hilbert analysis as applied to the three corrupted testsignals of Fig. 4.10 has produced some interesting anomalies. The shorterwindows appear to favor the Hilbert technique over Prony analysis; bothmethods appear to suffer as the level of distortion increases. It could reasonablybe argued that some form of noise filtering of a signal, prior to application ofeither assessment method, would result in more accurate modal parameterestimation, and ultimately improved stability assessment and correction.

4.6.3 Changes in Underlying System Dynamics

A key property of a transmission network affecting modal identification is thecontinual change in network conditions, as evidenced by variations in loading,generation, and switching conditions. To obtain an indication that a mode changehas occurred, two options present themselves: direct use of themodal identificationtechnique, or use of external signals such as breaker statuses. The latter option is tobe preferred, as the external signals can be expected to be available.

The initial part of the test signal [4] given in Fig. 4.13 is constructed from thesingle-mode signal developed in Section 4.4. At t =16 s, however, the singlemode changes from 0.27Hz to 0.40Hz. The intent is to represent a change in thepower system as might occur in a switching event. Figure 4.14 isolates the

Fig. 4.13 Test signal with mode change at t¼ 16 s. Reproduced in part, with permission, from[4]. # 2008 IEEE


interval 10–20 s, which includes the mode change, and includes an EMD and

Hilbert analysis over this interval. As the first IMF, IMF1, dominates and

accounts for the bulk of the energy in the signal, Hilbert analysis is confined

to this IMF.In particular, the slope of the analytic signal angle plot changes at the t =16 s

breakpoint. Because this slope represents the modal frequency, this effect is

expected. As the Hilbert analysis technique requires a least-squares estimate of

the slope of the analytic signal angle plot, the mode change will not become

apparent until the next 10 s intervalwhen the plot can be expected to track a single

straight line.Whilst techniques for identifying the change in slope are feasible, the

alternative – namely, relying on external signals to indicate a system change and

reset the modal analysis window – is a simpler and more elegant solution.

4.7 Application to Physical Signals

The foregoing discussion has applied EMD to synthetic signals, demonstrating

behavior that might be expected of power systemmeasurements. In this section,

recorded power system measurements are employed, in order to provide real-

world examples of how EMD and Hilbert analysis can be applied.The two signals in Fig. 4.15were recorded during a disturbance in theMexican

Interconnected System [31]. The upper graph gives a bus frequencymeasurement,

Fig. 4.14 Test signal of Fig. 4.13 (upper plot), along with empirical mode decomposition intotwo IMFs (center plot; residue suppressed) and the angle of the analytic signal for IMF1 (lowerplot). Reproduced in part, with permission, from [4]. # 2008 IEEE


the lower a tie-line power flow measurement. Both measurements are represen-tative of the signal types which might be expected to accommodate Hilbertanalysis under disturbance conditions. The disturbance is evident from a cursoryinspection of the two graphs.

Immediate application of the Hilbert analysis to either of these signals,without consideration of windowing effects noted throughout this chapter, isunlikely to yield meaningful estimates of, for example, modal damping para-meters. Again it is stressed that the least-squares estimates in the present Hilbertanalysis are most reliable when the signal under test can be decomposed intoIMFs closely resembling damped sinusoids. The substantial nonlinear effectsapparent in the signals of Fig. 4.15 implies that such a condition is unlikely.

Figure 4.16 gives the bus frequency signal over three successively shorterintervals, making sinusoidal behavior more apparent. Hilbert analysis of theshortest interval, from 126 to 136 s, yields only one mode: 0.23Hz of unstabledamping ratio –0.003. Although Prony analysis identifies a stable (dampingratio 0.001) mode at the same frequency, the discrepancy is minor: the Pronyversion is sufficiently close to instability to warrant correction. Prony alsoidentifies three further modes in the frequency band corresponding to inter-area oscillations, but their amplitudes are sufficiently small as to be likelyartifacts of the decomposition rather than actual modes excited by the physicalsystem.

Figure 4.17 applies a similar analysis to the tie-line power flow signal. Again,neither Hilbert nor Prony analysis on the longest window given is logical: thereis no obvious way for the signal to be decomposed into damped sinusoids. For

Fig. 4.15 Bus frequency and tie-line flow signals recorded during a disturbance in theMexicantransmission network. Reproduced in part, with permission, from [4]. # 2008 IEEE


Fig. 4.16 Recorded bus frequency signal over three successively shorter time intervals.Reproduced, with permission, from [4]. # 2008 IEEE

Fig. 4.17 Recorded tie-line power flow signal over three successively shorter time intervals.Reproduced in part, with permission, from [4]. # 2008 IEEE


illustration purposes, however, such an analysis is attempted on the second

interval. Figure 4.18 gives the three component IMFs for this second

interval.The tie-line flow signal decomposition given in Fig. 4.18 highlights well the

difficulties faced by a future online modal identification technique. Of the three

IMFs given, none bears realistic resemblance to a traditional linear mode. The

theoretical basis for the Hilbert and Prony analysis techniques cannot apply to

this signal. Any attempt to identify modal parameters is not readily meaningful.

Instead, analysis in terms of amplitude-modulated sinusoids might be more

appropriate.However, for the same signal over the shorter interval of 126–136 s, as given

in the lower trace of Fig. 4.17, modal analysis appears to warrant application.

The Hilbert technique yields – in addition to a large pseudo-DC component – a

dominant mode at 0.201Hz with a (stable) time constant of 3.62 s and damping

ratio of 2.14. Two slightly unstable contributions, of sufficiently low relative

magnitude to be suspect, are also identified, at 0.16Hz and at 0.37Hz. Prony

analysis provides very different results: a dominant lightly damped (time con-

stant 67 s) 0.23Hz component, a lightly damped (time constant 3.8 s) 0.63Hz

component, and an unstable 0.41Hz component.For this shorter 10 s long interval, neither set of results is necessarily wrong;

rather, each reflects a different objective in fitting the recorded data to a

theoretical ideal. Further, the substantial changes which occur in the behavior

Fig. 4.18 Decomposition of tie-line flow test signal into (dimensionless) three most significantintrinsic mode functions over the 120–180 s interval. Reproduced, with permission, from [4].# 2008 IEEE


of the signal soon after this short interval render the analysis immaterial:

changes in the system mean that modes excited during the interval between

126 and 136 s are not identical to the modes exhibited a short time later.

4.8 Conclusions

For assessment of modal parameters, the Hilbert analysis offers an alternative

approach to the more standard Prony techniques. Practical considerations such

as window selection and zero padding have been shown to influence the accu-

racy of results significantly.Considerable differences exist in the principles underpinning the twomethods.

Prony analysis is fundamentally directed toward extracting modal parameters

from stationary signals recorded in linear systems. The Hilbert method, in

principle, is applicable to more general conditions and forms a time–frequency

analysis technique. However, seeking numerical values for modal parameters

requires that stationarity is assumed to be an implicit signal property, removing

the flexibility introduced by the Hilbert method.Scope exists for improvement in both the EMD algorithm and the Hilbert

transform calculation. Nevertheless, the Hilbert analysis has demonstrated the

ability to identify modal parameters in many of the test signals examined,

especially where noise conditions are not excessive and modal frequencies are

well separated.

References

1. I. Kamwa, R. Grondin, Y. Hebert, ‘‘Wide-area measurement based stabilizing control oflarge power systems – a decentralized/hierarchical approach,’’ IEEE Trans. Power Syst.,vol. 16, no. 1, pp. 136–153, Feb. 2001.

2. I. Kamwa, L. Gerin-Lajoie, G. Trudel, ‘‘Multi-loop power system stabilizers using wide-area synchronous phasor measurements,’’ Proc. American Control Conference, vol. 5, pp.2963–2967, June 1998.

3. J. F. Hauer, ‘‘Application of Prony analysis to the determination of modal content andequivalent models for measured power system response,’’ IEEE Trans. Power Syst., vol. 6,no. 3, pp. 1062–1068, Aug. 1991.

4. T. J. Browne, V. Vittal, G. T. Heydt, A. R. Messina, ‘‘A comparative assessment of twotechniques for modal identification from power system measurements,’’ IEEE Trans.Power Syst., vol. 23, no. 3, pp. 1408–1415, Aug. 2008.

5. N. E. Huang, Z. She, S. R. Long, M. C.Wu, S. S. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, H.H. Liu, ‘‘The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,’’ Proc. Royal Society of London, vol. 454, pp. 903–995, 1998.

6. G. Rogers, Power System Oscillations. Boston: Kluwer, 2000.7. J. Hauer, D. Trudnowski, G. Rogers, B. Mittelstadt, W. Litzenberger, J. Johnson, ‘‘Keep-

ing an eye on power system dynamics,’’ IEEE Comput. Appl. Power, vol. 10, no. 4, pp.50–54, Oct. 1997.


8. C. E. Grund, J. J. Paserba, J. F. Hauer, S. Nilsson, ‘‘Comparison of Prony and eigen-analysis for power system control design,’’ IEEE Trans. Power Syst., vol. 8, no. 3, pp.964–971, Aug. 1991.

9. BPA/PNNL, Dynamic Systems Identification (DSI) Toolbox, [Online], available: ftp://ftp.bpa.gov/pub/WAMS_Information/.

10. D. Sundararajan, The Discrete Fourier Transform: Theory, Algorithms and Applications.New York: World Scientific Publishing Company, 2001.

11. E. Chu, Discrete and Continuous Fourier Transforms: Analysis, Applications, and FastAlgorithms, Boca Raton, FL: CRC Press, Taylor and Francis Publishing Co., 2008.

12. C. D.McGillem, G. R. Cooper,Continuous and Discrete Signal and System Analysis, NewYork: Prentice Hall, 1995.

13. G. R. Cooper, C. D. McGillem, Methods of Signal and System Analysis, New York:Prentice Hall, 1997.

14. A. V. Oppenheim, A. S. Willsky, S. Hamid, Signals and Systems, New York: Prentice Hall,1996.

15. P. O’Shea, ‘‘The use of sliding spectral windows for parameters estimation in powersystem disturbance monitoring,’’ IEEE Trans. Power Syst., vol. 15, no. 4, pp.1261–1267, Nov. 2000.

16. K. P. Poon, K. C. Lee, ‘‘Analysis of transient stability swings in large interconnectedpower systems by Fourier transformation,’’ IEEE Trans. Power Syst., vol. 3, no. 4, pp.1573–1581, Nov. 2007.

17. S. L. Hahn, Hilbert Transforms in Signal Processing. Boston: Artech House, 1996.18. R.N. Bracewell,The Fourier transform and its applications. 3rd ed. Boston:McGraw-Hill,

2000.19. E. Bedrosian, ‘‘A product theorem for Hilbert transforms,’’ Proc. IEEE, pp. 868–869,

May 1963.20. G. D. Cain, ‘‘Hilbert transform relations for products,’’ Proc. IEEE, pp. 673–674, May

1973.21. J. W. Brown, R. V. Churchill, Complex Variables and Applications. 6th ed. New York:

McGraw-Hill, 1996.22. G. Rilling, P. Flandrin, P. Goncales, ‘‘On empirical mode decomposition and its algo-

rithms,’’ IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (NSIP-03), Grado (Italy), Jun. 8–11, 2003.

23. L. Cohen, Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1995.24. MathWorks, Matlab, http://www.mathworks.com/products/matlab25. P. Kundur, Power System Stability and Control. Palo Alto: EPRI, 1994.26. B. Boashash, ‘‘Estimating and interpreting the instantaneous frequency of a signal – Part

1: Fundamentals,’’ Proc. IEEE, vol. 80, no. 4, pp. 520–538, Apr. 1992.27. R. Ceravolo, ‘‘Use of instantaneous estimators for the evaluation of structural damping,’’

J. Sound and Vibration, vol. 274, no. 1–2, pp. 385–401, Jul. 6, 2004.28. P. Flandrin, Empirical Mode Decomposition, [Online] available: http://perso.ens-lyon.fr/

patrick.flandrin/emd.html, Mar. 2007.29. N. Senroy, S. Suryanarayanan, P. F. Ribeiro, ‘‘An improved Hilbert-Huang method for

analysis of time-varying waveforms in power quality,’’ IEEE Trans. Power Syst., vol. 22,no. 4, pp. 1843–1850, Nov. 2007.

30. J. J. Sanchez-Gasca, V. Vittal, M. J. Gibbard, A. R. Messina, D. J. Vowles, S. Liu,U. D. Annakkage, ‘‘Inclusion of higher order terms for small-signal (modal) analysis:Committee report – Task force on assessing the need to include higher order termsfor small-signal (modal) analysis,’’ IEEE Trans. Power Syst., vol. 20, no. 4, pp.1886–1901, Nov. 2005.

31. A. R. Messina, V. Vittal, D. Ruiz-Vega, G. Enrıquez-Harper, ‘‘Interpretation and visua-lization of wide-area PMU measurements using Hilbert analysis,’’ IEEE Trans. PowerSyst., vol. 21, no. 4, pp. 1763–1771, Nov. 2006.


Chapter 5

A Real-Time Wide-Area Controller

for Mitigating Small-Signal Instability

Jaime Quintero and Vaithianathan (Mani) Venkatasubramanian

Abstract This work proposes a real-time centralized controller for addressingsmall-signal instability-related events in large electric power systems. The pro-posed system ismeant to be a safety net type control strategy that will detect andmitigate small-signal stability phenomena as they emerge in the system. Speci-fically, it will use wide-area monitoring schemes to identify the emergence ofgrowing or undamped oscillations related to inter-area and/or local modes.

The damping levels of the associated inter-area and local oscillatory modeswill be estimated by analyzing predefined sets of signals using multi-Pronymethod. Rules are developed for increasingmulti-Pronymethod’s observabilityand dependability. These rules are applied to simulated signals, but also to realnoisy measurements.

Rules for operating the SVC (static VAR compensator) controls in thedamping enhancement mode and for the application of the multi-Prony algo-rithm on detecting the onset of the oscillations are proposed and tested in a two-area power system and in large-scale simulation example. The controller isshown to be effective on a validated western American large-scale power systemmodel of the August 10, 1996 blackout event.

5.1 Introduction

In modern day power systems, the power flows across distant portions of thetransmission network have been growing steadily to accommodate growingconsumer demands. Moreover, owing to deregulation, the power transfershave also become somewhat unpredictable as dictated by market price fluctua-tions. As a result, the system operation can find itself close to or outside thesecure operating limits under severe contingencies. Recent occurrences of large-scale blackouts all over the world reinforce the significance of developing safety

J. Quintero (*)Faculty of Engineering, Universidad Autonoma de Occidente, Cali-Valle, Colombiae-mail: [email protected]


127

net type control mechanisms, which are specifically designed to handle suchunforeseen operating conditions [1, 2, 3].

Many efforts have beenmade in order to increase the damping of rotor modesunder a wide range of operating conditions, allowing larger steady-state powertransmissions. Classical lead–lag and bang–bang compensators have beendesigned as auxiliary controls on static VAR compensators (SVCs) [4, 5, 6].Also, robust tuning algorithms has been used like the linear quadratic Gaussianfor SVCs in [7] and for thyristor-controlled series capacitors in [8], or H1 and m-synthesis designs in [9, 10], respectively. Because the interactions between unco-ordinated damping controllers can be destabilizing, robust MIMO (multiple-input multiple-output) algorithms have also been developed for coordination ofmultiple power system stabilizers (PSSs) [11, 12, 13], or multiple flexible ACtransmission systems (FACTS) [14], or various PSS units and FACTS together asin [15, 16]. With the recent implementation of synchronized phasor measure-ments, wide-area measurement-based coordinated control techniques have alsobeen used [17]. Specific comparative studies may be found in [18, 19].

Although, all of these techniques achieve very good results on improving themode shape of the system and many achieve robust stability and robust per-formance over a wide range of operating conditions, none of them is specificallydesigned for the emergency condition when the systemmay become small signalunstable. Therefore, the main purpose of this work is to design a reliableemergency control strategy when the system despite all these well-designedcontrols gets close to or goes into small-signal instability.

Themain difference in our approach from the previous research works is thatthe controller as proposed here will initiate targeted damping control actionsonly when it detects the emergence of small-signal instability in the powersystem. Therefore, this controller is aimed at being a safety net type automaticsupervisory mechanism which is aimed toward handling growing or sustainedoscillations from poor damping of inter-area modes under unforeseen highlystressed operating conditions. It is meant to buy the system operators some timeto react, by damping out the oscillations in the interim, while the operators canpossibly initiate other corrective actions such as reducing the tie-line flows and/or initiate load shedding to relieve the system stress.

Our controller is motivated toward preventing oscillatory instability events suchas the August 10, 1996 western blackout that was caused by the negative dampingof the 0.25Hzwestern inter-areamode (Fig. 5.1). Such oscillatory instability eventstypically take a minimum of four to five swings of growing oscillations before theoscillations become large enough to lead to system separation. Therefore, for suchsmall-signal stability-related instability events, there does exist sufficient time for anautomatic controller such as the one postulated inFig. 5.2 to detect the event and totake corrective actions before the oscillations become critical.

On the other hand, Prony analysis [20], has been successfully applied foroff-line signal and model identification in power systems [21, 22, 23], and awide variety of fields [24, 25, 26]. It has been shown that comparing withFourier transform-based techniques, Prony’s method is more powerful when

128 J. Quintero and V. (Mani) Venkatasubramanian

it works [27]. Issues such as system’s nonlinearities [28], power system’s high-

order model [29, 30], noise, low signal-to-noise ratios (SNRs), and defining the

time window or data length [31, 32] have limited its use in real-time applications.

However, the problem suggested by [29] was already solved by one upgrade in

[33], and the high-frequency modes [30] are not an issue in small-signal stability

studies, as the low-frequency nature of inter-area and local modes. Reference

[32] analyzes the issues of sample size selection, data length selection, and noise

in Prony method. However, as seen later in this chapter with simple rules and a

SVC SVC

SVC LOAD

LOAD

LOAD

LOAD

CENTRAL CONTROL

P

PP

Fig. 5.2 Real-time small-signal stability centralized control strategy

Fig. 5.1 Growing oscillations on the California–Oregon tie-lines during the August 10, 1996blackout

5 A Real-Time Wide-Area Controller for Small-Signal Instability 129

smoothing filter, these issues can be strongly reduced in power system’s small-signal stability studies.

In [28], the authors apply the Hilbert spectral analysis obtaining the instan-taneous frequency and damping. In [34], the authors use matrix pencil methodand Hankel total least-squares method to determine also the frequency anddamping of power system modes. They results are comparable to Pronymethod. Moreover, in [34] the authors use the same principles presented inthis work to increase the reliability of the oscillation monitoring system (OMS).

The controller framework is summarized in Fig. 5.2. Rules for applying theright power swing phase compensation by strategically located SVCs are devel-oped. Also, they are demonstrated in the Kundur’s two-area power system [35]and briefly in the Western Systems Coordinating Council (WSCC) large-scalepower system. Practical rules for applying multi-Prony analysis successfully onreal-time basis for detection of low damped oscillations are developed. Off-linerules for selection and use of the appropriate signals for improving the accu-racy, the observability, and the reliability of multi-Prony analysis are obtainedthrough simulations in the two-area power system. Later, these rules are shownwith real noisy data and final online rules are proposed and proved with theWSCC power system for a validated case of the August 10, 1996 blackout [36].

The proposed approach is based on previous works [37, 33, 38] and tested withreal data and a validated simulation case [36]. Reference [37] introduces theapplicability and usefulness of Prony in power system, while [38] improves sub-stantially the accuracy of Prony’s estimations. Also, the idea of grouping signals bydominant modes to increase the observability of certain modes was first suggestedin [33]. We have taken these previous results and developed some more rules toimprove the accuracy, the reliability, and the observability of the multi-Pronyanalysis for the purpose of a real-time controller. The framework has beenextended into an OMS at Washington State University [34] and a prototypeversion ofOMShas been recently implemented at Tennessee ValleyAuthority [39].

Finally, the main objective of this work is on designing a practical real-timewide-area controller, independent of system configuration, to work as an auto-matic emergency controller to improve dynamic security in the small-signalsense. The main contributions are as follows:

� Targets small-signal instability.� Improves Prony’s method accuracy, observability, and reliability.� Analyzes the effects of full SVC PSDC (power swing damping control)

location and compensation on inter-area modes.� Makes use of available resources.

5.2 The Controller

The centralized control strategy introduced in this work (Fig. 5.2) will con-stantly monitor the inter-area modes as well as the local modes presented in thepower system. Data from predetermined signals with relevant content of local


and inter-area modes are sent from phasor measurement units (PMUs) atdifferent control areas and intertie lines to the central control unit via suitablecommunication channels. The central controller analyzes these signals in theform of different sets by using multi-Prony’s method grouped by dominantmodes, in order to estimate the power system’s mode shape. If there is one ormore underdamped inter-area modes, the central unit will issue triggers topreviously identified SVCs located at the affected areas and or along the intertielines. The SVCs will switch from the voltage regulation control mode to the fullPSDCmode until safe damping ratio values �i and safe operating conditions arereached again. Constant monitoring will assure that other local nor inter-areamodes are not weakened during the control action.

The two-area power system presented by Kundur [35] will be used in thiswork (Fig. 5.3). Although this is a small test system, the system has beenproven to be of great value for the study of inter-area oscillations in real powersystems.

5.3 The Central Control Unit

Selected signal measurements, received from dedicated PMUs, grouped bydominant modes are processed by the central control unit using multi-Pronyanalysis [38].

The data suitable for Prony analysis are the ones collected after a small- ormedium-scale disturbance. The initial period of fast and very nonlinear oscilla-tions just after the disturbance will give bad estimations, as Prony’s method is alinear approach.We present techniques for distinguishing between useful versusunreliable Prony estimations of modal data, from real-time measurement data.Also, we show that two or three swings of data are sufficient to obtain a fairapproximation of the mode shape.

P

G1

G2

G3

G4

1

2

3

4

5 6 7 8 9 10 11

Area 1 Area 2

SVC7SVC9C7

C9

SVC8

C8

Fig. 5.3 Two-area power system


It has been demonstrated in [38] that multi-Prony analysis of various signals ismore robust than single Prony analysis. Also, it was noted in [33] that the Pronyaccuracy is improved by having different signals with the same dominant modeanalyzed simultaneously. Moreover, our tests have shown that the observability onthe estimation of inter-area and localmodes is increased by analyzing different sets ofsignals grouped by dominant modes. In extension, for the common case of havingtwoormore inter-area or local areamodeswith almost the same frequency, it is likelythat the Prony’s methodwill have difficulties on differentiating these modes if we tryto extract the information froma single signal or a single set of signals, as shownnext.We have addressed this problem by analyzing the signals in different sets, previouslycharacterized by dominantmodes. That is, the signals with the same dominantmodeare grouped together, and we form a few such different sets to ensure redundancy.

Moreover, it is not unusual in Prony’s analysis to make bad estimations froma signal set at any time. Crosschecking results from different sets of data willmake sure that our estimations are reliable and will provide a certain redun-dancy level. In case of bad estimations, the analysis will be repeated over thosesets using another least-squares solution. If the results are still not consistent,the total estimations would be disregarded, and a complete new estimationprocess should be applied over the next time window.

Based on the estimated mode (eigenvalue) parameters, if the central con-troller determines that one or more inter-area modes are under damped ornegative damped, it will send triggers to corresponding SVCs.

Prony’s method, or multi-Prony’s method, estimates parameters of modesassociated with a signal, or certain set of signals. These are mode frequency fi,amplitude Ai, phase �i, and damping ratio �i. The Prony’s method which wasoriginated in a very early century [20, 37], calculates signal estimations byapproximating in the least-squares sense to a certain set of equally sampleddiscrete data, a linear exponential function of the form

yðtÞ ¼Xm

i¼1Ai expð�itÞ cosð2pfitþ �iÞ for t � 0: (5:1)

This real equation may be written in a complex exponential form as

yðtÞ ¼Xm

i¼1Bi expðlitÞ þ B�i exp l�i t

� �for t � 0 (5:2)

where m is the number of modes, Bi ¼ Ai=2ð Þ expðj�iÞ, li ¼ �i þ j!i, and*

indicates complex conjugate. Moreover, in a simplified manner (5.2) could beexpressed as

yðtÞ ¼Xp

i¼1Bi expðlitÞ for t � 0 (5:3)


where p is the number of the estimated eigenvalues. Without loss of general-ization, consider that for Eqs. (5.2) or (5.3) the Bi’s and the li’s are distinct. LetyðtÞ be N samples evenly spaced by �t such that

yðtkÞ ¼ yðkÞ for k ¼ 0; 1; . . . ;N� 1:

The Prony’s estimated output signal at time tk ¼ k will be

yðkÞ ¼Xp

i¼1Bi expðlik�tÞ for k ¼ 0; 1; . . . ;N� 1:

For convenience, define zi ¼ expðli�tÞ, then

yðkÞ ¼Xp

i¼1Biz

ki for k ¼ 0; 1; . . . ;N� 1: (5:4)

As can be seen, the objective with Prony’s method is to find the values of Bi

and Zi that produce

yðkÞ ¼ yðkÞ for all k: (5:5)

Multi-Prony analysis [38] is a vector–matrix extension of Prony analysis,considering multiple outputs at the same time.

5.3.1 Setting Up the Central Unit – Off-line Rules

Here, our control strategy is applied to the two-area power system as shown inFig. 5.3. A series of empirical rules for selecting the power system variables to bemonitored by the central control unit are developed and are explained throughnumerical simulations.

5.3.1.1 Defining the Time Window

Because Prony analysis is a linear approximation, there is a risk of not gettinggood estimations during the first strong nonlinear oscillations just after thedisturbance has been cleared. When this happens, the estimations obtainedfrom the different signals sets will not agree and the estimation will be disre-garded. The multy-Prony method will then be applied to the next time window.

For the two-area system, Table 5.1 shows relative energy Eri values of inter-area and local modes for two intertie line signals at different time windows. Itcan be seen that the most dominant mode, the inter-area mode, increasesits energy about three times with respect to the second most dominant modewhen the time windowmoves from 4.1–8.1 to 5.0–9.0 s. Also, simulations like inFig. 5.4 show how the SNR of these two most dominant modes is considerably


increased for the same time window change. As a consequence of these results,

in this case a time window after the fifth second will allow more accurate

estimations than a time window just after the disturbance.On the other hand, good parameter estimations were achieved by multi-

Prony analysis with only two complete swings of data. Table 5.1 and Fig. 5.4

show no significant advantages on using a longer time frame in the two-area

system. However, in a real system, due to more complex dynamics, three to four

cycles are a safer option.

5.3.1.2 Grouping Signals by Dominant Modes

Prony or multi-Prony analysis of signals with dominant modes that have very

close frequencies will likely estimates only one equivalent mode. This could be

seen in Table 5.1 on every single Prony estimation for the 5.1Hz local area 1 and

area 2 modes in the two-area power system. Inter-area active power flow P78

and voltage magnitude at Bus 8 V8 are signals that have inter-area and local

Table 5.1 Prony’s Ringdown analysis

Symbol Signal Mode fi �i Eri

Time window: 4.1–8.1 sec

P78 Interarea activepower flow

InterareaLocal Area 1 or 2

0.5091.088

0.00000.0835

1.00000.0189

V8 Bus 8 voltagemagnitude


0.5081.110

�0.00480.0685

1.00000.0069

Time window: 5–9 s



0.5081.088

0.00290.0822

1.00000.0066



0.5051.106

0.00190.0677

1.00000.0028

Time window: 5–13 s



0.5081.087

0.00020.0827

1.00000.0035



0.5091.107

0.00150.0697

1.00000.0015

The two–area power system configuration is as shown in Fig. 5.3, with an SVC at Bus 8operating in voltage regulation form, and with P78 ¼ 330 MW.Dynamic simulation was performed using Extended Transient Midterm Stability Program(ETMSP) from the Electric Power Research Institute (EPRI).One line from Bus 8 to Bus 9 is removed at 4.00 s and reconnected at 4.08 s. Samplingfrequency is 62.5 Hz.Prony signal estimation was calculated using the Ringdown GUI program from BPA/PNNLDynamic System Identification (DSI) Toolbox. Signal mean values were removed.Mode parameters calculated using the Multi–Area Small-Signal Stability Program (MASS)from EPRI are as follows interarea mode, fi ¼ 0.506 and �i ¼ �0.0007; local area 1 Mode,fi ¼ 1.078, and �i ¼ 0.0638; and local area 2 mode, fi ¼ 1.111, and �i ¼ 0.0574.


area mode content of the two local modes. However, Prony analysis is not able

to distinguish between the two local modes.Table 5.2 presents a multi-Prony analysis of two different groups of signals.

Both groups include signals that have different dominant modes. In the first

group, the generator 2 rotor angle �2 has local area 1 mode and inter-area mode

as dominant modes, while the generator 4 rotor angle �4 has local area 2 mode

and inter-areamode as dominantmodes. In the second group, line 6 to 7 current

magnitude I67 has local area 1 mode and inter-area mode as dominant modes,

and, line 9 to 10 current magnitude I910 has local area 2 mode and inter-area

mode as dominant modes. However, multi-Prony analysis at the first two rows

fails on the estimation of the two local modes and gives an equivalent estimation

of only one mode.The estimation problem of modes with very close frequencies may be

addressed by analyzing signals that are predominantly affected by only one

of these modes. For this case, we may see in Table 5.3 how analyzing

signals grouped by local area modes and by inter-area modes give a clear

estimation of each mode. Therefore, observability is improved by classify-

ing signals by dominant modes in groups and analyzing them

independently.

4.1 4.6 5.1 5.6 6.1 6.6 7.1 7.6–4

–3

–2

–1

0

1

2Bus 8 Voltage Magnitude Swings (pu)

Time (sec)

Measured DataProny's Approximationsnr = 2.0

5 5.5 6 6.5 7 7.5 8 8.5–1.5

–1

–0.5

0

0.5

1

1.5

2× 10

–3× 10

–3

× 10–3

Bus 8 Voltage Magnitude Swings (pu)

Time (sec)


5 6 7 8 9 10 11 12–1.5

–1

–0.5

0

0.5

1

1.5

2Bus 8 Voltage Magnitude Swings (pu)

Time (sec)


Fig. 5.4 Prony’s Bus 8 voltage magnitude signal approximation


Table 5.2 Multi–Prony analysis – Case 1 – Signals with different dominant modes combined

Signals Least-square solution Mode fi �i

�8, �2, and �4 Singular value Interarea 0.521 0.0164

decomposition Local area 1 or 2 1.104 0.0710

I78, I67, and I910 Singular value Interarea 0.564 �0.1920decomposition Local area 1 or 2 1.079 �0.0560

I78, I67, and I910 Total least squares Interarea 0.525 0.0143

Local area 1 1.066 0.0332

Local area 2 1.302 0.0948

The two–area power system configuration is as shown in Fig. 5.3, without SVCs, with onlytwo lines from Bus 8 to Bus 9, and with P78 ¼ 400 MW.Dynamic simulation was performed using Extended Transient Midterm Stability Program(ETMSP) from the Electric Power Research Institute (EPRI).One line from Bus 8 to Bus 9 is removed at 4.00 s and reconnected at 4.08 s. Sample data aretaken from 5.00 s to 9.00 s. Sampling frequency is 62.5 Hz.Prony andmulti–Prony signal estimation were calculated using theRingdownGUI program fromBPA/PNNL Dynamic System Identification (DSI) Toolbox. Signal mean values were removed.

Table 5.3 Prony mode parameter estimation – Case 1 – signals grouped by dominant modes

Symbol Signal Ai fi �i Ari

Interarea modea 0.523 0.0111

I78 Interarea current magnitude 0.224 0.524 0.143 1.000

P78 Interarea active power flow 0.156 0.522 0.0177 1.000

Q78 Interarea reactive power flow 0.061 0.520 0.0183 1.000

V6 Bus 6 voltage magnitude 0.015 0.524 0.0176 1.000



�10 Bus 10 voltage angle 2.461 0.522 0.0178 0.761


�6 Bus 6 voltage angle 0.076 0.559 �0.0143 0.068

Local area 1 modea 1.090 0.0694

�1 Generator 1 rotor angle 0.163 1.084 0.1113 0.145

I67 Line 6 to 7 current magnitude 0.009 1.081 0.0558 0.033



V1 Bus 1 voltage magnitude 2e–4 1.041 0.0529 0.027


V6 Bus 6 voltage magnitude 1e–4 1.075 �0.0032 0.007

Local area 2 modea 1.118 0.0704

I910 Line 9 to 10 current magnitude 0.011 1.060 0.0356 0.071






V4 Bus 4 voltage magnitude 3e–4 1.226 0.1551 0.004aValues calculated using theMulti–Area Small-Signal Stability Program (MASS) from EPRI.Same signal measurements and programs as Table 5.2


5.3.1.3 Using Mode Content: Group Using Ai and Ari

It is clear from Table 5.3 that signals with the greatest dominant mode contentare the more likely to give the best estimations. However, even with thesesignals, it is still possible to have bad estimations when using Prony analysis.It has been demonstrated in [38] that using multi-Prony will increase theaccuracy of the estimations, but still due to the nonlinear nature of powersystem signals we may have bad estimations. Our goal with this rule is makingas many subgroups as we want in order to increase the redundancy level in ourestimations divided by dominant modes.

When using Prony’s method, amplitude Ai is a good approximation of modei content in the processed signal. However, signals have very different valueranges even in per unit basis. Therefore, it is also important considering therelative amplitude Ari when judging the real mode content on a signal. In Table5.3, signals like voltage magnitudes V1,V2,V4, andV6, and bus voltage angle �6are between the signals with the lowest Ai and Ari values at each group and arethe ones with the largest estimation differences with respect to the dampingratio calculated with the Multi-Area Small-Signal Stability (MASS) Program.

Combining signals that have high Ai and Ari values with fewer signals thathave lowAi orAri values may improve the general estimation, as it is the case forthe group formed by �10, �8, and �6, and the group formed by �2, I67, and V6. Inmost cases, relative amplitude Ari is increased when using multi-Prony analysisof signals with similar dominant modes, as seen in Table 5.4. On the other hand,multi-Prony’s method for analyzing signal sets where all signals have very lowAi and Ari values does not improve the estimation, as it is the case for the groupof voltage magnitudes V1, V2, and V6 in Table 5.4.

5.3.1.4 Validating the Groups

With this rule we want to select the final subgroups that are going to be used formonitoring the mode shape of our power system in real time. While we want tohave a good redundancy level on the estimation of each mode, we also want tomake sure that the chosen subgroups has the ability of providing good estima-tions of the excited modes when a small or medium disturbance occurs in thepower system.

Because Prony or multi-Prony analysis is a linear approach applied to non-linear signals, we should expect certain margin of error in estimations fromdifferent signals during the same time window. Therefore, an acceptable errormargin � should be defined in order to admit or not the estimations. In Table5.4, for example, an accuracy level � of�0.01 with respect to the mode dampingratio values found with the MASS Programmay be selected. Consequently, theestimation from the group of voltage magnitudes V1, V2, and V6 will beconsidered as a bad estimation and will be eliminated as a subgroup.

Cross-checking damping ratio values obtained from subgroups selectedaccording with rules 1, 2, and 3, is another way for determining the finalsubgroups if no actual mode value is known. According to our results in the


two-area and the WSCC power systems, a difference of �2% damping ratio

value between the minimum and the maximum estimation for one damping ratio

may be acceptable. Bad groups, or groups with signals with weak contain of

certain mode will be discarded as they are detected with the cross-checking

process.Nevertheless, it is important to keep in mind that good data that fulfill the

best conditions of the above rules may give bad results. This can be seen in

Table 5.2 for the second group of signals I78, I67, and I910, where in spite of being

part of the signals with higher amplitudes and relative amplitudes at each group

in Table 5.3, we do get completely unacceptable results at the first attempt of

solving the least-squares problem. The results have shown that changing the

least-squares solution method originally used for the bad estimation will

Table 5.4 Multi–Prony mode parameter estimation – Case 1 – signals grouped by dominantmodes

Symbol Signal Ai fi �i Ari

Interarea modea 0.523 0.0111

V6, 0.015 1.000

V8, 0.018 0.522 0.0199 1.000

and V10 0.010 1.000

�10, 2.488 0.202

�8, 1.534 0.522 0.0191 0.124

and p78 0.152 1.000

�10, 2.454 1.000

�8, 1.490 0.525 0.0189 0.815

and �6 0.175 0.095

Local area 1 mode a 1.090 0.0694

V1, 9e–5 0.009

V2 7e–5 1.058 �0.0107 0.005

and V6 9e–5 0.006

�2, 0.173 0.117

I 67, 0.011 1.082 0.0765 0.040

and �6 0.040 0.028

�2, 0.173 0.145

I67, 0.011 1.083 0.0765 0.041

and V6 2e–4 0.012

Local area 2 mode a 1.118 0.0704

�4, 0.153 1.120 0.0718 0.056

and V3 3e–4 0.046

I910, 0.012 0.085

�3, 0.177 1.077 0.1048 0.057

and V10 7e–4 0.074

I910, 0.018 0.122

�3, 0.086 1.140 0.0705 0.030

and �10 0.021 0.009

Same signal measurements and programs as Table 5.2.


contribute to solve the problem for these cases as shown in Table 5.2, and later

in the WSCC application example.

5.3.2 Monitoring and Control – Online Rules

Here, measured data from the WSCC power system will be used in order to

illustrate the online rules. Signals from BCH-Boundary, BCH-Custer, Malin-

Round Mountain, Grand Coulee, and Tacoma were selected from a set of

measurements provided from Bonneville Power Administration [33]. These

measurements were taken during the events previous to the August 10, 1996

blackout (Fig. 5.1).According with the above off-line rules, two subgroups were created in order

to monitor the California–Oregon inter-area mode. The online rules will be

described, for the time window shown in Fig. 5.5 after the Keeler–Allston line

opens.

5.3.2.1 Activation Deactivation Criteria

Multi-Prony monitoring will be triggered only after a small or medium distur-

bance has been detected. If not, triggers are issued and if no any other distur-

bance has occurred, the monitoring will be cancelled after 7–14 swings. Also, if

triggers have been issued and based on the monitoring damping ratios are going

wrong, the controller will be deactivated.

–340.3 –330.3 –320.3 –310.3 –300.31320

1330

1340

1350

1360

1370

1380

1390

1400

Time in Seconds

Filtered Malin-Round Mountain #1 MW Data

Fig. 5.5 First oscillatory stage. Case 1


5.3.2.2 Validating Criteria

Cross-checking results from the same time window and for consecutive timeintervals will increase reliability in the online estimations. Therefore, two cri-teria are developed in order to validate the Prony estimations:

1. A Prony damping estimate is considered valid only if the net variation of thedamping estimates from the different subgroups is less than a threshold, say�1. Then, the average damping estimate for the current time window isdefined as the average of the estimates from the subgroups.

2. When we consider the estimates from a consecutive set of moving timewindows, the damping estimates are used for control purposes: (a) if eachof the damping estimates in the different time windows is valid (i.e., they allobey the �1 rule) and (b) if the net variation of the average dampingestimate from the different time windows is less than a predefined thresholdsay �2.

These rules are explained through Table 5.5. Here �1 has been set to 2%. Asshown, only window five is out of the threshold and is considered not valid. Thiscan be understood because the Keeler–Allston line opening is a small distur-bance and Prony’s method is very accurate when there are no strong nonlinea-rities presented in the signals. On the other hand, for control purposes, considerthe estimates of two consecutive time windows and �2 set at 1.5%. It may beseen in Table 5.5 that the difference between the average damping ratio esti-mates of window 1 (5.3%) and window 2 (4.95%) are within the acceptedmargin difference of less than 1.5% (5.3 – 4.95% ¼ 0.35%). However, thisdifference is out of the threshold for windows 2 and 3 (4.95 – 1.35% ¼ 3.6%)and windows 3 and 4. Window 5 is not a valid estimate and windows 6 and 7agree within the threshold.

Finally, if a cutoff value of 2% damping ratio may issue triggers, theywould be issued only after the third and fourth windows agree. If the cutoff

Table 5.5 Multi–Prony COI mode damping–Case 1a – Estimation in percent

Time windowbPmrl, Pbound,

and ftacoma

Pcuster, Vmalin,and Pcoulee Average Validity

�330.3 to �318.3 5.0 5.6 5.3 Yes

�328.3 to �316.3 4.5 5.4 4.95 Yes

�326.3 to �314.3 0.4 2.3 1.35 Yes

�324.3 to �312.3 �0.4 0.0 �0.2 Yes

�322.3 to �310.3 0.3 5.0 2.65 No

�320.3 to �308.3 1.6 1.0 1.3 Yes

�318.3 to �306.3 3.0 2.3 2.65 YesaMeasured data taken just after Keeler-Allston line trips at �332.3 s.bTime with respect to Ross–Lexington line trip.Sampling frequency is 20 samples/s. Prony estimation was calculated using the RingdownGUI program from BPA/PNNLDynamic System Identification (DSI) Toolbox. Signal meanvalues were removed. A smoothing filter with 1 Hz cutoff frequency was used.


value is 3%, triggers may be issued after the fourth window or after theseventh window.

5.3.2.3 Selecting the SVC

An SVC priority list corresponding to each inter-area mode may be built usingthe criteria demonstrated next in Section 5.4.4.1 for the SVC local units. Thecentral unit will make sure that the selected SVC will be available and workingunder the expected conditions by checking the actual mode content in thesignals coming from the SVC location.

5.3.2.4 Determining the Phase Compensation

Additional to the trigger, the central unit will determine the type of phasecompensation to be applied for the SVC with respect to the direction of thecorresponding major inter-area power flow. This rule is demonstrated next inSection 5.4.4.2.

5.4 The SVC Local Unit

This control strategy assumes that some of the SVC units have been earmarkedfor potential operation under the proposed scheme, and that these SVCs arenormally operating in the voltage regulation mode. Specifically, when the SVCreceives the external trigger from the central control unit, the SVCwill switch tofull PSDC form.

The SVC mode switch occurs only when the oscillations are already so largeas to threaten the integrity of the large system. Under such highly stressedconditions, we assume that the SVCs can be switched to the full dampingcontrol mode to quickly damp out the otherwise growing (or sustained) oscilla-tions in the system. The damping control at the SVCs will be designed by usingthe active power flow from a specific intertie line as the control input signal.Details will be discussed next.

As we will show in Section 5.4.3, the damping control actions of an SVCworking on full PSDC located at the sending or at the receiving sides of anintertie line are more effective than those of an SVC located close to the middleof the intertie line. Additionally, as seen in our results, an SVC located at theactive power sending area offers the most effective location for full PSDC whenthe respective intertie active power flow is used as the control input for the SVCdamping control.

In this work, we are interested on making an analytic study of the effects ofthe SVC location with respect to an intertie line, on the associated inter-areamode. In order to keep a low-order model for our analysis purposes, generatorsare modeled for this part using a classical representation or swing equation


(5.6), which includes only two state variables per generator, rotor angle �, androtor speed deviation �!. Because inter-area modes are mainly associatedwith these two state generator variables, inter-area modes are still presentedand their characteristics are preserved when using this simple classical model foranalytical study. Later, we show that the design rules that are derived from theclassical model appear to be valid in simulations of detailedmodels as well as fora large power system model.

5.4.1 The Classical Power System Model

The swing equation for a generator may be written as

_� ¼ �!

� _! ¼ !s

2HPm � Pe � KD

�!

!s

� �� (5:6)

where �! ¼ !� !s and !s is the synchronous speed, Pm is the shaft mechanicalpower in pu, Pe is the air-gap power in pu, and KD is a damping factor thatincludes mechanical and electrical damping components.

The classical representation can be derived from detailed power systemmodels bymaking the assumptions stated at [40].With generators characterizedby the transient reactance machine model and loads by constant impedances,the interconnected transmission network represented by

I ¼ YV (5:7)

may be portioned as

IG

0

� �¼

YGG YGR

YRG YRR

� �VG

VR

� �(5:8)

where subscript G relates to all internal generators nodes and subscript Rcomprises the remaining nodes with zero injection current. Reducing Eq. (5.8)by eliminating VR, we obtain

IG ¼ YGG � YGRY�1RRYRG

� �VG: (5:9)

Defining YGEN as

YGEN ¼ YGG � YGRY�1RRYRG (5:10)


we may write

IG ¼ YGENVG (5:11)

or

~IGi ¼Xng

j¼1

~YGENij~VGj for i ¼ 1; . . . ; ng (5:12)

where ng is the number of generators.From (5.12) and considering that air-gap power Pe is equal to generator

active output power P, neglecting stator resistance we may write that

Pi ¼Xng

j¼1EGiEGjYGENij cos �i � �j � ’GENij

� �for i ¼ 1; . . . ; ng: (5:13)

Initial values may be found from a pre-transient power flow load study.

5.4.2 The Linearized State–Space Classical Modelfor a Reduced Two-Area Power System

Here we reduce the original power system of four generators presented byKundur, as in Fig. 5.3, to only two generators. The compensator model usedfor the SVC in full PSDC operating mode, in this modeling case, is given onlyfor a low-pass filter followed by a phase compensation filter. Then, the PSDCmodel equations are

_eLP ¼1

T2LPP79REF � P79 � eLPð Þ

_BSVC ¼1

T2K eLP þ T1 _eLPð Þ � BSVC½ �

(5:14)

where P79 is the inter-area active power flow

P79 ¼V7V9

X79sinð�7 � �9Þ (5:15)

~V7; ~V9 bus voltages may be obtained approximately by

~V7 ffi E1ff�1 1� ~YGEN11 þ ~YGEN12

� jX0d1 þ jXT1

þ jX56 þ jX67

� �h i(5:16)


~V9 ffi E2ff�2 1� ~YGEN22 þ ~YGEN21

� jX0d2 þ jXT2

þ jX11�10 þ jX10�9� �h i

:

The nonlinear state–space model for the reduced two-area power system isdefined by the above equations, and the state–space variables are

x ¼ �1 �!1 �2 �!2 eLP BSVC½ �T:

An equilibrium point for the generated nonlinear state–space model can becalculated by

�10 ¼ cos�1Pm1 � YGEN110E

21 cosð’GEN110Þ

YGEN120E1E2

� �þ �20 þ ’GEN120

�!10 ¼ 0

�20 ¼ cos�1Pm2 � YGEN220E

22 cos ’GEN220ð Þ

YGEN210E2E1

� �þ �10 þ ’GEN210

�!20 ¼ 0

eLP0¼ P79REF �

V70V90

X79sin �70 � �90ð Þ

� �

BSVC0¼ KeLP0

:

(5:17)

Assuming that the equilibrium point stays away from the singularity of thenetwork equations, the linearized state–space model may be written as

� _x ¼ J�x: (5:18)

Therefore, the Jacobian matrix J around the equilibrium point (5.17) for thistwo-area power system, may be written as

J ¼

0 1 0 0 0 0

J21 � KD1

2H1J23 0 0 a

0 0 0 1 0 0

J41 0 J43 � KD2

2H20 b

J51 0 J53 0 � 1T2LP

c

J61 0 J63 0 d e

26666666664

37777777775

(5:19)

where the following J parameters depend mostly on the initial values and thepower system configuration. If we neglect variations in voltage magnitudes,these parameters are constant for the different PSDC phase compensatorstested


J21 ¼!sYGEN120E1E2

2H1sin �10 � �20 � ’GEN120ð Þ

J23 ¼ �J21

J41 ¼ �!sYGEN210E1E2

2H2sin �20 � �10 � ’GEN210ð Þ

J43 ¼ �J41:

(5:20)

The next J values are affected by the low-pass filter design included in thePSDC function, but independent of the other PSDC variables:

J51 ¼ �1

T2LPX79

@V7

@�1V90 sin �70 � �90ð Þ þ @�7

@�1V70V90 cos �70 � �90ð Þ

� �

J53 ¼ �1

T2LPX79

@V9

@�2V70 sin �70 � �90ð Þ � @�9

@�2V70V90 cos �70 � �90ð Þ

� �:

(5:21)

For convenience in analysis, we set the gain K as K ¼ T2=T1 so that thecompensator transfer function is of the pole-zero formGcðsÞ ¼ ðsþ 1=T1Þ=ðsþ 1=T2Þ. We study the impact of the phase-lead andphase-lag provided by the compensator on the damping of the inter-areamode, as they were the dominant contributors to the overall effectivenessof the compensator. The compensator gain has a lesser impact on thedamping as compared with the phase, as we have been able to test throughour simulations. With the gain K ¼ T2=T1, the parameters satisfy thefollowing property:

J61 ¼KT1

T2J51 ¼ J51

J63 ¼KT1

T2J53 ¼ J53:

(5:22)

On the other hand, the following parameters are going to change accordingto the SVC location:

a ¼ !s

2H1

� @YGEN11

@BSVCE21 cos ’GEN110ð Þ

þ @’GEN11

@BSVCYGEN110E

21 sin ’GEN110ð Þ

� @YGEN12

@BSVCE1E2 cos �10 � �20 � ’GEN120ð Þ

� @’GEN12

@BSVCYGEN120E1E2 sin �10 � �20 � ’GEN120ð Þ

2

666666666664

3

777777777775

(5:23)


b ¼ !s

2H2

� @YGEN22

@BSVCE22 cos ’GEN220ð Þ

þ @’GEN22

@BSVCYGEN220E

22 sin ’GEN220ð Þ

� @YGEN21

@BSVCE2E1 cos �20 � �10 � ’GEN210ð Þ

� @’GEN21

@BSVCYGEN210E2E1 sin �20 � �10 � ’GEN210ð Þ

2

666666666664

3

777777777775

(5:24)

c ¼ � 1

T2LPX79

@V7

@BSVCV90 sin �70 � �90ð Þ

þ @V9

@BSVCV70 sin �70 � �90ð Þ

þ @�7@BSVC

V70V90 cos �70 � �90ð Þ

� @�9@BSVC

V70V90 cos �70 � �90ð Þ

2666666666664

3777777777775

: (5:25)

Finally, it is clear that the last two parameters d and e are affected mostly bythe values of the PSDC design

d ¼ K

T21� T1

T2LP

� �(5:26)

e ¼ KT1

T2c� 1

T2:

Next, we will show how the parameters J21, J23, J41, J43, J51, J53, J61, and J63remain constant while the parameters a, b, and c vary when the SVC locationchanges. We will also see that the d and e values are modified mostly by thePSDC phase compensation type. By analyzing the parameters a, b, and c, wecan then derive rules on the placement of the SVC. Similarly, by studying theproperties of the parameters d and e, we can derive rules on the lead versus lagdesign principle of the PSDC compensator.

5.4.3 Numerical Results

Figure 5.6 shows the Jacobianmatrices JSVCB7_LP, JSVCB8_LP, and JSVCB9_LP of thereduced two-area power system above with an SVC located at Bus 7, Bus 8, andBus 9, respectively. The SVCs are in the PSDCmode and the controller includes alow-pass filter anda zero phase compensator (5.14).Note that as described in (5.19)and (5.20), the submatrix related with the original system without SVCs does notchange with the addition of the SVCs. This is because the change in the admittanceYGEN matrix is too small when an SVC is added. Therefore, values included into


this submatrix do not have any significant effect in the mode shape variation that

takes place when an SVC working in full PSDC is included into the system.On the other hand, the values of parameters a, b, and c change with respect to

the SVC location at any of the two areas or along the intertie line. Table 5.6 shows

these parameter value variations when an SVC is located in Bus 7, Bus 8, and Bus

9. From (5.23), (5.24), and (5.25) and the approximations we havemade, changes

in a, b, and c parameter values, when moving from one side to the other, depend

mostly on the changes suffered by admittance sensitivities. Between them, the

parameter c value is the one that experienced the smaller changes and, therefore,

has little effect on the observed differences in the inter-area mode damping.Admittance sensitivities for an SVCatBus 7, Bus 8, andBus 9were found for a

positive step in its BSVC bus output value (5.14). These sensitivities are shown in

Fig. 5.7, as well as the variation suffered by each of the four right terms in a and b

formulations, (5.23) and (5.24). If we compare corresponding sensitivities in a

and b terms, sensitivities @’GEN11=@BSVC and @’GEN22=@BSVC are the

only sensitivity factors with opposite slopes when moving the SVC from the

sending area to the receiving area, through the intertie. Therefore, they are the

main cause of the value difference between second right terms of a and b expres-

sions, called here a2 and b2, and ultimately for the value difference between terms

a and b. We may say here, that the difference between the a and b values is a

deterministic characteristic of the SVC location in the PSDC mode.

Fig. 5.6 The J matrix with SVC compensation at Bus 7, Bus 8, and Bus 9. The SVC controlincludes a low-pass filter and a 08 phase compensator. KD=0.5. Matlab model


Table5.6

Two-areapower

system

a. .(M

atlabresults)

SVClocation

PSDCcompensation(lowpass

filter

+)

ab

cd

el 1

,2f

�

NoSVC

––

––

––

�0.020�7.214i

1.148

0.003

Bus7

LPfilter

only

�6.847�3.418�0.233

0.033

�6.516�0.378�6.914i

1.100

0.055

LPfilter

+158lag

2.083

�4.995�0.580�6.995i

1.113

0.083

LPfilter

+308lag

5.514

�3.567�0.817�7.230i

1.151

0.112

LPfilter

+458lag

10.416

�2.614�0.989�7.559i

1.203

0.130

LPfilter

+158lead

�1.488

�8.567�0.214�6.913i

1.100

0.031

LPfilter

+308lead

�2.916�11.998�0.084�6.965i

1.108

0.012

LPfilter

+458lead

�3.869�16.900�0.020�7.029i

1.119

0.003

Bus8

LPfilter

only

�5.406�5.967�0.411

0.033

�6.694

0.031�7.262i

1.156�0.004

LPfilter

+158lag

2.083

�5.173

0.058�7.257i

1.155�0.008

LPfilter

+308lag

5.514

�3.745

0.100�7.242i

1.152�0.014

LPfilter

+458lag

10.416

�2.792

0.152�7.217i

1.149�0.021

LPfilter

+158lead

�1.488

�8.745

0.009+

7.261i

1.156�0.001

LPfilter

+308lead

�2.916�12.176�0.009�7.254i

1.154

0.001

LPfilter

+458lead

�3.869�17.078�0.019�7.244i

1.153

0.003

Bus9

LPfilter

only

�3.263�7.429�0.580

0.033

�6.863

0.322�7.554i

1.202�0.043

LPfilter

+158lag

2.083

�5.342

0.489�7.54li

1.200�0.065

LPfilter

+308lag

5.514

�3.913

0.743�7.503i

1.194�0.098

LPfilter

+458lag

10.416

�2.961

1.059�7.474i

1.189�0.140

LPfilter

+158lead

�1.488

�8.913

0.185�7.542i

1.200�0.024

LPfilter

+308lead

�2.916�12.345

0.059�7.495i

1.193�0.008

LPfilter

+458lead

�3.869�17.247�0.014�7.431i

1.183

0.002

aThisisareducedtw

o-areapower

system

,withonly

twogenerators,theSVC

consistsonly

ofalow-pass

filter

andaleadorlagfilter,KD=

0.5,

p78=

90MW.

Resultswerecalculatedusingadesigned

Matlabmodel.


In this particular Matlab model, the value difference term, b – a, is positivewhen the SVC is at the sending side, almost zero if the SVC is located close tothe center of the intertie line, and negative when the SVC is at the receiving side.Also, we note that the greater the absolute value difference between a and b, thegreater the stabilizing or destabilizing capacity of the SVC in the full PSDCmode. This is based on several simulations and the eigenvalue analysis. In Table5.6, this phenomenon may be observed when comparing the stabilizing anddestabilizing effects of the SVCs in Bus 7 and Bus 9 with respect to an SVC inBus 8. Simulations using the MASS software package presented in [41] confirmthis SVC location relation with values a and b.

Additionally, we have parameters d and e (5.26) that depend on the SVCscontroller specifications and on the parameter c, which does not changemuch inthis study. Therefore, as seen in Table 5.6, parameters d and e have similarvalues at Bus 7, Bus 8, and Bus 9, as they vary mostly with the type of PSDCcompensation used and not with the SVC location. Note that in order to obtaincomparative results and to avoid the gain K effect in d and e values, we setK ¼ T2=T1. As may be seen in (5.26), withK ¼ T2=T1, the parameter d dependsmostly onT1 while the parameter e relates mostly toT2; the magnitudes of d ande vary inversely proportional to T1 and T2, respectively.

5.4.4 SVC Rules

Based on the above analysis and numerical results, we state that parameters aand b values have the biggest impact on damping inter-area modes with an SVCin the full PSDC operation. That is, the location of the SVC with respect to theintertie active power flow direction is the most important factor to consider forimplementing this type of control. Then, based on the SVC location, the inter-area active power phase compensation should be decided, for suitably settingthe d and e values.

7 8 9–0.05

0

0.05

0.1

0.15

0.2

BUS

Sensitivities for Parameters a and b

7 8 9–0.18

–0.14

–0.1

–0.06

–0.02

0.02Term Components for a and b

BUS

svc

GEN

B∂∂ϕ 11

svc

GEN

B∂∂ϕ 22

srasrb

Fig. 5.7 Sensitivities and term component variations in a and b


5.4.4.1 Selecting the SVC Location

The SVC priority list corresponding to each inter-area mode may be built usingthe criteria shown in Section 5.4.3 for the SVC local units. These criteria may bewritten as follows:

1. An SVC at the sending inter-area active power flow area has more steeringpower than one located at the receiving area.

2. Between SVCs at the same areas, the greater the corresponding SVCs b –aabsolute value difference the greater the steering power.

5.4.4.2 Determining the Phase Compensation

From Table 5.6, it is clear that an intensification of the intertie active power-based phase-lag compensation, or the d magnitude increment, is particularlybeneficial on increasing the inter-area mode damping for an SVC located at thesending area, as Bus 7, while it is disadvantageous when the SVC is located atthe receiving area at Bus 9. However, it is important to note that increasing dtoo much in Bus 7, could lead to another pole close to the imaginary axis tomove to the right-hand side attracted by a zero on this side. Alternatively, theintensification of the phase-lead compensation has a good effect on dampingthe inter-area mode, when the SVC is located at the receiving area. An oppositeeffect results, when it is located on the sending side. For an SVC close to themiddle of the intertie line at Bus 8, the effects are much lower.

This rule may be expressed as follows:

1. An SVC in full PSDC is operated as a phase-lag compensator if the SVC is atthe sending side for the active power flow of the major intertie line that isbeing used as the control input.

2. An SVC in full PSDC is operated as a phase-lead compensator if the SVC isat the receiving side for the active power flow of the major intertie line that isbeing used as the control input.

5.5 WSCC Power System Example

Measured data taken at BPA Dittmer Control Center during the events pre-vious to the August 10, 1996 WSCC Power System breakup [33] are analyzedhere in order to illustrated our approach. Also, a validated model [36] of thisphenomenon is used to simulate our control strategy.

FromFig. 5.1 and [36], two different oscillatory stagesmay be identified. Thefirst oscillatory stage initiates when the Keeler–Allston line trips. Ten secondslater, the multi-Prony analysis on recorded signals shows a center of inertia(COI) mode with a damping ratio around 1%. After this disturbance, measuredata are analyzed using the multi-Prony approach presented in this work. Thewhole analyzed time window is presented in Fig. 5.5.


The improved accuracy on the estimations of using multi-Prony by groups

over the single Prony, may be seen in Tables 5.7 and 5.8 with real data.

Estimations in Table 5.7 give a wide range of possible values for the COImode damping ratio for that time window. As seen, the estimated damping

ratio values have differences of more than 4%. Table 5.8 shows how groupingmeasurements according to the rules developed in Section 5.3.1 makes the

estimated damping ratio differences less than 2% for the same time window.Note that, according with the analysis developed in Section 5.4.4, as an SVC

located at Maple Valley at that time would be at the active power sending side,

the COI active power phase compensation should be lagging and this SVC is thefirst choice in the priority list. On the other hand, an SVC located in Adelanto

would be at the receiving active power side, the COI active power phasecompensation should be leading at that bus and it is our second choice.

Table 5.7 Prony mode parameter estimation – WSCC power systems – (Measured signalswith COI dominant mode)

Symbol Measured signalsa Ai fi �i Ari

COI mode

Pmr1 Malin-Round Mountain #1 21.57 0.259 0.0452 1.000

Pcuster BCH-Custer 23.08 0.263 0.0125 1.000

Pbound BCH-Boundary 8.372 0.259 0.0103 1.000

Pcoulee Grand Coulee Generation 4.041 0.244 �0.0152 1.000

VMalin Malin bus voltage magnitude 1.785 0.259 0.0019 1.000

fTacoma Tacoma bus frequency 0.006 0.233 �0.0028 0.885aTime window is from �320.3 to �308.3 s, Fig. 5.5. Ten seconds, after Keeler–Allston linetrips. Sampling frequency is 20 samples/s.Prony estimation was calculated using the Ringdown GUI program from BPA/PNNLDynamic System Identification (DSI) Toolbox. Signalmean values were removed.A smoothingfilter with 1 Hz cutoff frequency was used.

Table 5.8 Multi-Prony mode parameter estimation – WSCC power system – (Groupedmeasured signals)

Symbol Ai fi �i Ari

COI mode

Pmr1,

Pbound,and fTacoma

16.62

8.7080.006

0.254 0.0158 1.000

0.6010.830

Pcuster,

Vmalin,and PCoulee

(SVD) (No Good Estimations)

Pcuster,

Vmalin,and PCoulee

(QR Factorization) (No Good Estimations)

Pcuster,Vmalin,and PCoulee

(TLS) 24.751.8894.548

0.256 0.0097 1.0001.0000.533

Same signal measurements and programs as Table 5.7.


If due to historical or technical reasons, the full PSDC compensation is not

activated during the first oscillatory stage, the second stage which starts at

time zero in our plots after the Ross–Lexington line trips (Fig. 5.1) presents a

worse oscillatory scenario. At that time, a sequential tripping of generators

and governor actions began, introducing a stronger oscillatory behavior

and also stronger nonlinearities in the signals. A closer look is presented in

Fig. 5.8.Using the same measurements from BPA, we tried to estimate the damping

of these oscillations applying our multi-Prony approach by groups through

the time frame defined by the dashed lines in Fig. 5.8. Results are presented in

Table 5.9.

–10 0 10 20 30 401240

1260

1280

1300

1320

1340

1360

Time in Seconds

Filtered Malin-Round Mountain #1 MW Data

Fig. 5.8 Second oscilatory stage. Case 2

Table 5.9 Multi-Prony COI mode damping estimation in precent–Case 2a

Timewindow (0)

Pmrl, PBound

and fTacoma

PCuster, VMalin

and PCoulee Average Validity

2.0–14.0 �2.7 3.6 0.45 No

4.0–16.0 �1.3 �3.6 �2.45 No

6.0–18.0 1.0 1.1 1.0 Yes

8.0–20.0 �0.9 0.2 �0.35 Yes

10.0–22.0 0.0 0.8 0.4 Yes

12.0–24.0 1.2 1.6 1.4 Yes

14.0–26.0 3.5 1.9 2.7 Yes

16.0–28.0 1.1 0.8 0.95 Yes

18.0–30.0 �1.3 2.2 0.45 No

20.0–32.0 �2.1 �3.4 �2.75 Yes

22.0–34.0 �3.4 �3.1 �3.25 YesaMeasured data taken just after Ross–Lexington line trips at 0.0s.bTime with respect to Ross–Lexington line trip.Sampling frequency is 20 samples/s. Prony estimation was calculated using theRingdown GUI program from BPA/PNNL Dynamic System Identification(DSI) Toolbox. Signalmean values were removed.A smoothing filter with 1Hzcutoff frequency was used.


Due to the strong nonlinearities and with an �1 equal to 2%, the first

two time windows estimations are inaccurate and they are neglected. Later,

our results show that the COI mode is going lightly damped less than 1%

damping ratio. If for control purposes we consider the estimates of two

consecutive moving time windows, �2 equal to 1.5%, and a cutoff value of

1% damping ratio, the central controller will issue triggers to the assigned

SVC after 20 s. If we use three consecutive windows and �2 equal to 1.5%,

then the triggers would be issued at 22 s for the same cutoff damping

of 1%.However, if we want to be very conservative and wait for the mode going

really unstable we may set the cutoff value at say –2% damping ratio. Then, in

Table 5.9 considering only two consecutive time windows the PSDC is triggered

at 35 s. Figure 5.9 represents the simulated behavior for this case, provide the

right phase compensation as described in Section 5.4.4 for an SVC located at

Maple Valley, the active power sending area, and for an SVC located at

Adelanto, the active power-receiving area.Finally, Table 5.10 presents the frequency and damping ratios estimated

from a time window from 85 to 97 s for triggers issued at –15 s and at 35 s,

when the right compensation control actions are taken and when the wrong

compensation control actions are applied. Note that, as we showed in Section

5.4.3, an SVC at the sending side has more steering power than one at the

receiving side, so it could add more damping but also could be more destabiliz-

ing when right phase compensation is not applied.

0 10 20 30 40 50 60 70 80 90 1003800

4000

4200

4400

4600

4800

5000

0 10 20 30 40 50 60 70 80 90 1003800

4000

4200

4400

4600

4800

5000

Time in seconds

Maple Valley full PSDC Lag Compensation at 35 seconds

Adelanto full PSDC Lead Compensation at 35 seconds

Fig. 5.9 Simulated COI behavior for an SVC triggered at 35 s to full PSDC. At MapleValley with COI phase-lag compensation or at Adelanto with COI phase-leadcompensation


5.6 Conclusions

This work presents a powerful and practical real-time wide-area controller which is

specifically aimed at improving the dynamic security in the small-signal sense. This

real-time control strategy showed how, with increased observability and reliability, a

signal analysis tool likemulti-Prony (ormatrix pencil, orHankel TLS, or a combina-

tion of them as in [34]) can be successfully applied on real-time basis for detecting the

proximity andonset of small-signal instability phenomena,while avoiding thepitfalls

and the shortcomings widely known and reviewed in the literature.The controller then initiates aggressive mitigatory control actions in certain areas

by using commonly available resources such as the SVCs inmodern power networks.

The controller will come into play only under extremely stressed operating conditions

when the damping of some oscillatory modes becomes problematic.Recent blackout events such as the 1996 western American events, 2004

Noreastern disturbance and the 2004 Italian blackout have clearly demon-

strated that the power system will repeatedly find itself in highly stressed

unplanned for operating scenarios. We need to design new automatic control-

lers which are specifically targeted toward such unforeseen operating condi-

tions which mitigate the problem from cascading into big blackouts. This effort

proposes one such ‘‘safety net’’ controller for stabilizing the large system while

facing small-signal instability problems.

Acknowledgments This work was supported by funding from Power Systems EngineeringResearch Center (PSERC) and by Consortium for Reliability Technology Solutions(CERTS), funded by the Assistant Secretary of Energy Efficiency and Renewable Energy,Office of Distributed Energy and Electricity Reliability, and Transmission Reliability Pro-gram of the US Department of Energy under Interagency Agreement No. DE-AI-99EE35075with the National Science Foundation. Partial funding of the work from Bonneville PowerAdministration is also gratefully acknowledged. J. Quintero received partial support fromUniversidad Autonoma de Occidente and Colciencias–Fulbright–Laspau.

Table 5.10 Simulation COI mode results

PSDCb action atPhasecompensation Trigged after f �

No PSDC Action 0.210 �0.0144Mapple Valley(sending Bus)

458 lag

608 lead

Keeler–Allston trips c

Ross–Lexington trips d

Keeller–Allston trips c


0.181

0.183SystemSystem

0.0697

0.0908CollapsesCollapses

Adelanto(receiving Bus)

458 lag

608 lead





0.2400.2200.2290.230

�0.0216�0.02890.02840.0201

aTime window is from 85 to 97s.bSVCs are originally working in voltage regulation control.cFull PSDC function triggers at �15 s.dFull PSDC function triggers at 35.


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34. G. Liu, J. Quintero, and V. Venkatasubramanian, ‘‘OscillationMonitoring System BasedonWide Area Synchrophasors in Power Systems,’’ in Proc. Bulk Power System Dynamicsand Control, Aug. 2007.

35. M.Klein, G. J. Rogers, and P.Kundur, ‘‘A Fundamental Study of Inter-AreaOscillationsin Power Systems,’’ IEEE Trans. Power Systems,vol. 6, pp. 914–921, Aug. 1991.

36. D. N. Kosterev, C. W. Taylor, and W. A. Mittelstadt, ‘‘Model validation for the August10, 1996 WSCC system outage,’’ IEEE Trans. Power Systems, vol. 14, pp. 967–979, Aug.1999.

37. J. F. Hauer, C. J. Demeure, and L. L. Scharf, ‘‘Initial Results in Prony Analysis of PowerSystem Response Signals,’’ IEEE Trans. Power Systems,vol. 5, pp. 80–89, Feb. 1990.

38. D. J. Trudnowski, J. M. Johnson, and J. F. Hauer, ‘‘SIMO System Identification fromMeasured Ringdowns,’’ in 1998 Proc. American Control Conf., pp. 2968–2972.

39. V. Venkatasubramanian and J. R. Carroll, ‘‘Oscillation Monitoring Sytem at TVA’’,presentation at North American Synchrophasor Initiative meeting at New Orleans,


LA, March 2008, [Online] Available http://www.naspi.org/meetings/workgroup/2008_march/session_one/tva_oscillation_monitoring_venkatasubramanian.pdf

40. P. M. Anderson and A. A. Fouad, Power System Control and Stability.New York: IEEEPress, 1994.

41. J. Quintero, ‘‘A Real-Time Wide-Area Control for Mitigating Small-Signal Instability inLarge Electric Power Systems,’’ Ph.D. dissertation, Sch. of Electrical Eng. and Comp.Sciences, Washington State Univ., Pullman, 2005 [Online]. Avail.: http://www.disserta-tions.wsu.edu/Dissertations/Spring2005/j_quintero_011905.pdf


Chapter 6

Complex Empirical Orthogonal Function Analysis

of Power System Oscillatory Dynamics

P. Esquivel, E. Barocio, M.A. Andrade, and F. Lezama

Abstract Multivariate statistical data analysis techniques offer a powerful toolfor analyzing power system response from measured data. In this chapter, astatistically based, data-driven framework that integrates the use of complexempirical orthogonal function analysis and the method of snapshots is pro-posed to identify and extract dynamically independent spatiotemporal patternsfrom time-synchronized data. The procedure allows identification of the domi-nant spatial and temporal patterns in a complex data set and is particularly wellsuited for the study of standing and propagating features that can be associatedwith electromechanical oscillations in power systems. It is shown that, in addi-tion to providing spatial and temporal information, the method improves theability of conventional correlation analysis to capture temporal events and givesa quantitative result for both the amplitude and phase of motions, which areessential in the interpretation and characterization of transient processes inpower systems. The efficiency and accuracy of the developed procedures forcapturing the temporal evolution of the modal content of data from timesynchronized phasor measurements of a real event in Mexico is assessed.Results show that the proposed method can provide accurate estimation ofnonstationary effects, modal frequency, time-varying mode shapes, and timeinstants of intermittent or irregular transient behavior associated with abruptchanges in system topology or operating conditions.

6.1 Empirical Orthogonal Function Analysis

Empirical orthogonal function (EOF) analysis is a statistical method of findingoptimal distributions of energy from an ensemble ofmultidimensional measure-ments [1]. The essential idea is to generate an optimal basis for the

P. Esquivel (*)Department of Electrical and Computer Engineering, The Center for Research andAdvanced Studies, Cinvestav, Guadalajara, Mexicoe-mail: [email protected]


159

representation of an ensemble of data collected from measurements or numer-ical simulations of a dynamic system.

Given an ensemble of measured data, the technique yields an orthogonalbasis for the representation of the ensemble, as well as a measure of the relativecontribution of each basis function to the total energy with no a priori assump-tion on either spatial or temporal behavior.

The following sections provide a review of some aspects of the qualitativetheory of empirical orthogonal functions that are needed in the analysis of low-dimensional models derived from the technique.

We start by introducing the method in the context of statistical correlationtheory.

6.1.1 Theoretical Development

The proper orthogonal decomposition (POD) method is an optimal techniqueof finding a basis that spans an ensemble of data, collected from an experimentor numerical simulation [1–4]. More precisely, assume that uðxj; tkÞ, j=1,. . .,n,k=1,. . .,N, denotes a sequence of observations on some domain x 2 � where xis a vector of spatial variables and tk½0;T� is the time at which the observationsare made. Without loss of generality, the time average of the time sequence

umðxÞ ¼ u x; tkð Þh i ¼ 1

N

XN

k¼1u x; tkð Þ

is assumed to be zero [3]. Generalizations to this approach are discussed below.The POD procedure determines EOFs, jiðxÞ, i ¼ 1; . . . ;1, such that the

projection onto the first p EOFs (a low-order representation)

uðxj; tkÞ ¼Xp

i¼1aiðtÞjiðxÞ; j ¼ 1; . . . ; n; k ¼ 1; . . . ;N (6:1)

is optimal in the sense that the average least-squares truncation error, "j

"j ¼ uðxj; tkÞ �Xp

i¼1aiðtÞjiðxÞ

��

��

2* +; p � N (6:2)

is minimized for any p � N, where :h i denotes the ensemble average,fk k ¼ f; fh i1=2, and :k k denotes theL2norm over�. The ai’s are time-dependent

coefficients of the decomposition to be determined so that (6.1) results in amaximum for (6.2). These special orthogonal functions are called the properorthogonal modes (POMs) of the reduced basis for uðxj; tkÞ.

160 P. Esquivel et al.

Following [1], assume that the field is decomposed into a mean valueum xj; t� �

and a fluctuating part �uðx; tjÞ

uðxj; tkÞ ¼ umðxj; tkÞ þ �uðxj; tkÞ (6:3)

More formally, let L2 denote the space of square integrable functions. It thenfollows that, a normalized basis function j is optimal if the average projectionof uonto j is maximized, i.e., [2]

maxj2L2 ½0;1�ð Þ

jð�uðxj; tkÞ;jÞj2D E

subject to jk k2¼ 1 (6:4)

where the inner product is defined as U;Vh i ¼ pk¼0 UkV

�k ¼ VHU, and

jk k2¼ j;jh i ¼ jTj ¼Xn

j¼1j2j

The optimization problem can be recast in the form of a functional for theconstrained variational problem [3]1

J½j� ¼ ð�u xj; tk� �

;jÞ�� 2D E

� l jk k2�1� �

(6:5)

A necessary condition for the extrema is that the Gateaux derivative vanishesfor all variations jþ �c 2 L2 ½0; 1�ð Þ, � 2 <. This can be expressed as

dJ

d�jþ �c½ �

��¼0¼ 0; 8c 2 L2ð�Þ (6:6)

Consider now the Hilbert space of all pairs f; gh i where fand gare functionsof L2½0; 1�, i.e., square integrable functions of the space variable x on theinterval ½0; 1�, where :; :h i denotes the standard inner product on L2 defined by

f; gh i ¼Z 1

0

fðxÞg�ðxÞ dx (6:7)

and

j2�� ¼ j;jh i ¼

Z

�

j2 dx (6:8)

where � is the domain of interest over which uðxÞ are j are defined and theasterisk * denotes the conjugate transpose.

1 Given a function to maximize, fðPÞ subject to the constraints gðPÞ ¼ 0, the Lagrangefunction can be defined as FðP; lÞ ¼ fðPÞ � lgðPÞ.

6 Complex Empirical Orthogonal Function Analysis 161

It follows immediately from (6.6) that

dJ

d�½jþ �c�j�¼0¼ 0 ¼ dJ

d��u;jþ �cð Þ jþ �c; �uð Þ � l jþ �c;jþ �cð Þh i½ �

��¼0

¼ 2Re �u;cð Þ j; �uð Þh i � l j;cð Þ½ �(6:9)

where use has been made of the inner product properties.Noting that

�u;cð Þ j; �uð Þh i � l j;cð Þ ¼Z 1

0

�uðxÞc�ðxÞdxZ 1

0

jðx0Þu�ðx0Þdx0�

� lZ 1

0

jðxÞc�ðxÞdx

¼Z 1

0

Z 1

0

uðxÞu�ðx0Þh ijðx0Þdx0 � ljðxÞ �

c�ðxÞ dx ¼ 0

the condition for the extrema reduces to

Z

�

�uðxÞ�u�ðx0Þh ijðx0Þ dx0 ¼ ljðxÞ (6:10)

Equation (6.10) has a finite number of orthogonal solutions over jiðxÞ (thePOMs) with corresponding real and positive eigenvalues li. They are conse-quently called empirical orthogonal functions.

Defining

Rj ¼Z

�

�uðx; tÞ�u�ðx0; tÞh ijðx0Þ dx0 (6:11)

where

Rðx; x0Þ ¼ 1

N

XN

k¼1�uðx; tkÞ�uðx0; tkÞ (6:12)

the problem of minimizing (6.9) becomes that of finding the largest eigenvalueof the eigenvalue problem Rj ¼ lj, subject to jk k2¼ 1.

In practice, the observations that form the data are only available at discretespatial grid points herein called snapshots. In this case, the kernel Rðx; x0Þ canbe written as [5, 6]

Rðx;x0Þ ¼

Rðx1; x1Þ � � � Rðx1;xnÞ... . .

. ...

Rðxn; x1Þ � � � Rðxn;xnÞ

2664

3775


where n indicates that the number of measurement points, and

Rðxi;xjÞ ¼1

N

XN

k¼1�uðxi; tkÞ�uðxj; tkÞ; i; j ¼ 1; . . . ; n (6:13)

In other words, the optimal basis is given by the eigenfunctions ji of (6.13)whose kernel is the autocorrelation function R x; x0ð Þ ¼ �u xj; tk

� ��u x0j; tk

� �D E.

6.1.2 Discrete Domain Representation

Time series are usually recorded in discrete form even though the underlyingprocess itself is continuous. For discretely sampled measured data, the integraltime average can be approximated by a sum over the set of sampled data points[1]. In this case, the vectors

xj ¼ �ujðxj; tkÞ ¼ �uðxj; t1Þ �uðxj; t2Þ � � � �uðxj; tNÞ� T

; j ¼ 1; :::; n (6:14)

represent a set of snapshots obtained from the observed data at n locations.The set of data can then bewritten as theN� n-dimension ensemblematrix,X [6]

X ¼ x1 � � � xn½ � ¼

�uðx1; t1Þ � � � �uðxn; t1Þ... . .

. ...

�uðx1; tNÞ � � � �uðxn; tNÞ

2664

3775 (6:15)

where each column corresponds to the response at a specific time.Typically, n 6¼ N, so X is generally rectangular. Under these assumptions,

the actual integral (6.10) can be written as Cj ¼ lj, where

Cij ¼ 1 NPN

k¼1 �uðxi; tkÞ�uðxj; tkÞ.

. Assuming the EOFs to be of the form

ji ¼PN

l¼1 wilxi where wi

l is a coefficient to be determined, the problem

of minimizing (6.2) can be recast as the problem of finding the largesteigenvalue of the linear equation

Cj ¼ lj (6:16)

where C is the autocorrelation (covariance) matrix defined as

C ¼ 1

NXTX ¼ 1

N

xT1 x1 xT1 x2 � � � xT1xn

xT2 x1 xT2 x2 � � � xT2xn

..

. ... . .

. ...

xTn x1 xTn x2 � � � xTn xn

266664

377775¼ 1

N

Xn

i¼1xi � umð ÞT xi � umð Þ (6:17)


The resulting covariance matrix C is a real, symmetric ðCij ¼ CjiÞ positive,and semi-definite matrix. Consequently, it possesses a set of orthogonal eigen-

vectors ji; i ¼ 1; :::; n, i.e.,

jTi jj ¼

�ij; i ¼ j

0; i 6¼ j

�

Using standard linear algebra techniques, the covariance matrix can be

expressed in the form

C ¼ ULVT (6:18)

where U and V are the matrices of right and left eigenvectors and

L ¼ diag½l1 l2 � � � ln�.The eigenvalues computed from (6.18) are real and nonnegative and can be

ordered such that l1 � l2 � � � � � ln � 0. The eigenvectors of Care called the

POMs and the associated eigenvalues are called the proper orthogonal vectors

(POVs).For practical applications, the number snapshots N can be rather large,

leading to a very large eigenvalue problem. There are at least two methods to

solve the eigenvalue problem (6.16) [6]: the direct method and the method of

snapshots.The direct method attempts to solve the eigenvalue problem involving the

N�N matrix directly using standard numerical techniques. This can be

computationally intensive if the number of observations is larger than the

number of observing locations or grid points. Themethod of snapshots, on the

other hand, is based on the fact that data vectors ui and the POD modes jl

span the same linear space. The latter is explored here.The next section describes in more detail the nature of the approximation

employed here to construct the statistical representation.

6.1.3 The Method of Snapshots

The method of snapshots is based on the fact that the data vectors ui and the

POD modes span the same linear space [6]. In this approach, we choose the

eigenfunctions j to be a linear combination of the snapshots:

ji ¼XN

l¼1wilxi (6:19)

where the coefficients wil are to be determined such that u maximizes


maxj

1

N

Xn

j¼1

xj;j�� j;jh i (6:20)

These l functions are assembled into an n�Nmatrix, F, known as the modalmatrix. In matrix form Eq. (6.19) becomes

F ¼ XW (6:21)

where

F ¼" " " "j1 j2 � � � jn

# # # #

2

64

3

75; X ¼" " " "x1 x2 � � � xn

# # # #

2

64

3

75;W ¼" " " "w1 w2 � � � wn

# # # #

2

64

3

75

in which

w1 ¼

w11

w12

..

.

w1N

2666664

3777775; w2 ¼

w21

w22

..

.

w2N

2666664

3777775; . . . ; wn ¼

wn1

wn2

..

.

wnN

266664

377775

Substituting the expression (6.19) into the eigenvalue problem (6.16) gives

CXN

l¼1wilxi ¼l

XN

l¼1wilxi (6:22)

where Cij ¼ ð1=NÞ ð�ui; �ujÞ. This can be written as the eigenvalue problem ofdimension n

CW ¼ LW (6:23)

where

w ¼ w1 w2 � � � wn½ �

and L is a diagonal matrix storing the eigenvalues li of the covariance matrixC.In words, the first-order necessary optimality condition for j to provide amaximum in (6.20) is given by (6.16). This completes the construction of theorthogonal set j1 j2 � � � jnf g.

Once the modes are found using these equations, the flow field can bereconstructed using a linear combination of the modes


ukðxj; tkÞ ¼X1

i¼1aiðtÞjiðxÞ (6:24)

for some aiðtÞ 2 <, where the aiðtÞ are the time-varying amplitudes of the PODmodes jiðxÞ.

The truncated POD of uis

ukðxj; tkÞ ¼Xp

i¼1aiðtÞjiðxÞ þ " (6:25)

where p is the number of dominant modes and " is an error function.Having computed the relevant eigenmodes, the temporal behavior can be

evaluated as the inner product of the eigennmode (the POD mode ji) and theoriginal data. To ensure uniqueness of the solution, the normalization condi-tion of ji;jih i ¼ 1 is imposed.

The temporal coefficients are then expressed as

ai ¼ x;jih i= ji;jih i (6:26)

Note that the temporal modes are uncorrelated in time, i.e.,ðaiðtÞ; ajðtÞÞ ¼ �ijlj; where �ij ¼ 1 for i ¼ j; 0 else, and that the system (6.25) isoptimal in the sense that minimizes the error functions

"ðjÞ ¼Xp

l¼1�uðxj; tkÞ �

Xp

i¼1aiðtÞjiðxÞ

��

�� (6:27)

It should also be stressed that no conditions are imposed on the data set; thedata can be a sample of a stationary process or a sample of a nonstationary process.

Equation (6.24) is called the Karhunen–Loeve decomposition and the set jj

are called the empirical basis [5].

6.1.4 Energy Relationships

The use of the POD method leads naturally to a discussion of truncationcriteria. Several techniques to derive truncated expressions have been proposedin the literature. Here, we choose to reduce the residual terms, R ¼ ", such thatthe mean square value

R ¼ Olpþ1Pni¼1 li

� �(6:28)

be as small as possible.


Among all linear decompositions, the most kinetic energy possible for aprojection onto a given number of modes. Defining the total energy E, byE ¼ Trace ½Rðxi; xmÞ�; one obtains

E ¼Xp

i¼1li (6:29)

The associated percentage of total energy contributed by eachmode can thenbe expressed as

Ei ¼liPn

i¼1li

(6:30)

Typically, the order p of the reduced basis j such that the predeterminedlevel of the total energy E of the snapshot ensemble is captured, i.e., 99%. Thep-dominant eigenfunctions are then obtained as

Ppi¼1 liPnj¼1 lj

¼ 99% (6:31)

for the smallest integer p where E is an appropriate energy level.The key advantage of this technique is that allows extracting information

from short and often noisy time series without prior knowledge of the dynamicsaffecting the time series.

6.2 Interpretation of EOFs Using Singular Value Decomposition

A useful alternative method for estimating modal characteristics can bedeveloped based on the analysis of the response matrix X in (6.15).

Before outlining the procedure for singular value decomposition (SVD)analysis, we introduce some background information on singular valueanalysis.

6.2.1 Singular Value Decomposition

Let A be a real m � n matrix. The SVD theorem states that A can be decom-posed into the following form [7]:

A ¼ USVT (6:32)


where U ¼ col½u1 u2 � � � um� is an m � m orthonormal matrix VT ¼ U�1,S is an m � n pseudodiagonal and semi-positive definite matrix with diagonal

entries containing the singular values ,V ¼ col½v1 v2 � � � vn� and is an n� n

orthonormal matrix UT ¼ U�1. The columns of U and Vare called the left and

right singular vectors for A.Matrix S has the form

S ¼

�1 0 � � � 0 0 � � � 0

0 �2 0 � � � 0

..

. ... . .

. ... . .

. ...

0 0 � � � �m 0 � � � 0

2

66664

3

77775for m5n

or

S ¼

�1 0 � � � 0

0 �2 � � � 0

..

. ... . .

. ...

0 0 � � � �m

0 0 � � � 0

..

. ... . .

. ...

0 0 � � � 0

26666666666664

37777777777775

for m4n

Throughout this research, we will, consider only the case when m4n. The

diagonal entries of S, i.e., the �ii ¼ �i, can be arranged to be nonnegative and in

order of decreasing magnitude �1 � �2 � � � � � �m � 0.Equivalently, we can express the model as

A ¼ u1 � � � uk j ukþ1 � � � um½ �

�1 � � � 0 0

..

. . .. ..

. ...

0 � � � �k 0

0 � � � � � � 0

2

66664

3

77775

vT1

..

.

vTkvTkþ1

..

.

vTn

2666666666664

3777777777775

(6:33)

or

A ¼ Uj?U�

D 00 0

� �� V?V

�


where D is the diagonal matrix of nonzero singular values and U and V are the

matrices of left and right singular vectors, respectively, corresponding to the

nonzero singular values.It is clear that only the first r of the u’s and v’s make any contribution to A,

and can be expressed as an outer product expansion

A ¼Xr

i¼1�i uiv

Ti

� �(6:34)

where the vectors ui and vi are the columns of the orthogonal matricesU and V,

respectively. Techniques to compute the POMs based on SVD are next discussed.

6.2.2 Relation with the Eigenvalue Decomposition

An interesting interpretation of the POD modes can be obtained from the

singular value analysis of the response matrix X.Using the notation in Section 6.1.2 let the response matrix be given by

X ¼

�uðx1; t1Þ � � � �uðxn; t1Þ... . .

. ...

�uðx1; tNÞ � � � �uðxn; tNÞ

2

664

3

775 (6:35)

The SVD of the response matrix X may be written in compact form as

X ¼ USVT (6:36)

where U is an orthonormal N � N matrix whose columns are the left singular

vectors of X, S is N � n matrix containing the singular values of X along the

main diagonal and zeros elsewhere, andV is an n� n orthonormalmatrix whose

columns correspond to the right singular vectors of X. The response matrix, X,

is complex and symmetric and possesses a set of orthogonal singular vectors

with positive singular values.In terms of the notation above for SVD, it can be seen directly from (6.18)

that the correlation matrix defined previously is given by

XXT ¼ USVð Þ USVð ÞT¼ US2UT (6:37)

and

XTX ¼ USVð ÞT USVð Þ ¼ VS2VT (6:38)


Hence, (6.32) becomes

XXTU ¼ U

l1l2

. ..

ln

2

66664

3

77775¼ U

�21�22

. ..

�2n

2

66664

3

77775(6:39)

It follows immediately from (6.32), (6.33), (6.34), the singular values ofX are

the square roots of the eigenvalues ofXXT orXTX [2, 8]. In addition, the left and

right eigenvectors of X are the eigenvectors of XXT and XTX, respectively. Also

of interest, the trace of (6.39) is given by

XTX ¼Xr

i¼1�2i

The POMs, defined as the eigenvectors of the sample covariancematrixC are

thus equal to the left singular vectors ofX. The POVs, defined as the eigenvalues

of matrix C are the squares of the singular values divided by the number of

samples N.

6.3 Numerical Computation of POMs

In this section, a step-by-step description of the algorithm used to extract modal

information is presented. The procedure adopted to compute the POMs can be

summarized as follows:

1. Given an ensemble of measurements of a nonstationary process, computethe response matrix X. Form the complex time series matrix X ¼ Xþ jXH,where XH is the Hilbert transform of X

2. Compute the singular vectors U;V and the corresponding singular values �.3. Determine the time evolution of the temporal modes, ai. Extract standing

and propagating features using the complex SVD (singular value decompo-sition) formulation.

Figure 6.1 illustrates the proposed algorithm.For different events recorded at the same location, statistical averaging can

be employed to take advantage of the statistics of the data. In this case, the

snapshots can be thought of realizations of random fields generated by some

kind of stochastic process.The processing steps are detailed in the sections that follow.


6.4 Complex Empirical Orthogonal Function Analysis

Empirical orthogonal function analysis of data fields is commonly carried outunder the assumption that each field can be represented as a spatially fixedpattern of behavior. This method, however, can not be used to for detection ofpropagating features because of the lack of phase information. To fully utilizethe data, a technique is needed that acknowledges the nonstationarity andbehavior of the time-series data.

Let uðxj; tkÞ be a space–time scalar field representing a time series, wherexj; j ¼ 1; . . . ; n is a set of spatial variables on a space �k and tk; k ¼ 1; . . . ;N isthe time at which the observations are made. Provided u is simple and squareintegrable, it has a Fourier representation of the form [9]

uðxj; tkÞ ¼X1

m¼1½ajðmÞð!Þ cosðm!tkÞ þ bjðmÞð!Þ sinðm!tkÞ� (6:40)

where ajðmÞð!Þ and bjðmÞð!Þare the Fourier coefficients defined as

ajðmÞ ¼1

p

Zp

�p

uðxj; tkÞ cos m!tkð Þ d!

bjðmÞ ¼1

p

Zp

�p

uðxj; tkÞ sin m!tkð Þ d!

Fig. 6.1 Conceptual view of the proposed algorithm


This allows description of traveling waves. Equation (6.40) can be rewrittenin the form

ucðxj; tkÞ ¼X1

m¼1cjðmÞð!Þe�im!tk (6:41)

where cjðmÞð!Þ ¼ ajðmÞð!Þ þ ibjðmÞð!Þ [1] and i ¼ffiffiffiffiffiffiffi�1p

. Expanding (6.41) andcollecting terms gives

Ucðxj; tkÞ ¼X1

m¼1ajðmÞð!Þ cosðm!tkÞ þ bjðmÞð!Þ sinðm!tkÞ� � �

þ iX1

m¼1bjðmÞð!Þ cosðm!tkÞ � ajðmÞð!Þ sinðm!tkÞ� � �

¼ uðxj; tkÞ þ iuHðxj; tkÞ

(6:42)

where the real part of U is given by (6.40) and the imaginary part is the Hilberttransform of uðxj; tkÞ[10]

uHðxj; tkÞ ¼ �1

p

Z1

�1

h u xj; tk� ��

tk � xjdx (6:43)

This represents a filtering operation upon uðxj; tkÞ in which the amplitude ofeach Fourier spectral component remains unchanged while its phase isadvanced by p/2. The eigenvectors here are complex and can be expressedalternatively as a magnitude and phase pair.

In the proposed formulation u can be estimated more efficiently by perform-ing a filtering operation on u itself. Equation (6.43) can be rewritten in the formof a convolution as

uHðxj; tkÞ ¼XL

‘¼�Luðxj; tk � ‘Þhð‘Þ; L ¼ 1 (6:44)

where h sis a convolution filter with unit amplitude response and 908 phase shift.In practice, a simple filter that has the desired properties of approximate unit

amplitude response and p/2 phase shift is given by [11]

hðlÞ ¼2pl sin

2 pl2

� �; l 6¼ 0

0; l ¼ 0

((6:45)

where �L � l � L. As L!1, Equation (6.45) yields an exact Hilberttransform.


In what follows we discuss the extension of the above approach to compute

standing and propagating features.

6.4.1 Complex EOF Analysis

Drawing on the above approach, an efficient formulation to compute complextime-dependent POMs has been derived. Following Susanto et al. [12] assumethat X is a j (spatial points) by k (temporal points) ensemble matrix. From thepreceding results, it follows that

X ¼ USVH (6:46)

where VH is the conjugate transpose of V, the superscript H denotes a Hermi-tian matrix, and we assume that

UHU ¼ I

VHV ¼ I

Now, it can be easily verified that

XXH ¼ USVH� �

USVH� �H¼ USSTUH

XHX ¼ USVH� �H

USVH� �

¼ VSTSVH(6:47)

where ST denotes the transpose of S. As is apparent from Eq. (6.46), the

columns of U are the eigenvectors of X XH� T

, and that the columns of V are

the eigenvectors of XH� T

X. The n singular values on the diagonal of S are the

square roots of the nonzero eigenvalues of both X XH� T

and XH� T

Xwhere n is

the rank of X.Once the spatial eigenvectors are calculated, their corresponding time evolu-

tion is given by the time series AiðtÞ which is obtained by projecting the timeseries X onto the proper eigenvector ji, and summing over all locations:

AiðtÞ ¼Xn

j¼1Xðxj; tkÞjiðxÞ (6:48)

The original complex data field, Xðx; tÞ, can be reconstructed by adding thisproduct over all modes, i.e.,

Xðx;tÞ ¼Xn

i¼1AiðtÞjH

i ðxÞ (6:49)


Using the complex SVD, it is possible to compute the spatial amplitude andspatial and temporal phase functions as discussed below.

6.4.2 Analysis of Propagating Features

The time-dependent complex coefficients associated with each eigenfunction canbe conveniently split into their amplitudes and phases. From the complex EOFanalysis in (6.49), the ensemble of data can be expressed as the complex expan-sion [13, 14]

Xðx; tÞ ¼Xn

i¼1RiðtÞffqi SiðxÞfffi (6:50)

where RiðtÞ is the complex temporal amplitude function, SiðxjÞ is the complexspatial mode or eigenvector, and q, f are the phase functions corresponding toRiðtÞ and SiðxjÞ. These phase functions describe the propagation characteristicsof the ith mode.

Equation (6.50) can be rewritten as

uðxj; tÞ ¼Xn

i¼1

��RiðtÞ��SiðxÞ

��e j½qRiðtÞþfSi

ðxÞ�(6:51)

This effectively decomposes the data into spatial and temporal modes.Four measures that define possible moving features in uðx; tÞ can then be

defined [9].

1. Spatial distribution of variability associated with each eigenmode2. Relative phase of fluctuation3. Temporal variability in magnitude4. Variability of the phase of a particular oscillation

The following definitions introduce these concepts.

Definition 6.1 (Spatial amplitude function, SiðxÞ) The spatial amplitude func-tion, SiðxÞ, is defined as

SiðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijHi ðxÞjiðxÞ

q(6:52)

This function shows the spatial distribution of variability associated witheach eigenmode.

Definition 6.2 (Spatial phase function, ji) This function shows the relative phasefluctuation among the various spatial locations where u is defined and is given by


fiðxÞ ¼ tan�1ImfjiðxÞgRefjiðxÞg

� �(6:53)

This measure, for which an arbitrary reference value must be selected,

varies continuously between 0 and 2p. Together, Eqs. (6.52) and (6.53) give

a measure of the space distribution of energy and can be used to identifythe dominant modes and their phase relationships. Further, for each domi-

nant mode of interest, a mode shape can be computed by using the spatial

part of (6.50).

Definition 6.3 (Temporal amplitude function, Ri) Similar to the description ofthe spatial amplitude function in (6.52), the temporal amplitude function, Ri

can be defined as

RiðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAH

i ðtÞAiðtÞq

(6:54)

This function gives ameasure of the temporal variability in themagnitude of the

modal structure of the field u. As described later, the general form of theseequations is very amenable to computational analysis.

Definition 6.4 (Temporal phase function, qi) This temporal variation of phaseassociated with uðx; tÞ is given by

qiðtÞ ¼ tan�1ImfAiðtÞgRefAiðtÞg

� �(6:55)

For the simple case of a sinusoidal wave with fixed frequency and wave

number, �iðtÞ ¼ !t. In the more general (and interesting case), the space

derivative of the phase and frequency of the modal components can be calcu-lated from

ki ¼ dðfiÞ=dðxÞ

wi ¼ dðqiÞ=dðtÞ

ci ¼ wi=ki

(6:56)

where ci is the phase velocity of the function.Equations (6.52), (6.53), (6.54), (6.55), (6.56) provide a complete character-

ization of any propagating effects and periodicity in the original data field

which might be obscured by normal cross-spectral analysis. Finally, it mightbe remarked that, in the special case of real analysis, these expressions simplify

to the normal definitions.


6.5 Application to Time Synchronized Measured Data

To test the ability of the method to analyze complex oscillations, we analyze

data from time-synchronized measurements. The data used for this study were

recorded by phasor measurement units (PMUs) over a 4 s window during a real

event in northernMexico. A brief description of the data follows.More detailed

information on system measurements can be found in [4, 15].At local time 06:27:42 in the early hours of January 1, 2004, undamped

oscillations involving frequency, voltage, and power were observed throughout

the northern systems of the Mexican Interconnected System (MIS). The main

event that originated the oscillations was a failed temporary interconnection of

the Northwest regional systems to the MIS through a 230 kV line betweenMZD

(Mazatlar DOS) and TTE (Tres Estrellas) substations. It is noted that, prior to

this oscillation incident, the northwestern system operated as an electrical island.Oscillations in the northern systems with periods about 0.61, 0.50, and

0.27 Hz persisted for approximately 1.2 s before the northwestern system was

disconnected from the MIS. During the time interval 06:27:42–06:28:54 the

system experienced severe fluctuations in frequency, power, and voltage result-

ing in the operation of protective equipment with the subsequent disconnection

of load, independent generation, and major transmission resources.

Fig. 6.2 Schematic of the MIS system showing the location of the observed oscillations.Measurement locations are indicated by shaded circles


Figure 6.2 shows a geographical diagram of the MIS showing the PMU

locations and the location of the initiating event. For demonstration purposes,

three buses spread across the system are selected. To allow comparison with

previous work, bus frequency signals are used in the analysis.Figure 6.3 is an extract from PMU measurements of this event showing the

observed oscillations of selected bus frequencies. For the purpose of further

comparison to EOF analysis, the relevant time interval of concern is zoomed in.

Systemmeasurements in this plot demonstrate significant variability suggesting

a nonstationary process in both space and time. The most prominent variations

occur in the interval during which the oscillation start at 06:27:42 and the

interval in which the operating frequency is restored to the nominal condition

(60 Hz) by control actions (06:28:21).As discussed later, during this period the system experiences changes in

frequency (amplitude) content and mode shapes.

Fig. 6.3 Time traces of recorded bus frequency swings recorded on January 1, 2004 and detailof the oscillation buildup


As a first step toward the development of a POD basis, the observed records

are placed in a complex data matrix, Xðxj; tkÞ, as

Xðxj; tkÞ ¼ fHðtÞ; fMZDðtÞ; fTTEðtÞ½ � ¼

uðx1; t1Þ � � � uðxn; t1Þ... . .

. ...

uðx1; tNÞ � � � uðxn; tNÞ

2

664

3

775

where j ¼ 1; 2; 3 is the spatial position index (grid location), tis the time, andN is

the number of data points in the time series. For our simulations, 2021 snap-

shots are available, representing equally spaced measurements at three different

geographical locations.Each time series is then augmented with an imaginary

component to provide phase information and the EOF method is employed to

approximate the original data by a general, nonstationary, and correlatedfrequency model.

The following subsections describe the application of complex empirical

orthogonal function analysis to examine the temporal and spatial variability

of measured data.

6.5.1 Construction of POD Modes via the Method of Snapshots

The method of snapshots was applied to derive a spatiotemporal model of the

oscillations. In order to improve the ability of the method to capture temporalbehavior, the individual time series are separated into their time-varying mean

and fluctuating components. By separating the data into their mean and fluc-

tuating components, EOF analysis is able to selectively determine the temporal

behavior of interest. The method may also help reduce the detrimental effects of

crossing events, spatial aliasing, and random contamination.In the application of the proposed method to measured data, we assume that

each signal can be construed as a superposition of fast oscillations on top of a

slow oscillation (the time-varying instantaneous mean) [16]. The slow oscilla-

tion essentially captures the nonlinear (and possibly time-varying) trend while

the slow oscillations are the fluctuating parts.A two-stage analysis technique based on wavelet shrinkage is proposed to

determine the temporal properties of time-synchronized information. In thefirst step, the original system time histories are decomposed into their time-

varying mean speeds and fluctuating speeds through wavelet shrinkage. More

formally, the recorded time series are decomposed into their time-varying mean

frequencies, umðxj; tkÞ; and nonstationary fluctuating components, �uðxj; tkÞ; asfollows:

uðxj; tkÞ ¼ umðxj; tkÞ � �uðxj; tkÞ (6:57)


In the second stage, complex EOF analysis is applied to the fluctuating field

to decompose the spatiotemporal data into orthogonal temporal modes.Following Donoho [17], the two-stage approach can be used to reconstruct

an unknown function f from noisy data di ¼ fðtiÞ þ �zi; i¼ 0; . . . ;n� 1;wheredi is the observed data point (the noise-contaminatedmeasurement point), zi is astandard Gaussian white noise, ti are equispaced data points, and �is a noiselevel. The first stage is to find the estimates fð:Þ ¼ Tðy; dyÞ ð:Þ, where Tðy; dyÞ ð:Þis a suitable reconstruction formula with spatial smoothing parameters � and dyis data-adaptive choice of the spatial smoothing parameter. The separatingprocedure is carried out using the wavelet shrinkage method because of itsability to capture fast changes in the data.

The adopted separation/identification procedure can be summarized asfollows:

1. Expand the data uðxj; tkÞ into wavelet series. Using the wavelet decomposi-tion structure, estimate level-dependent thresholds for signal compressionusing suitable thresholding approach

2. Obtain a denoised compressed function, um, representing the instantaneousmean, via the wavelet shrinkage, and

3. Compute the fluctuation of the signal by calculating �uðxj; tkÞ ¼ uðxj; tkÞ �umðxj; tkÞ; where um is the time-varying mean of the signal. Among theexisting techniques, we use the Birge–Massart strategy in [18].

The algorithm is simple to implement and the computational requirementsare small.

6.5.2 Spatiotemporal Analysis of Measured Data

Based on the analytical procedure outlined in Section 6.5.1, the complex EOFmethod was used to determined dynamic trends and to analyze phase relation-

ships. In this procedure, the measured data are augmented with an imaginarycomponent defined by its Hilbert transform, and the temporal patterns areextracted by using the procedure in Section 6.4.

Complex EOF analysis was performed on the original time series, and thenonlinearly detrended (wavelet shrinkage model) time series. The analysis pre-sented here uses the Daubechies wavelet with a fixed decomposition level. Alevel-dependent threshold is then obtained using a wavelet coefficients selection

rule based on the Birge–Massart strategy [18].As seen in Fig. 6.4, this model very effectively describes the long-term

behavior of the data while also capturing transient fluctuations. This, in turn,results in improved characterization of system behavior.

Spectral analysis results for the leading POM in Fig. 6.5 show that the mainpower is concentrated in oscillations with frequencies about 0.61, 0.50, and0.27 Hz, which can be associated with major inter-area modes in the system [4].


Small peaks in Fig. 6.5 may indicate nonlinear interactions between frequency

components. From Fig. 6.5 it is also seen that the time-varying instantaneous

mean (bold gray line) acts as a high-pass filter. This approach can be used to

separate the slow from the fast components as well as to improve the numerical

accuracy of the method. This is discussed further in the next two subsections.Figure 6.6 presents the corresponding eigenspectrum of these modes com-

puted to capture 99.9% of the signal’s energy. In these plots, the horizontal axis

shows the number of modes required to attain 99% of the averaged total

energy; the vertical axis shows the energy in (6.13) captured by each POM.

The data set being analyzed corresponds to bus frequency measurements

observed with the PMUs at various geographical locations.

100 120 140 160 180 200 220 240 260 280 30058.5

5959.5

6060.5

Fre

quen

cy [H

z]

100 120 140 160 180 200 220 240 260 280 30058.5

5959.5

6060.5

Fre

quen

cy [H

z]

100 120 140 160 180 200 220 240 260 280 30059.9

60

60.1

60.2

60.3

Time (ms)

Fre

quen

cy [H

z]

Original signalTime−varying instantaneous mean

Fig. 6.4 Time-varying means and fluctuating components of the recorded bus frequencysignals. The time series are smoothed using wavelet shrinkage

0 0.2 0.4 0.6 0.8 1 1.2

–100

–50

0

50

Frequency (Hz)

Mag

nitu

de (

dB)

POD of original signalsPOD with instantaneous mean removedPOD of instantaneous mean0.61 Hz

0.49 Hz0.27 Hz

Fig. 6.5 Comparison of theFourier transform spectrumof the original signal and thespectra constructed with theinstantaneous meanremoved


Complex EOF analysis of the synchronized measurements of frequency in

Fig. 6.6 shows that wide-area system dynamics is well represented by three

modes; the two leading modes together account for 96.5% of the total energy.

Individually, these modes account for 72, 24.5, and 3.5% of the energy (see

Fig. 6.6 caption for details).Figure 6.7 compares the reduced solution using the POD basis functions to

the full solution obtained from the measured data for the Hermosillo signal in

1 2 30

10

20

30

40

50

60

70

80

Number of modes

Ene

rgy

(%)

Real EOF formulationComplex EOF formulation

Mode Energy (%) Cumulative energy (%)

1 72 722 24.5 96.53 3.5 100

Fig. 6.6 Energy captured as a function of the number of modes. The percentage of energylocated in the jth mode is measured by E ¼ 100lj 100lj

� Pni¼1 li

120 140 160 180 200 220 24059

59.5

60

60.5

61

Freq

uenc

y (H

z)

120 140 160 180 200 220 24059

59.5

60

60.5

61

Freq

uenc

y (H

z)

120 140 160 180 200 220 24059

59.5

60

60.5

61

Time (ms)

Freq

uenc

y (H

z)

Original time seriesPOM 1

Original time seriesPOM 1 + POM 2

Original time seriesPOM 1+ POM 2 + POM 3

Energy = 44 %Cumulative = 44 %



Fig. 6.7 Reconstruction of the original data using the three leading POMs. Solid linesrepresent the original time series and dotted lines represent the composite oscillation obtainedby adding the temporal modes


Fig. 6.4. As observed in this plot, using only three modes we are able toaccurately approximate the measured data over the entire observation period.The agreement between the reduced order model and the observed behaviorillustrates the high degree of accuracy that is possible with a simplified model.

On the basis of these results, we conducted detailed analysis aiming atdisclosing hidden information in the data. For clarity of exposition, the analysisof temporal and spatial patterns will be presented separately.

6.5.3 Temporal Properties

In order to reveal the hidden wave signatures in the time series, we examine bothamplitude and temporal patterns in system behavior in the light of complexorthogonal function analysis. Figure 6.8 illustrates the temporal evolution(amplitude and phase) of the dominant mode.

These functions display a number of interesting features. As discussed below,the analysis identifies two periods of interest; a transient period associated withthe interconnection of the systems (06:27:42–06:28:21) and a nearly stationaryperiod in which the frequency of the interconnected system is restored to itsnormal value (06:28:54–06:29:39).

The first interval manifests particularly strong temporal activity as can be seenin Fig. 6.8a. In interpreting these results, we remark that break or changes in thetemporal functions may signal different physical regimes or control actions.

An examination of the temporal phase in Fig. 6.8b, on the other hand,reveals a nonstationary behavior in which the phase (frequency) contentchanges with time. Here, the slope of the spatial phase function represents theinstantaneous frequency. The slowly increasing trends indicate periods ofessentially constant frequencies.

6.5.4 Frequency Determination from Instantaneous Phases

Additional insight into the frequency variability of the observed oscillations canbe obtained from the analysis of instantaneous frequencies. Recognizing thatthe instantaneous frequency is the time derivative of the temporal phase func-tion, �, the instantaneous frequencies can be estimated from (6.55) for eachmode of concern.

The study focuses on POM 1 which is the mode that captures most of thevariability in the signal. Figure 6.9 gives the instantaneous frequency of POM 1for the interval of interest in this study. Also plotted, is the instantaneous meanfrequency (nonlinear trend) determined using wavelet shrinkage analysis above.

Nonstationary features are evident in this plot. Analysis of these plots showstwo modal components: a 0.27 Hz component associated with the steady-statebehavior of the system, and a 0.64 Hz component associated with the transient


system fluctuation following the system interconnection. The 0.27 Hz compo-

nent captures the slow ambient swings previous to the onset of system oscilla-

tions and the steady behavior of the system. The results are consistent with

those based on nonstationary analysis of the observed oscillations giving valid-

ity to the results.

120 140 160 180 200 220 2400

0.2

0.4

0.6

0.8

1

1.2

Time (ms)

Mag

nitu

de 06:27:42

06:28:06 06:28:21

06:28:54 06:29:39

a) Temporal amplitude

120 140 160 180 200 220 240−200

−150

−100

−50

0

50

100

150

200

Time (ms)

Phas

e (d

egre

es)

06:27:42

06:28:06

06:28:21

06:28:54

06:29:39

b) Temporal phase

Fig. 6.8 Temporal patterns of variability associated with the dominant mode


6.5.5 Mode Shape Estimation

One of the most attractive features of proposed technique is its ability to detectchanges in the shape properties of critical modes arising from topology changes

and control actions. Changes in mode shape may indicate changes in topologyor changes in load/generation and may be useful for control decisions and the

design of special protection systems. This is a problem that has been recentlyaddressed using spectral correlation analysis [19].

Using the spatial phase and amplitude, the phase relationship between key

system locations (the mode shapes) can be determined. In this analysis, wedisplay the complex value as a vector with the length of its arrow proportional

to eigenvector magnitude and direction equal to the eigenvector phase.Figure 6.10 shows the mode shape for the three intervals of interest above

(06:27:42–06:28:06, 06:28:06–06:28:21, and 06:28:21–06:28:54 ) computed using

the spatial function (6.50). It is interesting to note that the dominant POMmode shape changes with time. The effect is more pronounced for the time

interval 06:28:06–06:28:21 in which several control actions take place in the

system. This information may be useful to identify the dominant generatorsinvolved in the oscillations, and ultimately devise control mechanisms to damp

the observed oscillations.These results are in general agreement with previously published results

based on real EOF analysis and Prony results [4]. The new results, however,

provide clarification on the exact phase relationships between key systemmeasurements as a function of time.

120 140 160 180 200 220 240−1

−0.5

0

0.5

1

1.5

2

Time (ms)

Freq

uenc

y (H

z)Instantaneous frequencyTrend

0.64 Hz

0.27 Hz

Fig. 6.9 Instantaneous frequency of POM 1. Time interval 06:27:42–06:29:39


6.5.6 Energy Distribution

In the previous section it was shown that a linear combination of an individual

eigenmode can accurately reconstruct the temporal behavior of simultaneous

measurements at different geographical (spatial) locations. A key related ques-

tion of interest is that of finding a small number of measurements that will

provide a good estimation of the entire field of interest.Based on the decomposed EOFs, complex EOF analysis was used to deter-

mine the locations with the most energy. Figure 6.11 shows the participation of

each location to the total energy of the record. The x-axis shows spatial sensor

location and the y-axis shows the energy value. From Fig. 6.11, it is evident that

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

a) 06:27:42−06:28:06 b) 06:28:06−06:28:21

c) 06:28:21−06:28:54

Hermosillo

Mazatlan Dos

Tres Estrellas

Fig. 6.10 Mode shape of POM 1 for various time intervals of interest

Fig. 6.11 Energy distribution


modes 1 and 2 are quite prominent at the Mazatlan Dos and Hermosillosubstations while mode 3 is more strongly evident at the Tres Estrellas substa-tion. This is consistent with conventional analysis (not shown). However, theproposed approach provides an automated way to estimate mode shapes with-out any prior information of the time intervals of interest.

6.6 Concluding Remarks and Directions for Future Research

In this chapter, a new method of temporal representation of nonstationaryprocesses in power systems has been presented. Complex empirical orthogonalfunction analysis provides an efficient and accurate way of looking at thetemporal variability of transient processes while at the same time providingspatial information about each one of the dominant modes with no a prioriassumption on either spatial or temporal behavior. The main advantage of thisapproach is its ability to compress the variability of large data sets into thefewest possible number of temporal modes.

Complex empirical orthogonal function analysis is shown to be a usefulmethod for identifying standing and traveling patterns in wide-area systemmeasurements. Using wide-area frequency information, spatiotemporal analy-sis of time-synchronized measurements shows that transient oscillations maymanifest highly complex phenomena, including nonstationary behavior.Numerical results show that the proposed method can provide accurate estima-tion of nonstationary effects, modal frequency, time-varying mode shapes, andtime instants of intermittent transient responses. This information may beimportant in determining strategies for wide-area control and special protectionsystems. The identified systemmodes from the decomposition may also serve toreveal relevant, but unexpected structure hidden in the data such as that result-ing from short-lived transient episodes. Other issues such as the effect ofnumerical approximations on modal estimates will be investigated in futureresearch.

A generalization of this theory is also needed to treat statistical data from anensemble of nonstationary oscillations. This is an aspect that warrants furtherinvestigation. Finally, the generalization of the proposed technique to deter-mine the most suitable locations for phasor measurement devices and theanalysis of modal coherency are topics worthy of further investigation.

References

1. Philip Holmes, et al., Turbulence, coherent structures, dynamical systems and symmetry.New York: Cambridge University Press, 1996.

2. Gaetan Kerschen, et al., ‘‘The method of proper orthogonal decomposition for dynamicalcharacterization and order reduction of mechanical systems: An overview,’’ NonlinearDynamics, vol. 41, 2005, p. 147.


3. S. S. Ravindran, ‘‘Reduced-order controllers for control of flow past an airfoil,’’ Interna-tional Journal for Numerical Methods in Fluids, vol. 50, 2006, p. 531.

4. A. R. Messina, et al., ’’Extraction of Dynamic Patterns from Wide-Area Measurementsusing Empirical Orthogonal functions,’’ IEEE Trans. on Power Systems, vol. 22, no. 2,May 2007, p. 682.

5. G. Kerschen, et al., ‘‘The proper orthogonal decomposition for characterization andreduction of mechanical systems,’’ Nonlinear Dynamics, vol. 41, 2005, p. 147.

6. Lawrence Sirovich, Turbulence and the dynamics of coherent structures,’’ Quarterly ofApplied Mathematics,vol. XLV, no. 3, October 1987, p. 561.

7. Dan Kalman, ‘‘A singularly valuable decomposition: The SVD of a matrix,’’ The CollegeMathematics Journal, vol. 27, no. 1, January 1996, p. 2.

8. G. Kerschen, et al. ‘‘Physical interpretation of the proper orthogonal modes using thesingular value decomposition,’’ Journal of Sound and Vibration, vol. 249, no. 5, 2002,p. 849.

9. T. P. Barnett, ‘‘Interaction of the Monsoon and pacific trade wind system at interannualtime scales. Part I: The Equatorial zone,’’Monthly Weather Review, vol. 111, 1983, p. 756.

10. S.L. Hahn, Hilbert transforms in signal processing, The Artech House, Signal ProcessingLibrary, 1996.

11. A. V. Oppenheim, and R. W. Shafer, Discrete-Time Signal Processing, 2nd Edition,Prentice Hall, 1998.

12. R. Dwi Susanto, et al. ‘‘Complex Singular Value Decomposition Analysis of EquatorialWaves in the Pacific Observed by TOPEX/Poseidon Altimeter,’’ Journal of Atmosphericand Oceanic Technology, vol. 15, 1998, p. 764.

13. J. D. Horel, ‘‘Complex principal component analysis: Theory and examples,’’ Journal ofclimate and Applied Meteorology, vol. 23, 1984, p. 1660.

14. James K. Kaihatu, et al., ‘‘Empirical orthogonal function analysis of ocean surfacecurrent suing complex and real-vector methods, ‘‘Journal of Atmospheric and OceanTechnology,’’ August 1998, p. 927.

15. A. R.Messina, et al., Leading-Edge Electric Power System Research, CianM. O’Sullivan(editor), Nova Science Publishers, Inc., 2008. New York.

16. L. Chen, et al.,’’ Proper orthogonal decomposition of two vertical profiles of full-scalenonlinearity downburst wind speeds,’’ Journal of Wind Engineering and Industrial Aero-dynamics, vol. 93, 2005, p. 187.

17. Davd L. Donoho, and Iain M. Johnstone, ‘‘Ideal spatial adaptation by wavelet shrink-age,’’ Biometrika, vol. 81, no. 3, 1994, p. 425.

18. M. Misiti, et al. ‘‘Wavelet Toolbox for use with Matlab,’’ The Mathworks Inc., 1996.19. Daniel J. Trudnowski, ‘‘Estimating Electromechanical Mode Shape from Synchrophasor

measurements,’’ IEEE Trans. on Power Systems, vol. 23, no. 3, August 2008, p. 1188.


Chapter 7

Detection and Estimation of Nonstationary Power

Transients

Gerard Ledwich, Ed Palmer, and Arindam Ghosh

Abstract This chapter looks at issues of non-stationarity in determining whena transient has occurred and when it is possible to fit a linear model to anon-linear response. The first issue is associated with the detection of loss ofdamping of power system modes. When some control device such as an SVCfails, the operator needs to know whether the damping of key power systemoscillation modes has deteriorated significantly. This question is posed here asan alarm detection problem rather than an identification problem to get a fastdetection of a change. The second issue concerns when a significant disturbancehas occurred and the operator is seeking to characterize the system oscillation.The disturbance initially is large giving a nonlinear response; this then decaysand can then be smaller than the noise level ofnormal customer load changes.The difficulty is one of determining when a linear response can be reliablyidentified between the non-linear phase and the large noise phase of thesignal.The solution proposed in this chapter uses ‘‘Time-Frequency’’ analysis tools toassistthe extraction of the linear model.

7.1 Introduction

For much of the time a power system operates in near steady state with gradualchanges in the overall loading and pattern of generation. There are continualchanges as different customers switch loads ON and OFF but when the systemamalgamates the loads of millions of customers the effect is one of continual smallperturbations. Occasionally, dramatic disturbances happen, lightning causes afault on a line which is then tripped, a boiler tube failure causes a generator togo off-line or a secondary systems fault causes a static VAR compensator (SVC) togo off-line. Even rarer is the cascading failure such as in the USA/Canada in 2003.

G. Ledwich (*)Faculty of Built Environment and Engineering, Queensland University of Technology,Brisbane, Australiae-mail: [email protected]


189

Here lines sagging into trees tripped, which shifted load onto other lines whichthen tripped under overload, generators became isolated, and there was massiveloss of supply for millions of customers. For a correct response to a transient, itcritical to know how severe it is and whether drastic response is required.

To ensure that the control response is appropriate the operators need toknow whether a change in performance has occurred and what are the condi-tions of the new level the system is moving to. This chapter explores two aspectsof small-signal models: the detection of change in the system and the identifica-tion of the new operating parameters.

7.2 Modal Damping Change Detection

When a power system is operating normally there are continual customer loadchanges which excite modal responses. If the damping parameters of the powersystem suddenly deteriorate due to the loss of a control item such as an SVCwith stabilizer signals, then the response of the modes will change. In particular,a reduction of damping will increase the level of response in one or more modes.While the level of customer changes exciting the system will change for thedifferent portions of the day, it is not expected that there will be significantshort-term variations in the level of customer variation so an increase of energyof a mode can be used to signal a reduction of damping. If there was a majorpower system transient that stayed in the linear domain then Prony techniqueswould be applicable to perform fast identification of the modal parameters. Ifthe loss of the control unit itself did not generate a significant transient, then theonly route to modal identification would be from the response of the systemdynamics to the background customer load variations. For accurate identifica-tion of the system, changes using this technique can take from 10 to 30 min ofdata to reach different levels of confidence in the identification. If the systemchange was to a marginally stable operating point this would be too long beforedetection. The approach in this section of the chapter is to develop a fastdetection of significant change rather than precise estimation of the parameters.This will be able to alert operators to the deterioration while more preciseevaluation of the change is being identified.

In this section, there are two approaches to this detection of change. The firstapproach is to monitor the energy of the signal around the known systemmodes. When the relative energy level of the mode rises beyond some threshold,it indicates a significant loss of damping relevant to that mode. Because thedamping level can be mapped to the magnitude of response of a mode thenthreshold levels can be identified and alarming processes set.

The second approach is based on Kalman filtering. If the system has beenoperating in one condition for an extended time then the modal parameters canbe identified and an ideal whitening filter can be identified. The whitening filterwould process measurements such as angle difference from center of area, and

190 G. Ledwich et al.

when the modes are identified, the error between the measurement and theKalman filter prediction of measurement would be random white noise. Nobetter predictor of the next measurement would be possible. Now, if there hasbeen a significant reduction of modal damping, the whitening filter would nolonger be perfect and the residual error would no longer be white. In particular,when amodal response magnitude increased due to loss of damping, the residuewould show a rise of energy at this frequency.

There is a strong interrelation between the two approaches but the theore-tical advantage of the Kalman approach is that the properties of the white noiseresidue are well characterized so the degree of confidence in showing thatdamping change has occurred with a precise degree of confidence or specifiedprobability of false alarms becomes standard statistical analyses.

7.2.1 Energy Detection Approach

This energy detection approach seeks to characterize the expected energy in theresponse of the power system and thus detect sudden changes in damping. Thework presented here is drawn from [1].

7.2.1.1 Theory

This section of the chapter presents formulae for the probability density func-tion (PDF) of the energy of x(n) under normal operating conditions. Theseoperating conditions can be determined by the types of techniques described in[2]. The availability of the PDF enables reliable thresholds to be set so that onecan create alarms if the energy deviates too much from the normal operatingconditions. Typically, one will set the threshold for a false alarm rate of 10% orless. Because energy can be calculated fairly rapidly, the computation does notprovide a significant barrier to rapid detection of modal deterioration.

Once an alarm has been ‘‘raised’’ it is necessary to furthermonitor the energy.A series of sequential data windows are collected and statistical comparisonsare made with the energy PDF. Consistently high-energy readings will triggercorrective action.

7.2.1.2 PDF Derivation

Typically customer induced changes in the load are unpredictable in the shortterm. The power is modeled as the integral of white noise as shown in Fig. 7.1.The power PDF is first derived as shown below. The energy PDF is subse-quently determined from the power PDF via a simple scaling of the axes. Basedon the model in Fig. 7.2, the discrete spectrum of the output signal is

XðkÞ ¼ HðkÞWðkÞ (7:1)

7 Detection and Estimation of Nonstationary Power Transients 191

and the output power is

P ¼XN�1

k¼0XðkÞj j2 ¼

XN�1

k¼0HðkÞj j2 WðkÞj j2 (7:2)

whereH(k) is the discrete Fourier transform (DFT) of h(n) andW(k) is the DFT

of w(n). Note that W(k) is a complex random variable (RV) with real and

imaginary parts

W kð Þj j2¼ Real W kð Þf g2þImag W kð Þf g2 (7:3)

Now if the variance of w(n) is �2, then the left-hand side of (7.3) is a �2 RV

with two degrees of freedom and variance, �2/N, that is.

f W kð Þj j2n o

¼ N

2�2e�xN2�2 (7:4)

Fig. 7.1 Model for quasi-continuous modal disturbances in a power system

Fig. 7.2 Equivalent model for quasi-continuous modal oscillations in a power system


Using (7.1) and (7.4) the PDF of X(k) can be deduced to be

fx Xkð Þ ¼ 1

HðkÞj j2

��

��fwx

HðkÞj j2

!¼ N

2 H kð Þj j2�2e�x

N

2 H kð Þj j2�2(7:5)

From (7.2) it is evident that the power is obtained by summingNRVs whereeach RV represents the ensemble power at discrete frequency k. Furthermore,these RVs have PDFs given by (7.5). The PDF of the sum (i.e., of the totalpower) is obtained by convolving the PDFs of all the RVs being summed. Thatis, the PDF of the total power is given by

fX xð Þ ¼ fXN�1 xN�1ð Þ � fXN�2 xN�2ð Þ � � � � � fX0x0ð Þ (7:6)

Expanding, we get

fX xð Þ ¼ N

2�2H 0ð Þj j�2e�xN2 H 0ð Þj j�2��2 � H 1ð Þj j�2e�xN2 H 1ð Þj j�2��2

� � � � � H N� 1ð Þj j�2e�xN2 H N�1ð Þj j�2��2

" #(7:7)

where * denotes convolution.The PDF of the energy that will correspond to a particular observation

interval is then obtained from the power PDF in (7.7) by simply rescaling theaxes by a factor equal to the observation length.

From the PDF, the threshold for detection of change can be formulated.Establishing say the 10% false alarm rate is via the cumulative summation ofthe PDF area until the 90% point is determined.

7.2.1.3 PDF Verification

To verify the theoretically determined system output PDF, simulations wereconducted with known modal parameters. These simulations created a data-base of outputs that were then formulated into a histogram. The simulatedhistogram and the theoretical PDF were then compared directly.

A two-mode system with modal parameters

Mode 1. !1 = 1.7 rad/s, �1 = –0.4 s–1

Mode 2. !1 = 2.7 rad/s, �1 = –0.52 s–1

generated the PDF shown in Fig. 7.3. It is based on a 20 s data acquisitionwindow. The variance of w(n), the disturbance excitation, was set to 1.0.The verification procedure involved creating a histogram from10,000 simulation runs of random noise feeding the known modal system,based on a 20 s data window with a sampling rate of 5 Hz. Statisticalcharacteristics of the simulated histogram and of the theoretical PDFwere also calculated and compared. The statistical characteristics


examined were the first three central moments: mean, variance, and skew.

The percentage errors between the theoretical and simulated PDFs are

shown in Table 7.1. The errors are all comparatively low, inspiring

confidence in the fact that the derived PDF is correct.

7.2.1.4 Results

Simulations for detecting change have demonstrated promising results. The

following simulation is for a two-mode system with the following stationary

modal parameters:

Mode 1. !1 = 1.7 rad/s, �1 = –0.4 s–1

Mode 2. !1 = 2.7 rad/s, �1 = –0.52 s–1

Mode 1 (only) was changed in simulations in the following way:

� No change from normal operation (quiescent damping of �1 = –0.4 s–1)between 0 and 100 min.

� Deteriorating damping from –0.4 to –0.1 s–1 between 100 and 200 min.� Reset damping at –0.2 s–1 for 100 min.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

10

20

30

40

50

60

70

80

90

Energy-Joules

Output half energy histogram vs PDF

Fig. 7.3 Histogram versus PDF

Table 7.1 Comparison of theoretical and simulated moments

Time window 20 s 40 s 60 s

Noise variance 1.0 1.0 1.0

% Error �1 0.94 0.00 0.00

�2 0.28 1.26 0.06

Skewness 3.97 6.38 6.01


The energy levels for the simulated output are then compared to three setthresholds:

1. 10% false alarm rate.2. 5% false alarm rate.3. 1% false alarm rate.

Simulations were run for two data windows, 20 and 40 s with results shownrespectively in Figs. 7.4 and 7.5.

From these simulation results the following can be observed: when there isno deterioration in the damping then the rate of false alarms falls within anacceptable tolerance of false alarm. In both the 20 and 40 s simulations thealarm rate becomes significantly higher as the damping deteriorates. The alarmrates are shown in Tables 7.2 and 7.3 for different data windows.

It can be seen in the above results that during quiescent operation(0–100 min) the occurrences of false alarms happen at a generally expectedrate. However, as the damping linearly deteriorates (100–200 min), the numberof alarms rises dramatically. The two data window lengths (20 and 40 s) bothexhibit similar responses in alarming to the deteriorating damping, although the40 s data window does provide more tangible evidence of change (a higher rateof alarming). This is due to the increased statistical reliability which occurs withlonger data records. The longer the intended data analysis window, the moretruly Gaussian the formulated PDF tends to become. Hence, the skewed tail ofthe PDF in Fig. 7.3 (for a 20 s window) will be minimized for longer datawindows and alarm thresholds will be lower in respect to the PDF mean.However, as rapid detection is a major requirement for system control [3],

0 50 100 150 200 250 3000

0.005

0.01

0.015

0.02

0.025

0.03

0.035Half Output Energy vs Time

Time-Minutes

Ene

rgy-

Joul

es

Energy10% FAR5% FAR1% FAR

Fig. 7.4 The 20 s data window of energy measurements with false alarm rates


shorter data analysis windows have been used in this section of the chapter. It

can also be observed in Figs. 7.4 and 7.5 that the alarm rate is very high between

180 and 200 min. The alarming occurs as the deteriorating damping approaches

�1¼ –0.1 s–1. This characteristic is highly desirable in rapid alarming situations

where a large detrimental change occurs within the system dynamics.

0 50 100 150 200 250 3002

4

6

8

10

12

14

16× 10

–3 Half Output Energy vs Time

Time-Minutes

Ene

rgy-

Joul

es

Energy10% FAR5% FAR1% FAR

Fig. 7.5 The 40 s data window of energy measurements with false alarm rates

Table 7.2 The 20 second data window alarm rates

Alarm rates

For threshold set togive false alarmrate¼ 10% at�1¼ –0.4 s–1



% Alarms for0–100 min nochange (�1 ¼–0.4 s–1)

6.33 3.33 1.67

% Alarms for100–200 minramped change(–0.4 to –0.1 s–1)

23.33 17.33 9.33

% Alarms for200–300 min setdamping (�1 ¼–0.2 s–1)

28 19.67 9.67


Another point to note is in the 200–300 min data analysis window, where the50% reduction in Mode 1 damping is also ‘‘detected’’ by the rise in the rate ofalarming with respect to quiescent operating conditions.

7.2.2 Introduction to Kalman Approach

It is proposed that the change detection be performed using measurements frompower systems in normal operation [4]. This section of the chapter initially definesa stochastic model relating random ambient disturbance inputs in the powersystem, such as customer load changes, to the measured system output. Based onthis model a Kalman filter is then set up to estimate the output arising from thedisturbances. The innovation is then determined as the difference between themeasured output and the estimated output. It is well known that the ‘‘innovation’’from aKalman filter is spectrally white as long as the assumedmodel parametersare valid [5, 6]. By monitoring the whiteness of the innovation, therefore, one candetect if there are any changes in the model parameters [6].

7.2.2.1 Theory (See Development in [7])

This section uses the theory of optimal detection of random signals [8] fordetecting changes to individual modes. The impulse response, h(n), is consid-ered to be the sum of h1(n) and h2(n), with h1(n) corresponding to the currentmode of interest, and h2(n) corresponding to the other modal components ofh(n). The output, x(n) is considered to have two components, x1(n) and x2(n),with x1(n) being the output due to the mode of interest and x2(n) being theoutput due to the other modes. The detection algorithm involves passing the‘‘observed signal’’ power spectral density (PSD) and the ‘‘reference signal’’ PSDthrough a whitening filter and cross-correlating the outputs. The algorithm isoutlined in Fig. 7.6. The ‘‘observed signal’’ is considered to be x(n), while the

Table 7.3 The 40 s data window alarm rates

Alarm rates




% Alarms for0–100 min no change(�1 ¼ –0.4 s–1)

10.66 6.67 1.33

% Alarms for100–200 min rampedchange (–0.4 to–0.1 s–1)

34.67 22.67 17.33

% Alarms for200–300 min setdamping (�1 ¼–0.2 s–1)

36.67 27.33 16


‘‘reference signal’’ is taken to be x1(n). It is assumed that there are Nsamples inthe observation.

The detection algorithm uses a threshold which is determined from a PDF ofthe cross-correlation test statistic. The availability of the PDF enables reliablethresholds to be set so that one can create alarms if the modal response deviatestoo much from the normal operating conditions. Typically, a threshold for afalse alarm rate of would be 10% or less. In practice, the monitoring processinvolves applying the detection algorithm to all the modes individually.

Because the detection algorithm can be calculated fairly rapidly, the compu-tational overhead does not provide a significant barrier to monitoring all theindividual modes. It should be noted though that an n-mode power systemwould require n parallel detectors to monitor each mode.

Once an alarm has been ‘‘raised’’ it is necessary to further monitor the modes.A series of sequential data windows are collected and statistical comparisonsare made with the stationary condition PDFs. Consistently high readings willtrigger corrective action on the deteriorating mode.

7.2.2.2 Individual Mode Test Statistic Details

The following quantities are first defined:

xðnÞ)= XðkÞ ¼W kð ÞH kð Þ (7:8)

h1ðnÞ)=H1ðkÞ (7:9)

h2ðnÞ)=H2ðkÞ (7:10)

x1ðnÞ)=X1 kð Þ ¼W kð ÞH1 kð Þ (7:11)

x2ðnÞ)=X2 kð Þ ¼W kð ÞH2 kð Þ (7:12)

where = indicates discrete Fourier transformation [9].

Observation PSD Whiten

Reference PSD Whiten

Cross-correlate

Threshold and Detect

Fig. 7.6 The detection algorithm


Now to detect a change in Mode 1 (the mode of interest), we choose X1(k) asthe (frequency domain) reference signal. The remainder of the frequencydomain observation (X2(k)) becomes the interference signal. A whitening filteris created to whiten the interference according to

Hwh kð Þ ¼ H�12 ðkÞ (7:13)

This whitening filter is applied to both the reference and observation signals.The corresponding PSDs are then determined:

PSDref kð Þ ¼ H1 kð Þj j2 Hwh kð Þj j2E WðkÞj j2n o

(7:14)

PSDobsðkÞ ¼ jX kð Þj2jHwh kð Þj2

¼ jWðkÞj2jHobsðkÞj2jHwhðkÞj2(7:15)

where Efg denotes the expected value.Now cross-correlate (7.7) and (7.8) to obtain a test statistic �

� ¼XN�1

k¼0PSDrefðkÞPSDobsðkÞ (7:16)

To practically apply the test statistic (7.16) in a detection process, a thresholdlevel must be determined. To intelligently set the threshold, a PDF of theexpected test statistic is required. A threshold can then be set based on thePDF at a desired level of confidence.

7.2.2.3 PDF Derivation

The formulation of theMode 1 test statistic PDF is as follows. To derive the teststatistic PDF, (7.16) can be expanded using (7.14) and (7.15) to give

� ¼XN�1

k¼0E W kð Þj j2n o

H1 kð Þj j2 H kð Þj j2 HwhðkÞj j4 W kð Þj j2 (7:17)

The above equation can then be rewritten as

� ¼XN�1

k¼0jZðkÞj2jWðkÞj2 (7:18)


where Z(k) is defined as

Z kð Þ ¼ jH1 kð ÞjjH kð ÞjjHwhðkÞj2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE W kð Þj j2n or

(7:19)

Now the expression in (7.18) containsW(k) which is a complex RV with realand imaginary parts. Furthermore, the square magnitude of |W(k)|2 is

W kð Þj j2¼ Real W kð Þf g2þImag W kð Þf g2 (7:20)

Where Real{} and Imag{} denote the real and imaginary parts, respectively.

If the variance of w(n) is �2, then the left-hand side of (7.20) is a �2 RV with twodegrees of freedom and variance, �2/N. That is,

f W kð Þj j2n o

¼ N

2�2e�xN2�2 (7:21)

Using (7.18) and (7.21), the PDF of |Z(k)|2|W(k)|2 at discrete frequency k can

be deduced to be

f ZWð Þ ¼ 1

ZðkÞj j2

��

��fwx

ZðkÞj j2

!¼ N

2 Z kð Þj j2�2e�x

N

2 Z kð Þj j2�2(7:22)

From (7.18) it is evident that the test statistic is obtained by summingNRVs.

Furthermore, these RVs have PDFs given by (7.22). The PDF of the sum(i.e., of the test statistic) is obtained by convolving the PDFs of all the RVsbeing summed. That is, the PDF of the test statistic for Mode 1 is given by

f� zwð Þ ¼ fZWN�1 zwN�1ð Þ � fZWN�2 zwN�2ð Þ

� � � � fZW0zw0ð Þ

(7:23)

f� zwð Þ ¼ N

2�2

Z 0ð Þj j�2e�xN2 Z 0ð Þj j�2��2

� Z 1ð Þj j�2e�xN2 Z 1ð Þj j�2��2

� � � � � Z N� 1ð Þj j�2e�xN2 Z N�1ð Þj j�2��2

2664

3775 (7:24)

where * denotes convolution.


From the PDF in (7.24), the threshold for detection of change can be

formulated. To establish the 10% false alarm rate, the cumulative summation

of the PDF area is taken until the 90% point is determined.

7.2.2.4 Results

In practical systems; modal parameters are quasi-stationary and the values

determined via modal estimate algorithms [10, 11] applied to data measured

at optimal points within a power system as outlined in [12]. However current

algorithms for modal parameter estimation require significant data lengths for

accurate estimates. Therefore, to demonstrate the rapid detection ability of this

section’s algorithm, a 40 s data analysis window will be applied.The simulation in this section is for a two-mode system with the following

stationary modal parameters:

Mode 1. !1 = 1.7 rad/s, �1 = –0.4 s–1

Mode 2. !1 = 2.7 rad/s, �1 = –0.52 s–1

Mode 1 (only) was changed in the simulations in the following way:

� No change from normal operation (quiescent damping of �1 = –0.4 s–1)between 0 and 100 min.

� Deteriorating damping from –0.4 to –0.1 s–1 between 100 and 200 min.� Reset damping at –0.2 s–1 for 100 min.

The tests statistics for the simulated output are then compared to three set

thresholds

1. 10% false alarm rate.2. 5% false alarm rate.3. 1% false alarm rate.

Simulations were run for the 40 s data windows, involving change detection

of Mode 1 andMode 2. The results are shown respectively in Fig. 7.7–Table 7.4

and Fig. 7.8–Table 7.5.It can be seen in the above results that during quiescent operation

(0–100 min) the occurrences of false alarms are at a generally expected rate

for both Mode 1 (Fig. 7.6) and Mode 2 (Fig. 7.7), respectively. As the damping

of Mode 1 linearly deteriorates (100–200 min) the number of Mode 1 alarms

rises dramatically (Fig. 7.6). Importantly, it is shown in Fig. 7.4 and in Table 7.2

that the quiescent Mode 2 false alarms do not rise.It can also be observed in Fig. 7.7 that the alarm rate is quite high between

180 and 200 min. The alarming occurs as the deteriorating damping approaches

�1 = –0.1 s–1. It clearly indicates the Mode 1 change whilst Fig. 7.8 indicates

that there is no deterioration toMode 2. This characteristic is highly desirable in

rapid alarming situations where a large detrimental modal change occurs within

the system and the aberrant mode requires identification.


Another area of interest is within the 200–300 min data analysis window

where there is a 50% reduction in Mode 1 damping. The analysis provides

significant alarming to this event (Fig. 7.7), whilst still maintaining no signifi-

cant deterioration has occurred in Mode 2 (Fig. 7.8).

0 50 100 150 200 250 3000

5

10

15

20

25

30Neta vs Time

Time-Minutes

Test

Sta

tistic

Net

a

Neta10% FAR5% FAR1% FAR

Fig. 7.7 The 40 s data window of Mode 1 test statistic with false alarm rates shown

Table 7.4 The 40 s data window Mode 1 alarm rates

Mode 1 alarm rate

For threshold set togive false alarmrate¼ 10% at�¼ –0.4 s–1



% Alarms for0–100 min (nochange)

11.33 4.67 1.33

% Alarms for100–200 min Mode1: ramped change(–0.4 to –0.1 s–1)

34.67 30.67 20.67

% Alarms for200–300 min Mode1: set damping(�1 ¼ –0.2 s–1)

44.00 36.00 22.00


Other simulations (not shown here) involving three mode systems have

displayed similar successes in identifying individual modal change.

0 50 100 150 200 250 3001.5

2

2.5

3

3.5

4

4.5

5Neta vs Time

Time-Minutes

Tes

t Sta

tistic

Net

a

Neta10% FAR5% FAR1% FAR

Fig. 7.8 The 40 second data window of Mode 2 test statistic with false alarm rates shown

Table 7.5 The 40 s data window Mode 2 alarm rates

Mode 2 alarm rate




% Alarms for0–100 min (nochange)

7.33 2.00 1.33

% Alarms for100–200 min Mode1: ramped change(–0.4 to –0.1 s–1)

2.67 2.00 0.67

% Alarms for200–300 min Mode1: set damping(�1 ¼ –0.2 s–1)

4.67% 0.67% 0.67%


7.2.3 Application to Real Data [7]

Data were obtained from the Australian power system, comprising voltageangle measurements at the Adelaide, Melbourne, and Sydney measurementsites from 22:00 on 09/04/2004 to 03:05 on 10/04/2004. As mentioned earlier,voltage angle measurements were used rather than power signals because thepotential for modal information extraction is greater than when using powersignals [13]. It is generally understood that inter-area modes have frequencies inthe range 0.1–0.8 Hz and so this will be the region of focus in the PSD [14].

To formulate the state–space model and consequently the system Kalmanestimator, a knowledge (or at least an accurate estimate) of the power systemtransfer function must be available. Accordingly, for the work reported in thissection, a long-term estimator (LTE) was applied to estimate this transferfunction [15]. This LTE was determined using a 45 min window. Short-term(change) detection was then continuously applied to the PSD of the Kalmaninnovation.

The real data analysis was conducted in two parts. Part I closely examined305 min of the Melbourne measurement data difference from the center of areaof the connected system (called Melbourne-COA) and Part II briefly analyzedthe data collected at the Sydney and Adelaide sites. Part II also examined theopportunity for combining the multisite innovation power spectrum data toenhance the detector performance.

7.2.3.1 Part I: Analysis of the Melbourne Data

The LTE was determined from the data between 120 and 165 min after the startof the measurement record. The LTE quasi-stationary modal estimates wereobtained using the technique in [15] and are listed in Table 7.6. The LTE alsoprovides an estimate of the measurement site transfer function in Laplace form.As a result, an estimate of the system quasi-stationary frequency response forMelbourne-COA can be observed in Fig. 7.9. In Fig. 7.9, Mode 1 peak is quiteapparent at 0.33 Hz, whileMode 2 peak (estimated to be at 0.59 Hz) is harder todistinguish due to the relatively heavy modal damping.

Once the long-term estimate of system characteristics was established, theremaining 140 min were examined in 1 min intervals for any sudden detrimentalchanges to system modes. To demonstrate the significant information theinnovation sequence contains, one only has to compare the differentiatedangle measurements with the normalized innovation obtained after application

Table 7.6 Damping and frequency for longtime estimates over 120–135 min

Mode Damping Frequency (Hz)

1 0.2913 0.33

2 1.0083 0.59


of the Kalman estimator, shown in Fig. 7.10. The comparison appears to

support the result noted by Kailath in [5] – namely, that the Kalman filter

innovation sequence contains the same information as the system output

sequence, but in a less correlated form.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

1

3

× 10–3

Hz

Melbourne Frequency Response

Mag

nitu

de (

units

)

Fig. 7.9 Melbourne frequency response estimate from LTE at 165 min (portions reprinted,with permission, from [16]. # 2007 IEEE)

Fig. 7.10 Comparison of (a) system output and (b) normalized innovation (portions rep-rinted, with permission, from [16]. # 2007 IEEE)


It is apparent from inspection of Fig. 7.10 that the system output and the

innovation both demonstrate a deviation from a quasi-stationary operation

within the 197–201 min interval. To lend some numerical support to this visual

inspection a 60 s analysis windowwas applied to the innovation PSD before and

during the 197–201 min interval. The results of the analysis are provided in

Figs. 7.11, 7.12, and 7.13. The innovation is shown in Fig. 7.11(a) with the

196–197 min interval marked off by two vertical lines. The spectrum of the

196–197 min segment of innovation is shown in Fig. 7.11(b). To account for

the fact that with real data, there is less certainty than there was with simulated

data, a different threshold level will be used for the detection. Accordingly, the

99.999%CI threshold is shown as a dashed horizontal line. Even though such a

threshold could be regarded as high, the main focus is to only detect large

detrimental changes and minimize false alarms. In the single site analysis, the

data does exhibit a wide variance and hence a high threshold is required to

minimize the false alarms. However in the following section, this issue is

addressed and the ability to have more acceptable threshold CIs is presented

along with a measured number of false alarms.

Fig. 7.11 (a) Innovation sequence �(n) and (b) innovation PSD at 196–197 min (portionsreprinted, with permission, from [16]. # 2007 IEEE)


In examining Fig. 7.11, no part of the spectrum crosses the threshold.Figure 7.12 depicts the innovation segment and associated spectrum correspond-ing to the 197–198 min segment. For this segment the innovation spectrum crossesthe threshold. Moreover, the threshold is crossed at the 0.59 Hz frequency posi-tion, indicating a loss of damping for Mode 2. Figure 7.12 shows the innovationsequence and associated spectrum corresponding to the 198–199min segment. Forthis segment the innovation spectrum crosses the threshold in an even morepronounced way than it did in Fig. 7.13(b). Again, the threshold is crossed atthe 0.59 Hz frequency position, indicating a damping change for Mode 2.

Further analysis was performed which showed that the loss of damping wastemporary. By the 201st minute of the data record, the modes reset to theiroriginal characteristics.

7.2.3.2 Part II: Combining Multisite Data for Enhanced SNR and Detection

Similar data analysis was conducted for the other two measurement sites inSydney and Adelaide (with COA correction). A comparison of the results forthe 196–197 and the 198–199 min time frame can be seen in Figs. 7.14(a) and



7.15(a), respectively. Figure 7.14(a) demonstrates a spectrum for Sydney and

Adelaide, which does not cross the threshold prior to the event. Figure 7.15(a)

shows a detection of damping deterioration centered at 0.59 Hz at both the

Sydney and Adelaide sites. These results confirm the detection registered at the

Melbourne site for the same analysis window.Within these plots, all sites exhibit similar responses to detection of damping

deterioration. Therefore, a combinationof the innovation spectrum was exam-

ined to assess opportunities for an enhanced detector. The combined spectrum

was obtained by adding the complex innovation spectrums from all three sites.The threshold for the combined innovation spectrum will need to be set

differently to that for the individual innovation spectra. Ideally, if the individual

normalized innovation spectra are all uncorrelated with one another then the

samples of the combined innovation spectrum will again have a �2 PDF with 28of freedom, �2, but with three times the variance (assuming individual normal-

ized innovations have unity variance) [15]. In the case of the innovation spec-

trums from the multisite data as examined in this paper, the innovation spectra

will not be strictly independent, as the power system is interconnected.



Nonetheless, empirical experiments have indicated that a �2 distribution, withvariance 3�2 is a suitable way to model the PDF.

In the general case whereXmeasurement sitesmust be combined, the samplesof the combined normalized innovation spectrum would have a �2 PDF with 2degrees of freedom [17, 18], and variance X�2. Thus using the normalizedinnovation the generalized ensemble frequency PDF for a combination of X,N-point innovation spectra would be

f � kð Þf g ¼ N

Xe�� kð ÞNX (7:25)

Hence for the three measurement sites, X =3, (7.25) simplifies to

f � kð Þf g ¼ N

3e�� kð ÞN3 (7:26)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–50

–40

–30

–20

–10

0(b)Combination Normalised Innovation Spectrum

Frequency Hz

dB/H

z

PSD @ 196 mins

Innovation CombinationThresholdest Mode 1 frequencyest Mode 2 frequency

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–50

–40

–30

–20

–10

0(a)Individual Normalised Innovation Spectrum

Frequency Hz

dB/H

z

SydneyMelbourneAdelaideThreshold

Fig. 7.14 Normalized (a) Individual innovation PSDs for Sydney, Melbourne, and Adelaide(detection threshold and modal frequency estimates also shown) (b) Combination PSD at196–197 min showing new threshold with false alarm rate (FAR) of 99.9% (portions rep-rinted, with permission, from [16]. # 2007 IEEE)


where (7.26) is the expected ensemble PDF of the three combined spectra

assuming the power system is quasi-stationary and the resulting innovation, white.The resulting combined innovation spectra prior and during the disturbance

are shown in Figs. 7.13(b) and 7.14(b), respectively. It is important to note two

significant outcomes attributed to the combining of the normalized innovation

spectra. First, the combination of the three sites has lead to an improvement in

the signal-to-noise ratio (SNR) for detectable signals. The SNR analysis results

are shown in Table 7.7, in which the bold letters emphasize the key results. The

improved SNRs exhibit comparable values to an ideal theoretical improvement

of 4.72 dB, whereby an ideal theoretical improvement is one that would be

expected if three identical deterministic signals, with independent, equal var-

iance Gaussian white noise were spectrally combined. Hence, the resulting

theoretical expected improvement under these criteria for three sites is

SNRimprovement ¼ 10 log10 3sitesð Þ ¼ 4:77 dB (7:27)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–50

–40

–30

–20

–10

0(a)Individual Normalised Innovation Spectrum

Frequency Hz

dB/H

z

SydneyMelbourneAdelaideThreshold

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–50

–40

–30

–20

–10

0(b)Combination Normalised Innovation Spectrum

Frequency Hz

dB/H

z

PSD @ 198 minsAlarm @ 0.58333Hz

est Mode 1 frequencyThresholdInnovation Combination

est Mode 2 frequency

Fig. 7.15 Normalized (a) Individual PSD and (b) combination PSD at 198–199 min showingnew threshold with FAR of 99.9% (portions reprinted, with permission, from [16]. # 2007IEEE)


Although the result in (7.27) was verified both analytically and empirically,in practice however, the responses from the three sites are not perfectly inde-pendent and the detection signals exhibit similar statistical properties. Hence,the combining of the innovation spectra will exhibit slightly less than thetheoretically maximum improvement in SNR.

7.2.3.3 Summary of Kalman Approach

The simulations indicate that the method proposed in this section can effec-tively monitor individual modal changes in power systems. It does so with apredetermined level of confidence that sets the alarm at a desired false alarmrate. Although not shown here, longer time widows exhibit a greater degree ofconfidence in a detected change, and can be used in conjunction with the shortertime windows to confirm or disaffirm a need for remedial action.

In practice, the monitoring is done for all modes concurrently. It should benoted that the methods outlined in this section only apply to multimodesystems. Rapid detection of detrimental mode change within single-mode sys-tems can use the method discussed in [7].

7.3 Estimation ofModal Parameters fromNonstationary Response

7.3.1 Introduction

The estimation of power system modes, in particular mode shapes andeigenvalues, is of importance in the design of stabilization controls such aspower system stabilizers. The use of eigenvalues to describe modes is derivedfrom the concept of a linear model for a power system. A power systemhowever, like many systems is actually nonlinear and so the linear model isat best only a good approximation when the system is not subjected to largedisturbances.

In recent years, much research has focused on both active [19, 20], andpassive [21], estimation of modes using parametric, and in some cases, non-parametric signal processing techniques, as detailed in [22]. Active estimationhas involved subjecting the system to a disturbance and analyzing the powerflows that result using a number of linear estimation techniques. It is the active

Table 7.7 SNR improvement through combination of site analysis

Analysis window during alarm signal (min)

Parameter 197–198 198–199 199–200 200–201

Average single-site SNR (dB) 9.60 7.60 10.82 11.64

Combined SNR (dB) 13.98 12.00 15.27 15.80

SNR improvement (dB) 4.38 4.40 4.45 4.16


estimation of power system modes using large disturbances that this paper willfocus on. The results are also applicable to recorded large disturbance eventsthat may naturally occur within a power system from time to time.

The estimation of modes presupposes a linear model for a power system.Any disturbance applied to a power system as part of an active modalestimation study runs the risk of exciting nonlinear system behavior. Becausea linear model for a power system is basically a small-signal model theprobability of nonlinear behavior increases with the size of the applieddisturbance. The purpose of this section is to present a range of techniqueswhich can

1. readily detect if any nonlinear behavior is present in a disturbance record,and

2. if any nonlinear behavior is present which decays with time, obtain anaccurate modal estimate.

7.3.2 Estimation of Linear Power System Models

In state–space form the standard homogeneous linearized model of a powersystem is of the form

_x ¼ Ax (7:28)

where x is a state vector consisting of machine angles, velocities, andother machine-related variables such as fluxes and A is the state matrix.The eigenvalues of A give information on the damping and frequency ofeach mode, while the eigenvectors give information related to the parti-cipation of each state in the various modes. The latter is usuallyexpressed in terms of participation factors pki which represents the parti-cipation of the kth state in the ith mode. These are computed from theleft and right eigenvectors of A according to [23, p. 715].

From the model in (7.28), the oscillatory time-domain response of thelinearized power system with M complex eigenvalues of the form lI = �I �j2pfi is

y tð Þ ¼XM M2= 2

i¼1Ai exp �itð Þ cos 2pfitþ �ið Þ (7:29)

The estimation of the eigenvalues involves the estimation of the dampingfactors �i, modal frequencies fi, amplitudes Ai, and phase angles �i.

The estimation of the eigenvalues of A may be performed either parametri-cally or nonparametrically. For details of some of the nonparametric techni-ques that have been proposed, see [9, 24,25].


A parametric technique used by many workers in the analysis of powerswings following large disturbances is Prony’s method, [11, 19, 26–28]. Theuse of Prony’s method assumes a linear time-invariant system and using a linearprediction models fits a set of P complex exponential functions to a given datarecord. The linear predictor model of a set of N data samples of a discretizedsignal y(n) is

y nð Þ ¼XP

i¼1Ai exp �i þ j2pfið Þ n� 1ð ÞTþ j�ið Þ (7:30)

which maybe written as

y nð Þ ¼ �XP

i¼1a ið Þy P� ið Þ (7:31)

The discrete time eigenvalues zk are then found as the roots of the character-istic equation

� zð Þ ¼XP

m¼1a mð Þzp�mk (7:32)

The estimation procedure which is detailed in [11] is as follows:

1. Find the coefficients a(k) that best fit the data record.2. Form and determine the roots of (7.32).3. Determine the damping factors, modal frequencies, modal amplitudes and

modal phases using (7.33) below

�k ¼ln zkð ÞT

fk ¼tan�1 Im zkð Þ

Re zkð Þ

h i

T(7:33)

where T is the sampling period.

7.3.3 Time–Frequency Representations

The use of Prony’s method and other linear parametric/nonparametric methodsassumes the absence of nonlinear and time-varying dynamics. Time–frequencyrepresentations (TFRs) originally designed to analyze nonstationary signals area logical choice as a method of testing this assumption. Examples of nonsta-tionary signals occur in radar, swept sine wave testing, communications, and


biomedical phenomena. A simple piece of music is another example of a non-stationary signal.

The response of a linear system to a disturbance will consist of a linearcombination of modal frequencies which do not change with time. This is nottrue of a nonlinear system, and in many cases the response will be a nonsta-tionary signal.

The spectral variation of a nonstationary signal is measured by its instanta-neous frequency (IF). The IF is a measure of how the frequency of a nonsta-tionary signal changes with time. The IF fi is defined by Eq. (7.34),

fi ¼1

2pd�ðtÞdt

(7:34)

where z(t) = z(t)ej�(t) is the analytic version of the measured signal y(t)computed by taking its Hilbert transform, see [29, Chapter 1] and �(t) isknown as the instantaneous phase. If the signal is a constant frequencytone then �(t) = 2pf. A simple example of a nonstationary signal is a chirp,whose IF is a linear function of time. The analytic version of a chirp signalwould be given as

zðtÞ ¼ ej2p�t2

2 þf0t� �

and would have an IF given by fi(t) = � t + f0. An example of a linear chirpsignal where �= 50 and f0 = 0 Hz is illustrated in Fig. 7.16.

A time–frequency distribution (TFD) seeks to display both temporal andspectral data on a three-dimensional plot. The short-time Fourier transform(STFT) is perhaps themost intuitive of TFDs. The STFT computes a successionof spectra localized about a time t by first multiplying the signal by a windoww() centered about t = . Thus the STFT S(t,f) is computed as (7.35)

S t; fð Þ ¼Z 1

�1s ð Þw t� ð Þe�j2pf d (7:35)

The STFT, like all TFRs, represents a compromise between time-domainresolution and frequency-domain resolution. The longer the window w(), thebetter the frequency resolution but at the expense of reduced time-domainresolution and vice versa. It has been shown in [29, Chapter 2] that an optimalrectangular window length is

� ¼ffiffiffi2p dfi

dt

��

��1

2

(7:36)

where fi is the IF of the signal. Because this is unknown a priori, the STFT islimited in its usefulness.


The Wigner–Ville distribution (WVD) W(t, f), of a signal S(t) is defined by(7.37)

Wðt; fÞ ¼Z1

�1

w

2

� �w��2

� �s tþ

2

� �s� t�

2

� �exp �j2pfð Þ d (7:37)

where w(t) is the time-domain window applied to the signal. W(t, f) is basicallythe Fourier transform of the autocorrelation function of a signal in the lagvariable.

A discrete version of theWVD,W(n, k) where n is the time index and k is thefrequency index may be computed as

Wðn; kÞ ¼ 2XM�1ð Þ=2

m¼�ðM�1Þ=2w mð Þw� mð Þs nþmð Þs� n�mð Þe�j4pmk=M (7:38)

where w is an M point window centered at m ¼ 0. Algorithms for determiningthe WVD can be found in [29, Chapter 6].

TheWVD has the property of giving the optimal trade-off between temporaland frequency resolution for a linear FM signal, that is, a signal whose IF is alinear function of time. In the case of a chirp signal or a sinusoid that hasconstant IF the continuous time WVD consists of a series of delta functionslying along the IF law.

An example of the WVD plot for a chirp signal is shown in Fig. 7.17. The IFis described by a ridge in the time–frequency plane.

Fig. 7.16 Linear chirp signal with � ¼ 50 and f0 ¼ 0 Hz


Where there is only one component it is straightforward to compute theinstantaneous frequency as

fiðtÞ ¼ f :Wðt; fiðtÞÞ ¼ maxf

Wðt; fÞj j (7:39)

The WVD being a quadratic TFD suffers, however, from the presence ofcross-terms when more than one component is present in the signal. In otherwords, if s(t) ¼ s1(t) + s2(t), then the WVD of s(t) is

Wss t; fð Þ ¼Ws1s1ðt; fÞ �Ws2s2ðt; fÞ �Ws2s2ðt; fÞ �Ws2s1ðt; fÞ

These cross-terms appear at intermediate frequencies and may obscureaspects of the time–frequency behavior of a system. This effect can in part bemitigated by replacing the signal in (36) by its Hilbert transform [30].

To reduce these effects further, a number of TFRs have been proposed basedon filtered versions of the WVD. The general form of these other TFRs is givenin (7.39).

W t; fð Þ ¼Z1

�1

G t; ð Þ �ts tþ

2

� �s� t�

2

� �exp �j2pfð Þ d (7:40)

where �t denotes convolution in the time variable such that

Fig. 7.17 WVD of linear chirp signal shown in Fig. 7.16


G t; ð Þ �ts tþ

2

� �¼Z1

�1

G l; ð Þs t� lþ 2

� �dl

and G(t,) is a function of time and lag called the time–lag kernel. The time–lagkernel can be engineered to filter out the effect of the highly oscillatory cross-terms. For further details of this, the reader is referred to [29, Chapter 2].Various toolboxes exist for computing the WVD and other TFRs. The onesused in this paper may be downloaded from http://tftb.nongnu.org/ asMATLAB ‘.m’ files and form a package of time–frequency routines developedby Francois Auger, Oliver Lemoine, Paulo Gonclaves, and Patrick Flandrinunder the auspices of the Centre National de la Recherche Scientifique. Wheremore than one component is present and there is more than one phase functionthe meaning of IF needs to be refined. Some workers in the field use thefollowing definition, see [31]:

fi ¼

R1

�1fW t; fð Þ df

R1

�1W t; fð Þ df

(7:41)

which for monocomponent signals is the same as (7.34).

7.3.4 Application to Transient Stability Swings

In the presence of a small disturbance a power system should behave in a linearfashion characterized by a number of modes with time-invariant parameters. Inthe case of a large disturbance, however, the linear assumption is not valid andnot only do the modes lose their decoupling but variation in frequency of eachmode may occur. The coupling phenomenon that occurs was investigated in[32], while the effect onmodal estimates was been investigated in [27]. In [27] thenonlinearities of the transmission system, excitation system current limiting,load models and field saturation were investigated and shown to affect eigen-value estimates.

In this chapter, the transmission system nonlinearity is the only nonlinearitythat is focused on. While this was found to be not most significant nonlinearityin the cases considered in [27] it can be the most significant nonlinearity whenthe system is subjected to a severe disturbance. This is indeed the case with thesystem considered in this chapter.

This is illustrated for the 21 bus, six-machine system shown in Fig. 7.18 anddescribed in [33, 2]. The model was originally designed to model a typicalAustralian network which in general are not meshed networks and consequen-tial have very well-defined modes. The machine models used were fifth order


and the loads are modeled as constant impedances. Further details of the

parameters of this system may be obtained from either [33] or [34].A modal analysis of this system reveals the mode shapes detailed in Table 7.8.

The mode shapes were determined by examining the phase angles of the angular

velocity components of the eigenvectors corresponding to electromechanicalmodes.There are five electromechanical oscillation modes associated with this

system. Three of these modes are local modes only involving two machines.

The other two are inter-area involving coherent groups of generators. It is the

inter-area modes that are of principal interest as they typically have lower

damping than the faster local modes.A disturbance in the form of a three-phase fault to ground is applied at bus

14 which excites inter-area modes 4 and 5. This means that components of both

modesmay be present inmeasured data depending onwhere the data is taken. If

the data are taken from machine 5 terminals only one mode, mode 5, will be

present, while at the terminals of machine 3 both modes 4 and 5 will be present.

The local modes 1–3 will not be excited by this fault.If the fault is short this corresponds to a small disturbance whilst a prolonged

fault constitutes a large disturbance likely to excite transmission system non-

linearities. Ringdown plots of machine 5 angular velocities are shown in

68

75

192017

16

92

121

1514

1312

103

11

4

18

Fig. 7.18 Twenty-one bus test system

Table 7.8 Eigenanalysis of six-machine system

Mode no. Description Eigenvalue

1 Machine 3 vs. 4 �1:25� j16:1812 Machine 1 vs. 2 �0:86� j12:82

3 Machine 5 vs. 6 �1:33� j12:01

4 Machines 1 and 2 vs. 3 & 4 �0:37� j10:89

5 Machines 5 and 6 vs. rest �0:13� 6:41


Figs. 7.19 and 7.21. The variation in frequency with time for a large disturbance

is apparent from Fig. 7.21 but to properly quantify this, a TFD is necessary.

WVDs were computed for each case using the entire downsampled analytic

version of the original signal. The preprocessing of the data involved:

1. Downsampling the signal from a 50Hz sampling rate to a 5Hz sampling rate2. Computing the Hilbert transform of the downsampled signal to obtain the

analytic version of the downsampled signal.

The effect of the size of the disturbance on the time–frequency spectra of the

angular velocity of machine 5 is illustrated by Figs. 7.20 and 7.22. Figure 7.20

shows a constant modal frequency following a short disturbance indicating that

a linear model is an accurate representation of the system in this case over the

entire transient period.Figure 7.22, on the other hand, shows that a time-varying frequency appears

following a large disturbance eventually settling down to a constant frequency.

This phenomenon cannot be explained by a simple linear model and so it is clear

that the transmission system nonlinearities are having a significant effect. These

effects, however, become less significant as the angular velocity swings decay.

Hence, the IF in the large disturbance case eventually reaches a constant

frequency which is consistent with the predictions of the linear model.The advantage of the WVD over the STFT in representing the time-varying

frequency is illustrated in Fig. 7.23 that shows the STFT of the signal whose

WVD is shown in Fig. 7.22. It is clear from comparing Figs. 7.22 and 7.23 that

the WVD plot gives a sharper image of the variation of modal frequency with

time In other words, the WVD has better time and frequency resolution than

Fig. 7.19 Machine 5 ringdown following small disturbance (clearing time ¼ 0.20 s)


the STFT. In doing so, this enables better the user to estimate the time variation

of the modal frequency.In Table 7.8, it is shown that of the two inter-area modes, machine 5 only

participates in mode 5. Hence, theWVD computed from the angular velocity of

machine 5 consists of only one component and so cross-terms are not of

concern. The local mode in which machine 5 participates is not excited by the

applied disturbance. For initially illustrating the technique presented in this

section, the discussion will concentrate on the case where a single mode is

Fig. 7.20 WVD of machine 5 angular velocity – small disturbance

Fig. 7.21 Machine 5 ringdown after a large disturbance (clearing time ¼ 0.85 s)


dominant in the response. For this reason, the discussion will initially focus on

the response of machine 5.Application of Prony’s method to the entire data set obtained from the large

disturbance case will clearly produce erroneous answers, as is shown in Table 7.9.

As expected, the error incurred in the case of a large disturbance is significant.

Fig. 7.22 WVD of machine 5 angular velocity – large disturbance (clearing time ¼ 0.85 s)

Fig. 7.23 STFT of machine 5 angular velocity – large disturbance (clearing time ¼ 0.85 s)


7.3.5 Error Reduction by Time-Domain Windowing

As expected, the modal estimates obtained in the presence of the nonlinear

effect of the transmission system are erroneous, and this nonlinear effect is

clearly illustrated by theWVD. It is seen from theWVD that system does attain

a linear state with constant frequency once the amplitude of the oscillations

have died down. An obvious way of removing the nonlinear effects is to

determine a region in time where the nonlinear effects as described by the

WVD are negligible and only perform Prony analysis (or any other linear

modeling technique) over that region.The instantaneous frequency may be estimated by peak detection in the

time–frequency plane, in accordance with Eq. (7.38). Using this technique, it

is possible to generate a plot of IF over the length of the transient (Fig. 7.24).To determine the boundary of this region we note from Fig. 7.24 that the

time-varying frequency reaches 95% of the steady-state value after about 10 s.

Clearly, it is easier to determine the point where the modal frequency has

reached steady state using the WVD than the more intuitive STFT on account

of its superior time and frequency resolution.It should be noted that after about 25 s the transient has decayed below the

noise floor and the IF estimate from that point onward is meaningless.

Table 7.9 Prony analysis

True eigenvalue Small disturbance estimate Large disturbance estimate

–0.13 � j6.41 –0.1 � j6.28 –0.19 � j5.35

Fig. 7.24 Instantaneous modal frequency, fi


If the signal is truncated so that only the section between 10 and 25 s after the

onset of the disturbance is retained, then the Prony modal estimate is as shown

in Table 7.10.This process is further validated by projecting backward the predictions of a

linear model fitted to the portion of the transient following the point where the

IF has reached a constant value. It is seen from Fig. 7.25 that the linear model

fails to accurately predict the transient response in the period before the IF

becomes constant but is a good predictor over the latter part of the transient

when the nonlinear effects have become negligible.As can be seen there is a dramatic improvement in the Prony estimate by

simply windowing the signal between the time when the nonlinear aspect has

sufficiently decayed and the time when the signal has decayed to the noise floor.

Clearly, the discarding of data must be done with some degree of caution, and

so, with this in mind varying window sizes were investigated.Defining fpu(t) = fi(t)/fss to be the per unit instantaneous frequency relative

to the steady state frequency fss, the percentage error in eigenvalue estimates for

windows commencing at different values of fpu is depicted in Fig. 7.26.The plot in Fig. 7.26 clearly illustrates the improvement in the eigenvalue

estimate error from nearly 16% for an estimate based on the entire data set to

less that 2 % when the window commences at a time when the per unit

instantaneous frequency has reached 0.95.

Table 7.10 Windowed eigenvalue estimates

True eigenvalue Unwindowed eigenvalue estimate Windowed eigenvalue estimate

–0.13 � j6.41 –0.19 � j5.35 –0.13 � j6.33

Fig. 7.25 Back prediction of linear model. ***, prediction of linear model; —, machine 5angular velocity)


7.3.6 Discussion

A power system is by nature multimodal. Discussion so far has only concen-trated on a signal containing a single mode. A disturbance will in general exciteseveral modes and these will be apparent in the signal record depending uponthe location of the measurement site. In the 21-bus system considered here,there are two inter-area modes excited by the applied fault. Machine 5 onlyparticipates in one of these modes and hence there is only onemodal componentpresent in its angular velocity signal. Machine 3, on the other hand, participatesin both inter-area modes as is evident from the WVD plot of the angularvelocity for machine 3 shown in Fig. 7.27. Here the cross-term effect discussedin Section 7.2.3 is apparent. This may be removed at the expense of some loss ofresolution using a reduced interference TFR such as the Choi–Williams Dis-tribution (see [29, Chapter 2]). However, in this case it is obvious that compo-nent lying in frequency midway between modes 4 and 5 is the cross-term effect.

Now it should be noted that the damping factor of the two modes aresignificantly different, the damping of the higher frequency mode being larger.The windowed Prony technique can be used provided both modes have left thenonlinear region. If however, one mode has decayed significantly by the timethe other component has left the nonlinear region, then it would be preferable toapply different windows to the two components. This would avoid the loss ofinformation about the more heavily damped mode, which may occur by trun-cating the signal at the point in time where the lightly dampedmode’s IF law hasreached a constant value. This would necessitate the filtering off of the longerlasting component in order to identify the more heavily damped one.

Fig. 7.26 Error in eigenvalue estimate versus per unit instantaneous frequency at the start ofthe window


Also as the higher frequency mode is more heavily damped the effectiveduration of that component of signal is less, which in turn creates a greaterspread in the time–frequency plane of that component. This is evident inFig. 7.26 and would make the determination of an IF law for the more heavilydamped component more difficult.

If an analysis is made of the data from machine 3 it is found that the error inboth inter-area eigenvalue estimates is much greater when the entire data set isused. If a window is used which is truncated to begin at the point where aconstant IF is attained, errors in the identification of both modes of less than3% are obtained. This is quite a good performance and tends to validate theymethod for well-separated modes. The effect of varying window start points isshown in Figs. 7.28 and 7.29.

Of course, not all modes are within the resolution limit of the DFT. In suchcases if a time-varying IF law is not obvious from the standardWVD one couldattempt to use a high-resolution WVD as described in [35]. The high-resolutionWVD relies upon the use of high-resolution spectral estimation techniques ofthe sort described in [22] in place of the FFT in computing Eq. (7.37).

The computational load of such high-resolution processing is quite intensive;however, with modern computing power it may well be possible to pursue thisapproach further. Work is continuing in addressing the issue of multiple modes.

Another factor that could be of assistance in identifying the modal para-meters where multiple modes are excited is the use of multiple measurementsites. Some sites will bemore sensitive to certain modes than others, and it mightbe possible to use different sites for measurement of modes of interest. Methodsfor choosing such sites are described in [2].

Cross -Term

Mode 4

Mode 5

Fig. 7.27 WVD – machine 3 angular velocity – multimodal example


Finally, it may be possible to extend this work by developing a means of

modal identification directly from the WVD. While estimation of modal fre-

quency is reasonably straightforward provided the signal is not too highly

damped, the estimation of damping presents challenges.

Fig. 7.28 Error in eigenvalue estimate versus per unit frequency at the start of the window –lightly damped mode

Fig. 7.29 Error in eigenvalue estimate versus per unit frequency at the start of the window –heavily damped mode


In conclusion, the method should prove useful in a real power systemwhere asingle dominant mode is excited by a disturbance in

1. Check to see if a nonlinear effect is present2. If one is present, then place a limit on the range of the record that can be used

with any linear estimation technique.

Finally it should be noted that TFDs may be used in any application where atime-varying frequency or transient event may be present. For this reason,discrete control events such as line switchings or load shedding may be detect-able used time–frequency analysis of machine or bus angle variations. Similarly,inter-modal coupling as described in [32] should also be observable and quanti-fiable using a TFD. In this case, a reduced interference TFD would be recom-mended in order to remove cross-term effects.

In terms of computational burden, an N length data sequence computing aWVD requires o(8N2 log2 2N) real multiplications and o(10N2 log2 2N) real addi-tions, [36], thus giving a total of o(18N2 log2 2N) floating point operations.Depending on the number of windows used in computing the STFT the numberof floating point operations involved is o(MN log2N) whereM is the number ofwindows used. Hence, it can be seen that the computation of a WVD is morecomputationally intensive. However, this must be weighed against the improve-ment in time–frequency resolution that results. It is this improvement that makestheWVD a better tool for estimating the point in a transient where the nonlineareffects may be discounted and a linear model may be fitted with confidence.

7.3.7 Recommendations for Time–Frequency

TheWVD has been shown to be of benefit in estimating power systemmodes incases where large disturbances have been applied to the power system. It hasbeen shown that the application of a linear estimation technique such as Prony’smethod over the entire disturbance record can yield very inaccurate results onaccount of system nonlinearities being excited.

Provided modes are separable by the DFT, it should be possible to identifymodal parameters more accurately using the WVD to design a time domainwindow over which to apply a linear technique such as Prony (windowed Prony).

7.4 Conclusions

Time-varying effects are normal for power systems but make the application ofstandard analysis tools more difficult. The first contribution for this chapterwas to define when a significant change had occurred in the dynamics of thesystem by analyzing the dynamic responses to the continuous load changes.Using detection theory rather than identification, the probability of false alarms


was directly set. The second issue was the identification of system modes

following a major system change. Initially the disturbance is large and linear

identification would be flawed. When the disturbance becomes smaller linear

theory applies but the signal is now more difficult to distinguish from the

background noise. The contribution is to show how time–frequency analysis

can yield improved modal estimates in this time-varying context.

References

1. R. Wiltshire, P. O’Shea, and G. Ledwich ‘‘Rapid Detection of Deteriorating ModalDamping in Power Systems’’ AUPEC, 2004.

2. E. W. Palmer and G. Ledwich, ‘‘Optimal placement of angle transducers in powersystems,’’ IEEE Transactions on Power Systems, Vol. 11, pp. 788–793, 1996.

3. T. George, J. Crisp, and G. Ledwich, ‘‘Advanced tools to manage power system stabilityin the National Electricity Market,’’ AUPEC, 2004.

4. D. J. Trudnowski, ‘‘Estimating electromechanical mode shape from synchrophasormeasurements,’’ IEEE Transcations on Power Systems, Vol. 23, No. 3, pp. 1188–1195,August 2008.

5. T. Kailath, ‘‘An innovations approach to least-squares estimation – Part I: Linear filter-ing in additive white noise,’’ IEEE Transactions on Automatic Control, Vol. 13,pp. 646–655, 1968.

6. R. K. Mehra and J. Peschon, ‘‘An innovations approach to fault detection and diagnosisin dynamic systems,’’ Automatica, Vol. 7, pp. 637–640, 1971.

7. R. A. Wiltshire, P. O’Shea, and G. Ledwich, ‘‘Monitoring of Individual Modal DampingChanges in Multi-Modal Power Systems,’’ AUPEC, 2004.

8. H. VanTrees, Detection, Estimation and Modulation Theory, Part 1. New York: JohnWiley, 1968.

9. P. O’Shea, ‘‘A High Resolution Algorithm for Power System Disturbance Monitoring’’IEEE Transactions on Power Systems, Vol. 17, No. 3, pp. 676–680, Aug. 2002.

10. V. Vittal, ‘‘Consequence and impact of electric utility industry restructuring on transientstability and small-signal stability analysis,’’ Proceedings of the IEEE, Vol. 88,pp. 196–207, 2000.

11. J. F. Hauer, ‘‘Application of Prony analysis to the determination of modal content andequivalent models for measured power system response,’’ IEEE Transactions on PowerSystems, Vol. 6, pp. 1062–1068, 1991.

12. N. Uchida and T. Nagao, ‘‘A new eigen-analysis method of steady-state stability studiesfor large power systems: S matrix method,’’ IEEE Transactions on Power Systems, Vol. 3,pp. 706–714, 1988.

13. G. Ledwich and E. Palmer, ‘‘Modal estimates from normal operation of power systems,’’IEEE Power Engineering Society Winter Meeting, 2000.

14. M.Klein, G. J. Rogers, and P. Kundur, ‘‘A fundamental study of inter-area oscillations inpower systems,’’ IEEE Transactions on Power Systems, Vol. 6, pp. 914–921, 1991.

15. P. Z. Peebles, Probability, random variables, and random signal principles, 4th Ed. NewYork: McGraw Hill, 2001.

16. R. A. Wiltshire, G. Ledwich, and P. O’Shea ‘‘A Kalman Filtering Approach to RapidlyDetecting Modal Changes in Power Systems’’ IEEE Transactions on Power Systems, Vol.22, No 4, Paper No: TPWRS-2007.907529.

17. H. Urkowitz, ‘‘Energy detection of unknown deterministic signals,’’ Proceedings of theIEEE, Vol. 55, pp. 523–531, 1967.


18. R. A. Wiltshire, P. O’Shea, and G. Ledwich, ‘‘Monitoring of Individual Modal DampingChanges inMulti-Modal Power Systems,’’ Journal of Electrical & Electronics EngineeringAustralia, JEEEA, Vol. 2, pp. 217–222, 2005.

19. J. F. Hauer, C. J. Demuere, and L. J. Scharf, ‘‘Initial Results in Prony analysis of powerSystem Response Signals’’, IEEE Transactions on Power Systems, Vol. 5, No. 1,pp. 80–89, Feb. 1990.

20. N. Zhou and J. W. Pierre, ‘‘Electromechanical Mode Estimation of Power Systems fromInjected Probing Signals using a Subspace Method’’, Proceedings of the 35th NAPSConference, Rolla Mo, Oct. 2003.

21. J. W. Pierre, D. J. Trudnowski, andM. K. Donnelly, ‘‘Initial results in ElectromechanicalMode Identification fromAmbient Data’’, IEEE Transactions on Power Systems, Vol. 12,No. 3, pp. 1245–1251, Aug. 1997.

22. S. M. Kay, Modern Spectral Estimation, Englewood Cliffs, NJ: Prentice-Hall, 1988.23. P. Kundur, Power System Stability and Control, New York: McGraw-Hill, 1994.24. K. Poon and K. Lee, ‘‘Analysis of Transient Stability Swings in Large Interconnected

Power Systems by Fourier Transformation’’, IEEE Transactions on Power Systems,Vol. 3, No. 4, pp. 1573–1579, Nov. 1988.

25. P. O’Shea, ‘‘The Use of Sliding Spectral Windows for Parameter Estimation in PowerSystem Disturbance Monitoring’’, IEEE Transactions on Power Systems, Vol. 15, No. 4,pp. 1261–1267, Nov. 2000.

26. J. F. Hauer, ‘‘Application of Prony Analysis to the Determination of Modal Content andEquivalent Models forMeasured Power System Response’’, IEEE Transactions on PowerSystems, Vol. 6, No. 3, pp. 1062–1068, Aug. 1991.

27. D. J. Trudnowski and J. E. Dagle, ‘‘ Effects of Generator and Static-Load Nonlinearitieson Electromechanical Oscillations’’, IEEETransactions on Power Systems, Vol. 12, No. 3,pp. 1283–1289, Aug. 1997.

28. (2007) [Online] Available http://www.newcastle.edu.au/service/library/adt/public/adt-NNCU20070504.091618/index.html

29. B. Boashash, Time Frequency Signal Analysis and Processing – A Comprehensive Refer-ence, Oxford: Elsevier, 2003.

30. B. Boashash and P. J. Black, ‘‘An Efficient Real-Time Implementation of the Wigner-Ville Distribution’’, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol.35, pp. 1611–1618, Nov. 1987.

31. I. Kamwa and L. Gerin-Lajoie, ‘‘State-Space System Identification – Toward MIMOModels for Modal Analysis and Optimization of Bulk Power Systems,’’ IEEE Transac-tions on Power Systems, Vol. 15, No. 1, pp. 326–335, Feb. 2000.

32. J. J. Sanchez-Gasca, V. Vittal, M. J. Gibbard, A. R. Messina, D. J. Vowles, S. Liu, and U.D. Annakkage, ‘‘Inclusion of Higher Order Terms for Small-Signal (Modal) Analysis –Task Force onAssessing the Need to IncludeHigher Order Terms for Small-Signal (Modal)Analysis’’, IEEE Transactions on Power Systems, Vol. 20, No. 4, pp. 788–793, Nov. 2005.

33. E. Palmer, Multi-Mode Damping of Power System Oscillations, PhD thesis, The Univer-sity of Newcastle, 1998.

34. E. W. Palmer and G. Ledwich, ‘‘Optimal placement of Angle Transducers in PowerSystems’’, IEEE Transactions on Power Systems, Vol. 11, No. 2, pp. 788–793, May 1996.

35. H. Whitehouse, B. Boashash, and J. Speiser, ‘‘High resolution processing techniques fortemporal and spatial signals’’, in High Resolution Techniques in Underwater Acoustics,Lecture Notes in Control and Information Series, New York-Heidelburg-BerlinSpringer-Verlag, 1989.

36. B. Lovell, R. Williamson, and B. Boashash, ‘‘The Relationship between InstantaneousFrequency and Time-Frequency Representations’’, IEEE Transactions on Signal Proces-sing, Vol. 41, No.3, pp. 1458–1461, Mar. 1993.


Chapter 8

Advanced Monitoring and Control Approaches

for Enhancing Power System Security

Sergio Bruno, Michele De Benedictis, and Massimo La Scala

Abstract Enhancements of power system security can be achieved by develop-

ing an online environment where control center operators have the capability to

monitor in real time the power system dynamic behavior, recognize threats to its

integrity, evaluate, and implement suitable control actions. Themethodological

approaches proposed here are respectively a spectral analysis based on wavelet

transform and a response-based wide-area control for improving power system

dynamic behavior on the transient timescale. Themonitoring technique consists

in processing real-time data coming from phasor measurement units by adopt-

ing an approach based on wavelet spectral analysis. The final result is a diagram

that shows the amplitude of different modes usually associated to different

phenomena (electromechanical modes, inter-area oscillations, etc.) and their

related dampings. The methodology proposed for control assessment is based

on the solution of a dynamic optimization problem whose basic variables are

acquired through a wide-area measurement system.

8.1 Introduction

In recent years, drastic changes in the utility business and an intense sequence

of large blackouts on several major power systems occurred. Many major

countries involved in the restructuring of electric power industry have faced

emergency conditions and blackouts affecting large portions of national and

transnational grids.In the last decades, the management of the transmission systems was based

on well-known system behaviors, full identification of load/generation patterns

and configurations. In a restructured system, the environment is more uncertain

and the system operator (SO) has a scarce direct control over the generation

S. Bruno (*)Dipartimento di Elettrotecnica ed Elettronica (DEE), Politecnico di Bari, Bari, Italye-mail: [email protected]


231

side. National and transnational grids have beenmanaged with lack of data andin the presence of a large number of new uncertainties.

The main side effect of the electricity industry restructuring was pushinginvestments toward the generation side, core of the electrical energy business,neglecting security and transmission systems. The protection system, the silentsentinel that guarantees the integrity of the network and provides elementaryconditions for executing any financial and physical power exchange, often isnot upgraded taking advantage of the most updated technologies. In mostcases, transmission networks are exploited close to their capacity limits, withlack of transfer capacity due to the continuous growth of energy demand and todifficulties in building new transmission lines.

In some cases, transmission capability is not limited by localized systemresponse but by wide-region dynamic constraints (i.e., transient stability limits,inter-area oscillations, and voltage collapse). Because the time response is muchfaster than the one associated with thermal overloads, predicting, detecting, andcontrolling such phenomena requires real-time sensors, fast computing, fastcommunication units, and fast actuators sufficiently disseminated all across thegrid.

There are several countermeasures and approaches to system securitythat can be adopted: methodologies for automation and control [1], dynamicsecurity assessment (DSA) and communication systems for real-time dataexchanges [2], adaptive relays [3], FACTS (flexible AC transmission system)and HVDC (high-voltage direct current) technology, and real-time measure-ments and control systems (wide-area measurement systems – WAMS andwide-area measurement and control systems – WAMC) [3, 4]. By means ofphasor measurements units (PMUs) and Global Positioning Systems (GPS)technologies, it is possible to get a state estimate which is a true snapshot of thepower system. Real-time calculations can be performed for a correct estimationof grid capacity on critical sections. Other advanced technologies (FACTS orHVDC)may guarantee the development of flexible transmission systems givingthe SO more tools for controlling power and enhancing the capabilities oftransmission lines.

In the following developments, it is shown how the potentials of wide-areamonitoring and control can be exploited in at least two approaches oriented toimprove power system dynamics on the transient timescale.

WAMS can be adopted in power system monitoring, covering what is arelevant aspect of power system operation and control. For instance, by adoptingsimultaneous synchronizedwide-areameasurements, it is possible to perform real-time monitoring of power system dynamics. An effective monitoring of powersystem dynamic performances can be assessed through wide-area informationand distributed synchronized PMUs. In [5], the IEEE PES Power SystemDynamic Performance Committee had indicated, among the main causes for thepast cascading events, the lack of reliable real-time data oriented to themonitoringof system dynamics. In general, possible failures can be avoided or mitigated byperforming an accurate monitoring of the system [5], keeping track of oscillation

232 S. Bruno et al.

modes during the evolution of operating conditions and detecting the undesirableonset of poorly damped oscillations. Advanced technologies like WAMS, cansupport power system monitoring, providing real-time information about thestate of large portions of national and transnational grids [6].

The usual approach adopted for applying corrective control consists inarming corrective action schemes (remedial action schemes or RAS) whichare triggered only after that the occurrence of a specific fault is detected(event-driven actions). The arming of RAS is based on off-line (preventive)evaluations. Differently, response-based control schemes apply correctiveactions, basing on real-time measurements and on the actual system responseto disturbances and control actions [7].

Ongoing communication technology permits to monitor the status of largeportions of the network, whereas feedback control signals may permit to applyremedial actions in fractions of a second.With this response time, it is conceivableto control a significant number of unstable cases in potentially dangerous areas(critical corridors) [8, 9]. In order to implement these remedial actions, fastactuators are needed. Classically, load/generation shedding can be fast enoughto correct the undesirable transient behavior. More advanced fast devices such asFACTS or line switching can be adopted for the same purpose. In recent years,unified power flow controller (UPFC) has been recognized useful for theenhancement of the power system dynamic behavior [10, 11].

8.2 Monitoring Power System Oscillations by Wavelet Analysis

and Wide-Area Measurements

8.2.1 Approaches for Monitoring Power System Oscillations

Monitoring power system oscillations can be accomplished by means of severalmethodological approaches. Undoubtedly, the eigenvalue analysis is the mosteffective method for studying power system small-signal stability. Major limita-tions of this approach are related to the computational burden in the presenceof large networks.Moreover, the effectiveness of this method can be affected bythe unavailability of a detailed representation or by the onset of nonlinearbehaviors (saturations or relays’ triggering) during stressed conditions or inpresence of large disturbances.

In [12], data from transient stability simulations are processed by meansof Hilbert spectral analysis, in order to characterize the time evolution ofnonstationary power system oscillations, and eventually tracking the dynamics ofcritical system modes. Other feasible approaches for the identification of powersystemmodes of oscillation are based on the direct spectral analysis of power systemresponse.

The Fourier transform is largely adopted for studying stationary signals,where the properties of signals do not evolve in time. Nevertheless, this method

8 Advanced Monitoring and Control Approaches 233

is less efficient in tracking nonstationary signals, where any abrupt change mayhave effect on the whole frequency axis [13]. A usual solution is the short-timeFourier transform (STFT) that employs a time window for mapping a signalinto a two-dimensional function of time and frequency. The drawback is thatonce a particular size for the time window is chosen, that window width has tobe kept constant for all frequencies.

In [14], a technique for analyzing system oscillations by employing the Fouriertransformation is presented. The major frequency components are identified inthe frequency domain and the corresponding damping constants are determinedcomparing the magnitudes of the respective frequency components in differentsubsequent time windows. The Fourier transform is commonly employed inpower network analysis and has the advantage of not requiring a heavy compu-tational effort. On the other hand, this method can be efficient only when aimedat finding a mode of oscillation whose frequency is already known. Finding anunknown frequency can be difficult and require several attempts with windowshaving different widths [15].

The Prony method decomposes time-domain signals into a sum of dampedsinusoids, each characterized by four parameters: frequency, damping, ampli-tude, and phase. This method is able to deal with the nonlinearities of powersystems and can also be applied to field measurements. The size of the model isnot limited by the technique because only the output is analyzed [16]. A scientificvalidation of the Pronymethod as a valuable tool in estimating themodal contentof power system oscillations and its methodological improvements have beenshown in [17, 18].

A new approach to online assessment and control of transient oscillationsis proposed in [19] taking into account Prony analysis and improving itsapplicability and reliability. Comparisons of Prony and eigenvalue analysis, high-lighting potentials and drawbacks of eachmethod, have been presented in [16, 20].In [21], a technique for extracting power system modes based on spectral analysishas been proposed. The approach, adopting the Z-transform identification algo-rithm of Corinthios, permits the identification of the damping factor and thefrequency of signal containing multiple modes.

Wavelet spectral analysis is based on a variable-sized windowing technique.A major advantage of wavelets is the ability to perform local analysis, i.e., toanalyze a localized area of a larger signal. Wavelet transform (WT) algorithmsprocess data with different scales, stretching or compressing the basic waveletfunction, providing a multiresolution analysis in both frequency and time.Wavelet analysis allows using respectively long time intervals for capturinglow-frequency information, and shorter windows for high-frequency informa-tion so that the characteristics of typical nonstationary power disturbances canbe efficiently captured and assessed.

WAMS technology, applied to real-time supervision of power systems is atechnology at hand, but security monitoring of system dynamic performances iscurrently a challenge in terms of operator interface implementation. From anoperator viewpoint, an overall perspective of relevant processes is important

234 S. Bruno et al.

and useful in order to achieve the online supervision of power system dynamicperformances. A contribution to this issue is given in [22], where direct spectralWT-based analysis is proposed as man–machine interface (MMI) for dynamicperformance monitoring and DSA. Analogously, in [23], a WT-based spectralanalysis approach has been applied on real data measured by PMUs duringan experiment on large scale in the Russian Far East Interconnected PowerSystems. The approach allows the detection of the disturbance and themonitoringof the behavior of principal modes of oscillation.

8.2.2 Morlet-Based Wavelet Analysis

Wavelet analysis can be performed adopting different formulations of the WT.Each formulation is characterized by specific features (and suitable applica-tions); for example, the WT can be continuous or discrete, real or complex, andis characterized by the shape given to the mother wavelet function.

The continuous wavelet transform (CWT) offers a good resolution, althoughit requires a significant computational burden and memory usage. Very often inpractical application, the discrete wavelet transform (DWT) can be adopted. Itpermits to discretize scale and time parameters, and its transform is not acontinuous function of the frequency. The DWT is fast and reduces redundantinformation and computational efforts.

Real and complex wavelets give, in general, different results. A real waveletproduces high values of magnitude when the oscillation reaches a maximum (ora minimum), or in the presence of sharp discontinuities. On the other hand, thecomplex wavelet is able to produce magnitude all along the duration of anoscillating signal.

In the proposed approach, the complex continuous wavelet can be adoptedbecause fast filtering and signal reconstruction that usually require high compu-tational efforts are not necessary.Moreover, the availability of continuous valuesof magnitude all along the duration of the signal, increases the feasibility of theapproach in terms of MMI, avoiding the discontinuity created by real CWTs.

Regarding the formulation of the mother wavelet, several functions havebeen proposed in the literature (Gauss,Morlet,MexicanHat, Gabor, etc.), eachhaving its suitable applications [9, 24]. In this approach, the continuous com-plex Morlet WT has been adopted because it offers a very good resolution inboth time and frequency: its frequency resolution is adequate even at the lowestwave number. Furthermore, as shown in the followings, the Morlet waveletpreserves the information on damping in the time–frequency domain [25–27].

The CWT of a time domain signaly tð Þis defined as

CWTcy b; að Þ ¼ 1ffiffiffi

ap

Zþ1

�1

yðtÞc� t� b

a

� �dt (8:1)


where c� tð Þ is the complex conjugate of the analyzing (or mother) wavelet c tð Þ,a is the dilation parameter, whereas b is the parameter that sets the waveletfunction in the time domain.

The complex Morlet wavelet, defined in its curtailed form [27], is

cðtÞ ¼ e�t2

N e j!0t (8:2)

where !0 is the central frequency (wave number) of the wavelet and N (withN > 0) is a parameter that controls the shape of the mother wavelet.

The following discussion is aimed at showing how the Morlet WT is able topreserve damping information in the frequency–time domain.

A general harmonic signal can be formulated as

yðtÞ ¼ AðtÞ cos ’ðtÞð Þ (8:3)

where AðtÞ and ’ðtÞ are time-varying envelope and phase functions.In [27], it has been shown how, if the signal expressed in (8.3) decays to zero

at +1, and therefore,

Zþ1

�1

yðtÞj j2 dt5þ1 (8:4)

then its WT given by Eqs. (8.1) and (8.2) is

CWTcy ðb; aÞ ffi

ffiffiffiap

2

ffiffiffiffiffiffiffiNpp

AðbÞe�N4 a _’ðbÞ�!0ð Þej’ðbÞ (8:5)

A particular application of given formulas is the calculation of the WT of asignal y(t), representing the response of a system governed by a second-orderlinear differential equation:

yðtÞ ¼ A0e��t cos !dtþ ’0ð Þ (8:6)

where � ¼ �!n is the damping defined by the product of the natural pulsation !nand damping ratio �, !d ¼ !n

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

pis the pseudopulsation, and ’0 is the

initial phase value.In this case, for a fixed value of the dilation parameter a = ai, modulus and

phase of the CWT of the signal represented in (8.6) are

CWTcy ðb; aiÞ

�� ¼

ffiffiffiffiaip

2

ffiffiffiffiffiffiffiNpp

AðbÞe�N4 ai!d�!0ð Þ2 (8:7)

ff CWTcy ðb; aiÞ

h i¼ !dbþ ’0 (8:8)

236 S. Bruno et al.

By fixing two time instants b1 and b2, from (8.7) and (8.8) one derives:

CWTcy ðb1; aiÞ

��

CWTcy ðb2; aiÞ

��¼ A0e

��b1

A0e��b2¼ e� b2�b1ð Þ (8:9)

and hence:

� ¼ 1

ðb2 � b1Þlogn

CWTcy ðb1; aiÞ

��

CWTcy ðb2; aiÞ

��

(8:10)

Equation (8.10) shows how the WT of a signal y tð Þ allows identifying the

damping associated to each oscillation mode contained into the signal.The normalized dB value of the CWT magnitude can be expressed as

CWTðb; aiÞ ¼ 20Log10CWTðb; aiÞj jCWTmax

(8:11)

Considered two time instants b1 and b2, with b2> b1, themagnitude variation

�dB ¼ CWTðb2; aiÞ � CWTðb1; aiÞ can be expressed as

�dB ¼ 20Log10CWTðb2; aiÞj jCWTðb1; aiÞj j ¼

20

logn 10logn

CWTðb2; aiÞj jCWTðb1; aiÞj j (8:12)

Considered that logða=bÞ ¼ � logðb a= Þ, from (8.12) one derives:

lognCWTðb1; aiÞj jCWTðb2; aiÞj j ¼ �

�dB logn10

20(8:13)

If the analyzed mode can be represented as in Eq. (8.3) and has an envelope

decreasing exponentially with time, (8.13) can be substituted in (8.10), obtaining

Eq. (8.14):

� ¼ ��dB logn 10

20ðb2 � b1Þ¼ ��dB logn 10

20�b(8:14)

The wavelet function proposed in (8.2) has also the property of linearity and

therefore it is easy to extend the same WT approach to a signal yðtÞ composed

by a sum of different signals.


8.2.3 A WAMS-Based Monitoring Architecture

The proposed monitoring architecture is aimed at detecting large disturbancesoccurring in a control area or in external systems, evaluating the damping ofrelevant oscillation modes related to electromechanical transients.

A pictorial representation of the proposed centralizedmonitoring architectureis shown in Fig. 8.1.

PMUs, located in strategic points of the power system (substations, genera-tion buses, or important interconnectors), provide real-time measurements ofvoltage and current phasors (positive sequence). GPS provides synchronizingsignals for voltage and current samples. The PDC block represents the phasordata concentrator that collects, synchronizes, and normalizes all data sent byPMUs through the communication system, and exports them as a data stream.Dotted lines in Fig. 8.1 represent the communication links: for this architecture,both wired and wireless communication links can be adopted. In [28], an over-view of communication alternatives is given, assessing the related time delayswhich are in the range 100–700 ms.

As soon as measured trajectories are received in the Control Center, a WTdiagnosis is carried out for detecting and analyzing themodes of oscillation of thepower system. The final result is a wavelet mapwhich shows, in a suitable scale ofcolors, the amplitude of different frequencies (modes) usually associated todifferent phenomena (electromechanical modes, inter-area oscillations, etc.)and an assessment of the related damping. The proposed approach providesresults in a compact and concise form that can be adopted as useful tool forMMI. The same interface can be adopted in an off-line mode through a suitableOTS (operator training simulator) environment. By this tool, operators can betrained to identify and handle dynamic threats.

Fig. 8.1 Representation of the proposed monitoring architecture

238 S. Bruno et al.

The supervision of the system can be performed by analyzing only signals

provided by several PMUs located in the network. Real-time voltage trajectories,

differently from system state variables, can be directly measured and then adopted

formonitoring dominant oscillationmodes. In general, it is easy to imagine that the

choice of the PMUs’ location should be made basing on off-line calculations of

mode observability matrices or on the experience in system operation. Such selec-

tion is aimed at monitoring only those voltages that proved to be able to better

observe the oscillation modes that must be kept under surveillance.

8.2.4 Test Results A: Monitoring System Responseto Small Perturbations

This section is aimed at showing how principal modes of oscillation can be identified

bymeans ofWT-based spectral analysis and how the complexMorletWT preserves

the main properties of these modes (frequency and damping) in the time–frequency

domain. More explicitly, eigenvalue analysis is compared to the WT-based direct

spectral analysis in a test case where small-signal analysis is performed.The eigenvalue analysis and the dynamic simulation were carried out on a

test grid obtained bymodifying the IEEE 30-bus ‘‘New England’’ Dynamic Test

Case [29]. The main differentiation from the latter is that the adopted grid has

six generators instead of two generators and four synchronous compensators

(Fig. 8.2). Each generator is represented with a fourth-order model and is

equipped with a governor and an exciter.The eigenvalue analysis of the system for a specific operating point led to the

identification of several modes of oscillation. In Table 8.1, selected modes of

oscillation and their characteristics are reported.

Fig. 8.2 Scheme of the modified IEEE 30-bus test grid


In order to study the dynamic response to small signals, the system has been

perturbed by applying a 0.1% load increase at bus #18. The Complex Morlet

WT has been employed for analyzing the rotor angle trajectories subsequent to

the disturbance. Figures 8.3 and 8.4 show the wavelet chart of the rotor angle

signals for two generators, at bus #8 and at bus #11, respectively. TheWTof the

rotor angle signals showed that three modes of oscillation, namely, modes #3,

#5, and #6, are dominant. Mode #4 is characterized by frequency and damping

values close to mode #3.A comparison between the eigenvalue analysis and the spectral analysis can be

performed through the participationmatrix. In fact, the participationmatrix, obtained

by combining the left and the right eigenvectors, is able to measure the coupling

betweenmodes of oscillation and state variables. The elements of thismatrix are called

participation factors, and eachof themgives ameasure of the relative participationof a

state variable to a mode and vice versa. In Table 8.2, the participation factors of

principal modes, with respect the rotor angle state variables, are reported.

Table 8.1 Features of principal modes of oscillation

Mode EigenvalueFrequency(Hz)

Dampingratio � !n (rad/s)

1 –1.35� j 9.18 1.46 0.145 9.282 –1.20� j 8.07 1.28 0.147 8.163 –0.23� j 7.61 1.21 0.030 7.614 –0.38� j 7.30 1.16 0.052 7.315 –0.56� j 5.56 0.88 0.100 5.596 –0.18� j 3.89 0.62 0.046 3.897 –6.29� j 1.76 0.28 0.963 6.538 –6.23� j 1.73 0.28 0.964 6.47

Fig. 8.3 Wavelet spectral analysis of the rotor angle trajectory at bus #8

240 S. Bruno et al.

Thewavelet chart of a signal can be represented in a three-dimensional space:

frequency,time,andmagnitude.Mode#5,characterizedbya0.88Hzfrequency,isclearlyvisible inFig. 8.3 (rotorangle trajectoryatbus#8) and in theotherwavelet

charts (not shown). InFig. 8.4,mode#5 is less discernible, according to the lower

participationfactorwithrespect togeneratoratbus#11.Modes #3 and #6 are mostly related to the behavior of generator at bus #11,

and are neatly discernible only in Fig. 8.4. This is confirmed by the participation

factors that, for this generator, differ by an order of magnitude from the ones

related to the other generators. In particular, mode #6, being characterized byvery low participation factors with respect to the rotor angle and by high

participation factors with respect to regulators’ state variables, is related to

the local control of the machine at bus #11.The same oscillation modes can be identified by applying the WT to the

voltage magnitude signals. Figures 8.5 and 8.6 show the wavelet chart of the

voltage magnitude signals at the generator at bus #8 and the one at bus #11,

respectively.In Figs. 8.5 and 8.6, the same dominant oscillation modes can be identified.

The possibility of discerning amode, observing a specific trajectory, depends on

Fig. 8.4 Wavelet spectral analysis of the rotor angle trajectory at bus #11

Table 8.2 Participation factors of the principal modes with respect to the rotor angle statevariables

General bus Mode #3 Mode #5 Mode #6

#2 0.0287 0.0765 0.0012

#5 0.0248 0.0732 0.0015

#8 0.0496 0.1838 0.0036

#11 0.4858 0.0513 0.0439

#13 0.0824 0.0978 0.0022


the corresponding values in the mode observability matrix. In this case, mode

#6, previously hardly discernible by observing rotor angle trajectories at bus #8,

can be clearly identified by analyzing voltage trajectories at the same bus.By ‘‘cutting’’ the three-dimensional representation in correspondence of the

frequency of a specific mode, it is possible to represent the magnitude of that

mode as a function of time. Figure 8.7 has been obtained in correspondence of

0.88 Hz (the frequency of mode #5) by the WT previously shown in Fig. 8.4.As shown in Fig. 8.7, the magnitude of mode #5 increases in the first seconds

and, having reached a maximum, decreases exponentially with time (in the

Fig. 8.5 Wavelet spectral analysis of the voltage magnitude at bus #8

Fig. 8.6 Wavelet spectral analysis of the voltage magnitude at bus #11

242 S. Bruno et al.

figure the decrease is linear because the dB scale is a logarithmic scale).The

increasing behavior is due to the necessity for the algorithm to acquire the main

modes; consequently, damping evaluated in this phase cannot be utilized for

stability assessment.Measuring the steepness of the curve, it is possible to derive

the damping of the mode #5, already known by its eigenvalue.As demonstrated in Eqs. (8.11, 8.12, 8.13), if the mode decreases exponen-

tially, having a magnitude variation �dB in the time interval �b, the damping

can be evaluated with relation (8.15):

� ¼ ��dB logn 10

20�b� � 0:115�dB

�b(8:15)

Figure 8.7 shows how, when the mode decreases exponentially, a reduction of

14.6 dB is obtained in 3 s (from 4 to 7 s). From Eq. (8.15), the damping �=0.56.

This value is identical to the one evaluated through the eigenvalue analysis

(Table 8.1), confirming the fact that the complex Morlet WT preserves informa-

tion on oscillation modes’ damping.

8.2.5 Test Results B: Influence of Random Fluctuations Dueto Operating Conditions

Previous simulationswere performedon an ideal systemmodel not affected by noise:

in such case, all system parameters are assumed constant and deterministically

known. In real systems, operating conditions are in general not constant, as they

are affected by random fluctuations. It would be profitable, in this study, to explore

the performances of the proposed monitoring tool in presence of these random

fluctuations denoted here as ‘‘system noise.’’In general, representing the random variations of all system parameters is

not an easy task and many examples can be found in literature [30]. System

noise in the following tests has been represented simply by the fluctuation of

0 1 2 3 4 5 6 7 8 9Time

–25

–20

–15

–10

–5

0

dBN

orm

aliz

ed

–6.39 dB

–21.03 dB

Fig. 8.7 Spectral WT analysis of the rotor angle trajectory at bus #8, at 0.88 Hz


load demand and generation profiles. This choice derives by the rationale that

load demand and generation are major source of randomness in power systems.According to [31], load has beenmodeled as the sumof two terms: a deterministic

component plus a stochastic additive disturbance term. Even if this assumption

might seem too simplifying, certain stability properties can be preserved if the

disturbance term is represented by a stochastic process close to white noise

[31–33]. This is also true in the context of large deviation phenomena.In the following tests, active power demand has been represented as the sum

of a deterministic voltage-dependent term plus an additive stochastic time-

dependent term, modeled as Gaussian white noise. In practice, at each time

step of the integrating routine (0.02 s), each load has been considered affected,

conservatively, by a random load variation having Gaussian distribution and

variance 0.1% with respect to the deterministic average load term (evaluated at

1 pu). Reactive power is recalculated considering a constant power factor.In order to simulate the dynamic response of a full-scale power system, a

representation of the Italian national grid, characterized by an adequate degree

of detail, was adopted for these tests. Themodel includes detailed information on

the external systems and is characterized by about 1,333 nodes, 1,762 lines,

273 generators, and 769 transformers. The steady-state condition refers to a

system configuration during aWednesday (peak load day) at the first daily peak.The test consisted in the simulation of the transient subsequent to the

tripping of one of the two 400 kV circuits of the transmission line Latina–Gar-

igliano in Central Italy (see Fig. 8.8). The transmission line, at the moment of

the tripping, was carrying about 350 MW.The spectral analysis of the voltage trajectory at the Garigliano bus (Fig. 8.9)

shows how the insurgence of the event can be easily identified. The effect of system

noise is negligible with respect to voltage fluctuations due to the line tripping.

Fig. 8.8 Portion of the Italian 400 kV transmission system

244 S. Bruno et al.

The spectral analysis has been performed on the voltage trajectory at the

20 kV generating bus inGarigliano. All simulations start at time t¼ 0 s, whereas

the line tripping occurs at the time instant t¼ 10 s. Therefore, in the first 10 s the

system dynamic is affected only by the system noise. In Fig. 8.9, dominant

oscillation modes are clearly visible. In particular, the spectral analysis at

1.10 Hz, represented in two dimensions in Fig. 8.10, shows amode of oscillation

characterized by an exponential decreasing behavior and damping that, accord-

ing to Eq. (8.15), can be estimated in

Fig. 8.9 Spectral WT analysis of the voltage magnitude trajectory at the Garigliano bus(400 kV)

Fig. 8.10 Spectral WT analysis of the voltage magnitude trajectory at the Garigliano bus(400 kV) at 1.10 Hz


� � � 0:115ð�2Þ13:87� 12:79

� 0:213

Other tests, not represented here for the sake of brevity, showed howdisturbances like the tripping of a transmission line can be detected by analyzingthe voltage magnitude behavior at almost any node of the Italian power grid,with no regard to the actual geographical localization of the disturbance. Forexample, the tripping of a 400 kV transmission line in Central Italy (Villano-va–Larino), almost unloaded and carrying less than 50 MW, was found to berecognizable by performing the WT spectral analysis of the voltage magnitudeat the Priolo bus in Sicily (see Fig. 8.8).

Results are consistent with the experience and the modus operandi of controlcenter operators of the Italian transmission system, that usually, in order todetect the onset of large oscillations, keep under surveillance few selected nodesof the grid.

8.2.6 Test Results C: Influence of Measurement Noise

In this section, further simulations are presented, aiming at showing theperformances of the proposed method in the contemporaneous presence of‘‘system noise’’ and measurement noise.

The accuracy of a PMU can be estimated to be about 0.1% for voltagemagnitude [33]. Hypothesizing that voltage trajectories were measured by aPMU, simulated voltage trajectories were ‘‘spoiled,’’ overlapping the simulatedsignal with white noise, having variance 0.03%, with respect to the nominalmeasured value (around 1 pu).

The simulated trajectory at the 400 kV Garigliano bus, analyzed in theprevious section, was modified obtaining what is shown in Fig. 8.11. Thespectral WT analysis of the ‘‘measured’’ signal, shown in Fig. 8.12, is able tohighlight the dominating oscillationmodes (1.10 and 0.74Hz), showing how thedamping information is preserved.

The evaluation of the damping can be performed also in this case,representing the spectral analysis at 1.10 Hz, in two dimensions (Fig. 8.13).The represented mode is not characterized by an exponential decreasing beha-vior all through the transient. Therefore, the damping has been evaluated infirst part of the trajectory were the trajectory decreases linearly in the logarithmscale.

According to Eq. (8.15), the damping calculation provides a result close tothe one obtained in the previous section:

� � � 0:115ð�2Þ13:82� 12:78

� 0:221

246 S. Bruno et al.

8.3 Response-Based Wide-Area Control Approach

In this section, the effects of the implementation of a real-time corrective

control architecture is discussed. The scheme of the proposed control architec-

ture is quite simple, consisting of aWAMS, a Control Center, and a network of

fast actuators, all connected through a suitable communication system.

Fig. 8.12 Spectral WT analysis of the voltage magnitude trajectory at the Garigliano bus(400 kV), in the presence of measurement noise

Fig. 8.11 Voltage magnitude trajectory at Garigliano bus (400 kV), in the presence ofmeasurement noise


The main outline of such architecture is inspired by the special protection

schemes (SPSs) usually adopted in power systems in order to control critical

events. Such systems are designed for applying event-driven corrective actions,

that have been precalculated off-line for a limited number of postulated con-

tingencies, only after the occurrence of the fault. Such systems can be enhanced

taking advantage of real-time measurement system such as the one provided by

WAMS and PMU technology.The main idea is that real-time data acquired by PMUs are employed by the

Control Center in order to implement a response-based control strategy [7]. On

the basis of measured power system trajectories, when network integrity is

threatened, the Control Center elaborates response-based corrective control

actions, and is able to correct the system behavior after the actual occurrence of

large disturbances. Remedial actions may then be applied through any fast

actuator device (e.g., load shedding schemes or FACTS devices).The proposed wide-area control scheme can be defined as generalized

response-based global control architecture. In this context, global control

means that centralized control is performed observing static and dynamic

phenomena on a sufficiently wide-area range, allowing an integrated use of all

available control actions. The necessity of adopting a centralized control struc-

ture is based on the consideration thatmajor electrical infrastructure catastrophic

events are associated to dynamic phenomena (inter-area oscillations, voltage, and

transient instabilities) having wide-area geographical influences.The proposed control scheme can be designed in order to lay on an upper

hierarchical control level, overlapping local control systems. This upper level

works coordinately with lower levels, optimizing their functionalities (i.e.,

modifying protection and local control system set points). The upper control

Fig. 8.13 Spectral WT analysis of the voltage magnitude trajectory at the Garigliano bus(400 kV), at 1.10 Hz, in the presence of measurement noise

248 S. Bruno et al.

level improves performances of local controllers which are not able to evaluatewide-area geographical phenomena.

In this section, amethodology for assessingwide-area real-time corrective controlactions is presented. Such methodology takes advantage of the approaches devel-oped in [34, 35]. The methodology, based on nonlinear programming technique, isaimed at assessing real-time corrective control actions, in order to guarantee powersystem static and dynamic security. Real-time control actions are evaluated exploit-ing measured system trajectories in the solution of a dynamic optimization problemfor nonlinear systems, where static and dynamic inequality constraints are takeninto account. The approach is prone to be applied for improving the systemdynamicbehavior on different timescales. However, in this section we show the potentialsof the approach considering stability problems linked to the transient timescale.

The structure of the mathematical approach hereby presented is based on theapproach developed for event-based evaluation of corrective actions. In theevent-based approach, corrective actions are evaluated off-line on the basis ofprepostulated contingencies and applied after the actual occurrence of thefaults. Differently, in the response-based approach, corrective control has tobe operated online, basing calculations onmeasured values (based on the actualresponse of the system).

8.3.1 Mathematical Formulation

In general, power system behavior on a transient timescale can be describedthrough a set of nonlinear differential and algebraic equations (DAEs):

_xðtÞ ¼ fðxðtÞ;VðtÞ; uðtÞÞgðxðtÞ;VðtÞ; uðtÞÞ ¼ 0

(8:16)

where x is the state vector, u is the control variable vector, andV is the vector ofnodal voltages.

The DAEs set (8.16) can be discretized through a trapezoidal rule andwritten in implicit form as follows:

Hðy; uÞ ¼ 0 (8:17)

where

Hiðyi; uÞ ¼ 0; i ¼ 0; 1; 2; . . . ; nT (8:18)

yi ¼ xTi VTi

� �T(8:19)

y ¼ yT0 yT1 � � � yTi � � � yTnT

� �T(8:20)

H ¼ HT0 HT

1 � � � HTi � � � HT

nT

� �T(8:21)


with yi representing the composition of all state variable and voltage vectors atthe generic ith time step; consequently, y represents the discretization of thewhole trajectory of the system;Hi is the discretization of the DAEs set (8.19) atthe generic ith time step; nT is the total number of time steps relative to theintegration interval [0,T].

The methodology proposed in this chapter takes advantage of the oneadopted in [34, 36]. Corrective control actions are evaluated through theformulation and the solution of a dynamic optimization problem for nonlinearsystems, where static and dynamic inequality constraints are taken intoaccount. In order to solve it, the problem is reformulated in terms of staticoptimization (nonlinear programming) and solved by applying the Lagrangianmultipliers method (nonlinear programming).

Themain difference with regard to [34, 36] is that, instead of formulating andsolving a single optimization problem, this method aims at finding the optimalvalue of the control vector uw that solves the optimization problem given foreach time window tw by the equations

minuw

CU uwð Þ þ CO ywð Þ þ CP ywð Þð Þ (8:22)

subject to

H yw; uwð Þ ¼ 0 (8:23)

and

uw min uw uw max (8:24)

In (8.22), yw represents the system trajectory in the time window tw, CU

represents an objective function aiming at the minimization of the controllingeffort, and CO is the objective function whose scope is to improve the dynamicbehavior of the system. Typically, the objective function can be formulated asan integral norm of deviations of selected variables with respect to a desirablebehavior across time; CP is a penalty function that takes into account inequal-ity constraints. Usually, inequality constraints define a time-varying domainwhere the trajectories of the system should be contained in order to satisfypractical requirements about the dynamic performances of the system.Different objective and penalty functions have been tested during the lastdecade [34–37].

A commonly adopted objective function, aimed at minimizing the transientkinetic in the generic time window tw, can be expressed as follows:

CKTEðywÞ ¼ �KTE

X

i2twVi

KTE (8:25)

250 S. Bruno et al.

where ViKTE denotes the kinetic transient energy evaluated around the center of

inertia (COI) at ith time step, �KTE is a coefficient which takes into account therelative weight of this objective function.

A penalty function that takes into account voltage-related inequalityconstraints can be formulated as follows:

CVðywÞ ¼ �V

Pi2twPn

j¼1Vi;j� �Vlim

�i;j

� 2

ntwn(8:26)

where

�Vlim ¼VM

i;j if Vi;j4VMi;j

Vmi;j if Vi;j5Vm

i;j

Vi;j otherwise

8><

>:and �i;j ¼

Vi;j

1

if Vi;j5Vmi;j

otherwise

In (8.26), ntwrepresents the number of steps in window tw; Vi;j represents thejth bus nodal voltage at the ith time step; Vm

i;j and VMi;j are time-varying thresh-

olds fixed to ensure a desirable voltage transient; �i,j is a weight factor adoptedto avoid dangerous low voltage conditions such as the ones associated totransient voltage instability.

By applying the optimizationmethod of Lagrangianmultipliers, it is possibleto evaluate the solution of the problem stated above:

L ¼ CUðuwÞ þ CO ywð Þ þ CP ywð Þ þ �TH yw; uwð Þ (8:27)

From (8.27), the set of necessary conditions follows:

@L

@yw¼ @ðCO þ CPÞ

@ywþ �T @H

@yw¼ 0 (8:28)

@L

@uw¼ @CU

@uwþ �T @H

@uw¼ 0 (8:29)

@L

@�¼ H yw; uwð Þ ¼ 0 (8:30)

From (8.28) and (8.29) one derives:

@L

@uw¼ @CU

@uw� @ðCO þ CPÞ

@yw

@H

@yw

" #�1@H

@uw(8:31)


When the optimization is operated off-line adopting the event-basedapproach, such optimization problem is solved through an iterative algorithm.The control vector is updated through the equation

uneww ¼ uoldw þ �@L

@uw(8:32)

Then a new trajectory is simulated, and everything starts back from Eqs.(8.28, 8.29, 8.30, 8.31), beginning a recursive approach that stops when thesensitivity term @L=@uw is lower than a specific tolerance limit.

In the response-based approach, proposed here, there are two main differ-ences. First, the trajectory yw is acquired throughWAMSand cannot bemodifiedor resimulated. Thatmeans that the control vector, evaluated for optimizing suchtrajectory, can be applied only afterward the trajectory itself was acquired.Moreover, as calculations of response-based control actions must be performedwithin the shortest time, a research of the optimal solution by means of thesolution of the Eqs. (8.28, 8.29, 8.30, 8.31) is not suitable.

Moreover, it should be remembered that suboptimal solutions are acceptableprovided that stability is ensured. Consequently, at each time window, changeson control variables are evaluated through a simple sensitivity analysis (i.e.,it corresponds to the first step of the iterative process described before) andapplied in the subsequent time window:

uwþ1 ¼ uw þ �@L

@uw(8:33)

where

@L

@uw¼ � @ðCO þ CPÞ

@yw

@H

@yw

" #�1@H

@uw(8:34)

Because a sensitivity approach has been adopted, the function CU gives nocontribution to derivatives in Equation (8.33) at the first iteration. However,neglecting CU is conservative from the security point of view because, in thisway, the control action is evaluated without minimizing the controlling effort.

The basic idea for this approach is that, after calculations, response-basedcorrective control actions calculated with Eqs. (8.33) and (8.34) are applied withan overall time delay �. After the implementation of the control actions, the nextstep of the proposed approach starts when a new piece of trajectory is acquired.If necessary, new corrective actions are evaluated and applied on the next timewindow, and so on. A pictorial representation of this approach is given inFig. 8.14.

The overall time delay � takes into account the time necessary for dataacquisition from WAMS, data transmission to the Control Center, data

252 S. Bruno et al.

synchronization and CPU elapsing time for control action assessment, datatransmission to actuators, and triggering of the corrective control actions. Cleary,this delay is a key parameter for the feasibility of the approach and the roughquantification of its maximum acceptable value is addressed through simulationson a real representation of the Italian power systems.

8.3.2 Test Results

The proposed methodology has been tested simulating its functionality duringoperation of a real power system. Tests were carried out considering the samerepresentation and operating condition described in Sections 8.2.5 and 8.2.6,worsened introducing a 4% uniformly distributed load increase. The base caseconsisted in the simulation of the electromechanical transient subsequent to theoccurrence of a three-phase fault at the 400 kV Aurelia bus in Central Italy,cleared after 0.2 s through the tripping of the 400 kV Aurelia–Roma transmis-sion line.

For tests, two different corrective control schemes were chosen. In test #1control actions were implemented by means of load shedding schemes whereastest #2 has been carried out considering FACTS as actuators of correctivecontrol actions. Test #1 refers to a technology actually implemented on theItalian power system, whereas test #2 hypothesizes the adoption of an advancedtechnology such as the one provided byUPFCs. The adoptedmethodology and

Fig. 8.14 Schematization of the proposed response-based approach


the illustrated mathematical formulation are suitable for both tests. Never-

theless, the methodology implementation and the solution of the optimization

problem are slightly different due to a substantial difference in the composition

of the control variable vector u. Specific considerations for the solution of both

optimization method can be found in [35, 37].As already mentioned, estimating the maximum acceptable delay for

the implementation of response-based corrective control actions has crucial

relevance in assessing the feasibility of the proposed methodology. It also

permits to assess communication technical requirements, computational time,

and actuators’ speed of response. In order to estimate this parameter, in both

tests, simulations have been carried out considering increasing values of the

time delay �.As represented in Fig. 8.15, the chosen test case gives rise to the unstable

behavior due to the fault on the line Aurelia–Roma that produces negative

effects on the whole grid. In particular, generation bus at Termini I. power

station (in Sicily) shows a behavior clearly unstable (Fig. 8.15). The final

undesirable effect is the separation of Sicily from the National Transmission

Grid, and the loss of synchronism of most generators in Southern Italy.In test #1, the proposed algorithm for response-based control of the transient

phenomena allows evaluating the performances of a real-time corrective

control scheme based on load shedding. Beside the base unstable case, four

different simulations were carried out by changing the value of delay �.The values given to � during test #1 are 0 s (ideal case where the control

action is applied instantaneously), 0.2, 0.3, and 1 s. Results of the simulation

performed during test #1 are shown in Fig. 8.15 monitoring the behavior of the

rotor angle at the same generator bus (Termini I.). The generator behavior

when no control is applied, is clearly unstable, as is easily recognizable in the

Fig. 8.15 Rotor angle of a generator at Termini I. (test #1)

254 S. Bruno et al.

picture. The comparison of results, obtained by increasing the � values, showsthat only delays up to 1.0 s can ensure stability (Fig. 8.15).

During test #2, the corrective control actions have been implemented by two

400 kVAUPFCs installed, respectively, on Villanova–Larino and Valmontone–-

Presenzano transmission lines (therefore located along the two critical corridors

for power exchanges between Central and Southern Italy, see Fig. 8.8). The time

delays applied in this case are: 0.02, 0.1, and 0.3 s. Note that the time delay 0.02 s

is an ideal case where the control action is applied almost instantaneously

(with just one time step of delay). Results of the simulations performed during

test #2 are shown inFig. 8.16.Delays up to 0.3 s can ensure stability, whereas with

longer delays the control scheme is ineffective. This result is consistent with other

simulations not reported here for brevity.The two estimated values of the maximum acceptable time delay (1 s for test

#1 and 0.3 s for test #2) permit to compare the performance of the different

actuators adopted. Longer feasible time delays in test #1 can be explained

considering that the load shedding action is a more appropriate and strong

action with regard to the redistribution of power flows operated by the UPFC-

based control actions. The load shedding scheme proves to be an effective, even

if expensive, remedial action which permits to stabilize the unstable behavior of

the system with an overall time delay of about 1 s.The control action actuated byUPFC devices is not so drastic and incisive as the

load shedding in recovering unstable behavior. However, the exploitation ofUPFC

devices allows avoiding generation rescheduling or load curtailments even under

large perturbations by means of a power flows redistribution on the network.

Furthermore, there is no need for minimizing control efforts because variations

on UPFC reference signals do not yield, in general, further operative costs. It needs

just to verify that control actions are compatible with actual machine limitations.

Fig. 8.16 Rotor angle of a generator at Termini I. (test #2)


8.3.3 Computational and Communication TimeDelay Assessment

Previous simulations and test results showed how time delays not exceed-

ing 300 ms are compatible with the stability control of a critical section in

Italy. The feasibility evaluation of the whole proposed control architec-

ture should check if the overall delay due to PMU measurement acquisi-

tion, communication and computational timings do not exceed this

threshold.Even though the computational time might seem the main obstacle, it must

be remembered that, as trajectories are already known by WAMS, corrective

actions are evaluated just through the solution of the linear system (8.34). On a

100 ms time window, this action requires around 400 ms if run on a Compaq

Alpha Server XP1000, 667 MHz, 256 MB RAM.CPU timings are not a real obstacle as they can be made compatible with

response-based control requirements. By adopting more powerful computa-

tional resources (supercomputers, parallel computing environments) it is easily

possible to reduce computational time by a factor 4, or more. As an example, a

speed-up around 9 can be obtained through vectorization and implementing

this problem on a Cray Y-MP8. A speed-up around 6 can be achieved with

message passing machines or distributed machines (e.g., Alpha clusters)

equipped with 32 CPUs.It seems that the bottleneck in the control chain has to be associated with

communication system and actuators’ response. Potentials of WAMS architec-

tures and advanced stability control architectures, with also a special regard on

the performance of the communication system, were investigated during a full-

scale experiment [23, 38, 39]. The experiment program that benefited from the

collaboration of companies involved in PMU manufacturing and WAMS

development dealt with the installation and testing of PMU devices and with

the assessment of SPS performances, including telemetry, monitoring, and

wide-area detection systems. On the basis of data acquired during the above-

mentioned experiment, time performances of the communication

infrastructures have been assessed. The overall time delay for acquiring, trans-

mitting (to the Control Center), and retransmitting (to actuators) data has been

estimated in the range 70–100 ms (25 ms for each one-way data transmission).

This result is also consistent with the estimation of around 100 ms for transfer-

ring data measured by the PMUs to the Control Center in the Terna-WAMS

Italian project [40].Clearly, even though these timings are still on the edge of the feasibility,

technology can provide the right answers for overcoming such limitations. The

implementation of fast actuators such as FACTS devices and high-speed com-

munication infrastructures will certainly allow to meet the strict requirements

imposed by a centralized wide-area response-based control architecture in the

near future.

256 S. Bruno et al.

8.4 Conclusions

The chapter showed how a monitoring architecture based on WAMS technol-

ogy is able to achieve the online supervision of power system dynamic perfor-

mances, identifying poorly damped oscillation modes, and possible threats to

the security of the system. The feasibility of the approach has been proved

through simulations on test grids and on a detailed representation of the Italian

power system.The proposed approach also allowed the recognition of dominant modes of

oscillation. By adopting the complex Morlet WT, it was possible to preserve

damping information in the time–frequency domain. An experimental proof

of this property was obtained through some simple calculations that allowed

damping evaluation on wavelets maps. The feasibility of the approach had

also been proved in the presence of random fluctuations due to operating

conditions and measurement noise. The analysis of both simulated and actu-

ally measured trajectories proved that WT is able to filter out noise. A

suitable MMI for real-time operation and OTS can be addressed by the

proposed approach.An approach for wide-area response-based corrective control of dynamic

security was proposed. On the basis of test results the feasibility of the mathe-

matical approach was showed. Test results led also to the assessment of the

minimum requirements that are requested for this architecture in terms of

overall response time. Basing on performed test results, centralized wide-area

monitoring architectures showed to be quite compatible with response-based

control approach, as the delay associated to data acquisition and control action

implementation is comparable with the maximum acceptable delay.At first glance, it may seem that computation would be the bottleneck.

Nevertheless, because the elapsing time related to dynamic sensitivity calcula-

tions can be drastically reduced through high-performance computing, it is in

our belief that bottlenecks in response time of the control chain are still

associated with communication system and actuators. Fast actuators such as

FACTS devices andmore investments in high-speed communication infrastruc-

tures can provide the right answer to meet the strict requirements imposed by a

centralized response-based control architecture.

Acknowledgments The authors wish to thank the Italian Ministry of the Research forproviding financial support under Interlink Scientific Research Program 2004–2006.

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260 S. Bruno et al.

Index

A

Activation deactivation criteria, 139Actuators, 8, 232–233, 247–248, 253–257A-EMD algorithm, 78, 82, 91–92A-EMD method, 90, 96Air-gap power, 142–143Alarm rate, 194–195, 201–202Aliasing problem, 109Alpha clusters, 256Ambient noise, 7, 8, 10–11, 14, 20Analysis techniques, 1, 2, 34, 37, 102, 123Analysis window, 15, 17–18, 38, 117, 120,

195–196, 201, 205–206, 208Analytic signal, 40, 79, 107–109, 111,

117–118, 120Angular velocity, 39, 218–221, 223–225Aurelia–Roma line, tripping, 253–254Autocorrelation function, 7, 163, 215Autoregressive moving average (ARMA), 7Autoregressive moving average exogenous

(ARMAX) model, 8

B

Bang–bang compensators, 128Bimodal test signal, 113Black signal, 23Blackout, 127–130, 139, 154, 231Block processing algorithms, 7Blue signal, 23Bootstrap technique, 22, 26BPA Dittmer Control Center, 150

C

Canonical variate algorithm (CVA), 7Carrier signal, 46Cauchy principal value, 68Cellilo substation, 56

Center of inertia (COI), 150, 251Chirp signal, 214–216Choi–Williams Distribution, 224–225COA correction, 207Compensation control actions, 153Completeness, 71Complex Morlet WT, 235, 239–240,

243, 257Continuous wavelet transform (CWT),

235–237Convolution, 65, 79–80, 82, 102, 106–107,

109, 111, 161, 172, 193, 200, 216filter, 79, 172

Corrective actions, event-basedevaluation, 249

Corrective control actions, 250–254real-time, 249response-based, 248, 252, 254

Correlation matrix, 169Covariance matrix, 164–165, 170Creaker, 3, 119Crest factor, 11–12, 20Crosschecking, 132Cross-spectral density (CSD), 27–29, 31–33Curve-fitting technique, 102

See also Prony analysis

D

Damped sinusoids, 5, 10, 106, 112, 114,121, 234

Dampingcharacterization, 68coefficient, 69–70conditions, 16–17estimation algorithms, 5evaluation of, 246factor, identification of the, 234parameter, 105, 121, 190

261

Damping (cont.)ratio, 7, 26, 70–71, 90, 93, 103, 116, 121,

123, 131–132, 137–140, 150–151,153, 236

Daubechies wavelet, 179Decomposing capability test, 80Detection algorithm, 197Detection theory, 227Differential and algebraic equations

(DAEs), 249Discrete Fourier transform (DFT), 27, 65,

104–106, 109, 192, 198, 225, 227limitations of, 65

Discrete wavelet transform (DWT), 235Distortion, 38, 50, 110, 112, 119Double sideband modulated (DSB) signal, 46Dynamic security assessment (DSA), 232, 235Dynamic System Identification (DSI)

Toolbox, 134, 136, 140, 151–152

E

EDM algorithms, 63Eigenanalysis, 14, 29, 53, 64

six-machine, 218Eigenbasis matrix, 53Eigenspectrum, 180Eigenvalue, 5, 26, 54, 90, 103, 132–133,

149, 162, 164–165, 170, 173,211–212, 217, 222–226,233–234, 239–240, 243

analysis, 54, 90, 149, 233–234,239–240, 243

decomposition, 169–170estimate error, 223parameters, 132ralization algorithm (ERA), 5

Eigenvector, 26, 53, 164, 170, 172–173,212–213, 218, 240

16-machine, 29Electromagnetic stator–rotor interactions, 50Electromechanical dynamic effects, 2Electromechanical modes, 2–3, 5, 10, 26, 34,

218, 231, 238identification, 10

Empirical mode decomposition (EMD),37–40, 63–64, 66, 72, 75, 102

algorithm, 38–39, 77–78, 87, 89, 92, 98,115, 124

technique, 39, 59, 63–65, 95–97, 98Empirical orthogonal function, 159–187

complex analysis, 171discrete domain representation, 163

energy relationships, 166snapshots method, 164theoretical development, 160

Empirical transfer function estimation,10, 52

EOF, see Empirical orthogonal functionEstimating mode shape, 26–28

defining, 26–27estimating, 27–28

Extended Transient Midterm StabilityProgram (ETMSP), 134, 136

F

FACTS, see Flexible AC transmissionsystem

Fast Fourier transform (FFT), 38,43–45, 48–50, 52, 58–59, 75, 78,104–106, 109, 114, 225

Feedback control signals, 233Field current variations, 50–52Filter banks, 64, 79–80Filtering process, 77Flexible AC transmission system, 128,

232–233, 248, 253, 256–257Fourier-based techniques, 67, 128Fourier series, 103, 105Fourier spectral analysis, 64Fourier spectrum, 64, 75, 77–78Fourier technology, 103, 105Fourier transform, 38, 40–41, 66, 79, 102,

128, 180, 214, 233–234Fractional harmonics, 104Frequency deviation recorder (FDR),

55–58Frequency-domain

analysis, 12, 38calculation, 7decomposition (FDD), 8identification, 7

Frequency error, 32Frequency heterodyne technique, 45–48Frequency shifting, 45, 47

G

Garigliano bus, 244–248Gateaux derivative, 161Gaussian load variation, 14Gaussian white noise, 3, 13, 179, 210, 244Global Positioning Systems (GPS), 55,

232, 238Grid capacity, 1, 232

262 Index

H

Hankel total least squares (HTLS), 5, 130Harmonic series, 105Harmonic signal, 236Hermitian matrix, 173Hermosillo signal, 181Heterodyne detection, 46Heterodyne frequency, 46, 48, 59Heterodyne technique, see Frequency

heterodyne techniqueHigh-pass filter, 180High-temperature superconducting (HTS)

motors, 50High-voltage direct current technology, 56,

58, 232line tripping, 58

Hilbert analysis, 74, 78, 90, 97, 101, 103,115–117, 119–121, 123–124

Hilbert transform (HT), 38–42, 46, 49,54, 58–59, 63–64, 67–68, 79–80,82, 84, 98, 101–126, 170, 172,216, 219

Hilbert transform and analytic function, 106implementation, 108instantaneous frequency, 109modal parameters, 107properties, 106

Hilbert transformer, 46, 82Hilbert–Huang technique, 38–39, 48, 51–52,

59–60Hilbert–Huang transform (HHT) 37, 39–41,

48, 50, 52, 59–60, 63–65, 67–69,71–73, 75, 77, 79, 81, 83–87, 89, 93,95, 97–99

empirical mode decomposition, 39Hilbert transform, 40

HTS propulsion motors, 50–52HVDC, see High-voltage direct current

technology

I

Identification techniques, 102–103Fourier methods, 103prony analysis, 102

IF, see Instantaneous frequencyIMF, see Intrinsic mode functionIndex of orthogonality, 72Individual mode test statistic, 198–199Instantaneous damping, 67–71, 83, 93, 109

computaiton, 84–87Instantaneous frequency, 38–41, 49, 52, 54,

59–60, 68, 70, 77–78, 80, 85, 90,

92–93, 96, 109, 130, 182, 214, 216,222, 224

Instantaneous phase, 40–41, 52, 54, 58,182, 214

Inter-area modes, 3, 8, 10–12, 14, 31, 64, 87,90, 93, 109, 128, 130–132, 135, 142,149, 179, 204, 218–219, 220, 224

Inter-area oscillations, 38, 101, 121, 131,231–232, 238, 248

Intrinsic mode function, 37, 39–40, 42–49,51–56, 58–60, 64–68, 71–74, 76–78,80–83, 86–92, 94–98, 112–115,117–118, 120–121, 123

J

Jacobian matrix, 144, 146Jitters, 60

K

Kalman approach, summary of, 211Kalman estimator, 204–205Kalman filter, 190, 197, 205Karhunen–Loeve decomposition, 166Keeler–Allston line, 139–140, 150–151

L

Lagrangian multipliers, 250–251Laguna Verde (LGV) unit, 48, 87–88Lead–lag compensators, 128Leakage, 12, 79, 105Least-mean squares (LMS) method, 8Line switchings, 227Linear analysis method, 112Linear chirp signal, 214–216Linear decomposition, 72, 167Linear differential equation, 236Linear estimation techniques, 211, 227Linear modeling technique, 222Linear system theory, 103Load modeling, 13Load torque variation, 50, 52Load tripping schemes, 2, 26Long-term estimator (LTE), 204–205

M

17-machine system, 14, 17–18Man–machine interface (MMI), 235, 238, 257Maple Valley, 151, 153Masking frequencies, 50–51

Index 263

Masking signal, 44–46, 48–51, 53–54,58–59, 63, 65, 67, 75–78, 93,98, 114

modified, 92Masking technique, 65, 72–80

energy-based, 78Fourier-based, 75–77local Hilbert transform, 79–80standard EMD method, 72–75

MASS software, 149Matlab, 80, 147, 149, 217Matrix pencil method, 5, 130, 154Measurement noise, 3, 16–18, 246–248

influence of, 246–247Measurement-based modal analysis, 5Melbourne-COA, 204Mexican Interconnected System (MIS), 48,

102, 120, 176–177MIMO, see Multiple-input multiple-outputModal damping change detection, 190–211

application to real data, 204–211energy detection, 191–197Kalman approach, 197–203

Modal energy methods, 8Modal estimate algorithms, 201Modal identification, 101, 107–108, 119,

190, 226Modal matrix, 165Modal parameter estimation, 119, 201Mode-damping ratio, 26Mode estimation

algorithms, 2, 7, 17, 19analysis, 13examples, 12–24

field measured data, 19mode-meter performance, 15–19probing test, 20–24ringdown analysis, 14–15simulation system, 13–14

Mode meters, 5Mode observability matrix, 242Model validation and performance

assessment, 24–26model validation, 24–25performance assessment, 25–26

Mode-meter algorithm, 1–2, 8, 15, 17–18Modified Hilbert–Huang technique, 50,

59–60Modified Hilbert–Huang transform, 41–58

frequency heterodyne technique, 45–48limitations of EMD, 42masking signal-based EMD, 44power flow oscillations, 48–49

slow coherency analysis, 53–55torque and field current variations,

50–52wide-area measurement signals, 55–58

Monitoring architecture, 238, 257Monofrequency, 64, 68, 70–71, 74, 84–85, 88

signals, 68, 84Monte Carlo simulations, 16, 18Morlet wavelet, 235–236, 240, 243, 257Multi-Area Small-Signal Stability (MASS)

Program, 136–137, 149Multiple mode signals, 112Multiple-input multiple-output, 3, 64, 128Multi-Prony analysis, 130–135, 137, 150Multi-Prony monitoring, 139Multisine probing signals, 10

N

N4SID algorithm, 7, 8, 10, 17–18National Transmission Grid, 254Natural frequency, 54, 70, 103Natural pulsation, 236Network topology, 3Noise-contaminated measurement

point, 179Noise-corrupted test signal, 116Noise filtering, 119Noise signal, 12Noise tolerance, 116–119Nonlinear programming technique, 249Nonlinear state–space model, 144Nonlinear/Nonstationary signals, 80,

83–84, 86Numerical algorithm for subspace

state–space system identification,see N4SID algorithm

Numerical artifacts, 8

O

Off-line rules, 130, 133, 139Online modal identification technique, 123Online real-time software tools, 2Operator training simulator (OTS),

238, 257Optimization problem, 161, 231, 249–250,

252, 254Orthogonality index, 71–72, 98Orthogonality property, 71Orthonormal matrix, 168–169Oscillation frequencies, 52Oscillation monitoring system (OMS), 130

264 Index

P

Pacific Daylight Time (PDT), 56Pacific DC Intertie (PDCI), 8, 10–11, 19, 23,

25, 56Parameter identification techniques, 106Participation

factors, 55, 212, 240–241matrix, 240

PDC block, 238Performance assessment, 24–26Periodogram averaging methods, 28Phasor data, 32, 238Phasor measurement unit (PMU), 28, 32,

131, 176–177, 180, 232, 235,238–239, 246, 248, 256

Post disturbance setting, 2Power management units, 28Power spectral density, 7, 27–30, 32, 197,

204, 206–210Power swing damping control (PSDC), 130Power System Dynamic Performance

Committee, 232Power system electromechanical modes, 5Power system identification, 8–12

probing signal selection, 10–12Power system measurements, real-time

analysis, 101Power system oscillations, 233–247

approaches for monitoring, 233Morlet-based wavelet analysis, 235WAMS-based monitoring

architecture, 238Power system security, 231Power system stabilizer (PSS), 13, 48, 128, 211Priolo bus, 246Probability density function (PDF), 191Processing gain, 12Prony analysis, 5–6, 14, 19, 64, 83–84, 93, 97,

101–103, 108, 112, 116–117, 119,121, 123–124, 128, 130–133, 135,137, 222, 234

application of, 115matrix extension of, 133

Pronymethod, 102, 124, 127, 129–130, 133, 234application of, 221

Propagating features, 170–175Proper orthogonal decomposition (POD),

160, 164, 169, 178, 181Proper orthogonal vector (POV), 164, 170Propulsion motor, 48, 50–52PSD, see Power spectral densityPSDC compensation, 149, 152PSDC mode, 131, 143, 146–147, 149

Pseudo-DC component, 123Pseudoenergy, 17–18Pseudorandom

noise, 10probing, 20, 23signal, 11

Q

Quality instrumentation, 3Quantification, 253Quasi-stationary operation, 204

R

R3LS, see Recursive least-squaresRandom contamination, 178Random fluctuations, 243, 257Random variable (RV), 192Real-time analysis of power system

measurements, 101Real-time control actions, 249Real-time corrective control actions, 249Real-time sensors, 232Realtime-synchronized measurement

system, 2Realization algorithm, 5Real-zero (RZ) signal, 42Recursive least-squares, 8, 10Reference signal, 30–31, 33, 197–198, 255R-EMD (Refined EMD), 75, 95, 98, 87

algorithm, 77–78, 80, 82, 88–89, 95Remedial action schemes (RAS), 233Response-based control, 233, 248, 252, 254,

256–257Response-based corrective control actions,

248, 252, 254Response-based wide-area, 247–256

computational and communication timedelay assessment, 256

mathematical formulation, 249–253Response matrix, 167, 169–170Ringdown, 3–6, 12, 14, 18–19, 21, 38, 218–219

analysis, 5, 12, 14, 83, 134analyzers, 5

Ripple effect, 79RLS algorithm, 19Root mean square (RMS), 11Ross–Lexington line, 140, 152Rotor angle trajectories, 240, 242RRLS algorithm, 19, 22Russian Far East Interconnected Power

Systems, 235

Index 265

S

Sensor-induced jitters, see JittersSequential tripping, 152

See also TrippingShort-time Fourier transform, 106, 214–215,

233–234, 234Sifting, 60, 64–66, 76–77Signal processing algorithms, 2Signal processing methods for estimating

modes, 5–8mode-meter algorithms, 7–8ringdown algorithms, 5–6

Signal processing techniques, 1, 211Signal-to-noise ratio (SNR), 11–12, 14, 16,

25, 117, 129, 133, 207–211Simulation test system, 13Single-mode signal, 105, 107, 116, 119

application to, 109Single-mode systems, 211Single sideband modulated (SSB), 46Singular value decomposition, 151, 167,

169–170, 174Sinusoidal signal, 75, 85Sinusoidal wave, 175Six-machine system, 53, 21716-machine system, 14–16, 18–19, 21, 29Slow coherency, 48, 53–55

analyzing, 53–55Small-signal stability, 127–130, 233Snapshots, 162, 164, 170, 178

method, 159, 162, 164–166, 170,178–179

Spatial aliasing, 178Spatial amplitude function, 174–175Spatial phase function, 174, 182Spatiotemporal analysis, 179, 186Spatiotemporal model, 178Special protection scheme (SPS), 60,

248, 256Spectral analysis, 26–27, 29, 31–32, 38, 48,

64, 175, 179, 231, 233–235,239–242, 244–246

Spectral correlation analysis, 184Spectral estimation techniques, 225Spline fitting, 59Stabilization controls, 211Standard linear algebra technique, 164State–space model, 10, 144, 204Static VAR compensator (SVC), 128–132,

134, 136, 141–154, 170, 189Statistical correlation theory, 160STFT, see Short-time Fourier transformStochastic additive disturbance, 244

SVC local unit, 141–150classical power system model, 142–143linearized state–space classical model,

143–146SVC rules, 149–150

SVD, see Singular value decompositionSwing curves, 53–55Swing equation, 141–142Sylmar substation, 56Synchronized phasor measurements,

128, 159System identification

algorithms, 11theory, 11

System noise, 25, 243–246

T

Temporal amplitude function, 174–175Temporal behavior, 82, 98–99, 160, 166, 178,

185–186Temporal coefficients, 166Temporal phase function, 174–175, 182Testing mode estimation algorithms, 19Theory of optimal detection of random

signals, 197Three-phase fault, 218, 253Tie-line power flow signal, 115, 121–123Time-dependent decay function, 69Time-domain

algorithms, 7response, 4, 212signals, 234

Time–frequencydistribution (TFD), 214, 215, 219, 227domain, 235, 239, 257representation (TFR), 213–214, 216, 224resolution, 227

Time-synchronized data, 34, 159Time-synchronized information, 178Time-synchronized measured data,

176–186construction of POD modes, 178energy distribution, 185frequency determination, 182mode shape estimation, 184spatiotemporal analysis, 179temporal properties, 182

Time-synchronized measurements, 1, 26, 28,176, 186

phasor measurements, 93Time-varying effects, 227Torque, 48, 50–52

266 Index

Transfer functionequation, 10estimation, 10, 50, 52, 60

Transmission system nonlinearity, 217Tripping, 37, 48–49, 53–54, 56, 244–246, 253

U

Unified power flow controller (UPFC), 233,253, 255

V

Validating criteria, 140–141Vectorization, 256Verification procedure, 193Vibration analysis, free, 70Vibratory system solution, 70Voltage

angle, 17, 136, 204fluctuation, 244magnitude, 134, 137, 144, 241, 246trajectory, 244–245transient, 251vectors, 250

W

WAMS, see Wide-area measurementsystems

Wavelet analysis, 233–235Wavelet shrinkage model, 179

Wavelet transform (WT) algorithms, 234Welch periodogram, 7, 25Western North American power system,

2, 4, 8, 10, 12, 19–20, 23–24, 25,31–33

Western Systems Coordinating Council(WSCC), 130, 138–139, 150–151

White Gaussian noise, 116Whitening filter, 190, 197–199Wide-area frequency information, 186Wide-area measurement and control systems

(WAMC), 232Wide-area measurement signals, 48, 55–58Wide-area measurement technology, 234, 257Wide-area measurement systems, 37, 55–58,

232–234, 238, 247–248, 252,256–257

Wigner–Ville distribution (WVD), 215–217,219–222, 224–227

Windowed Prony, 224, 227Windowing technique, 234wNAPS, see Western North American

power system

Y

Yule–Walker (YW) algorithm, 7–8

Z

Zero padding, 112, 124Z-transform identification algorithm, 234

Index 267

Power Electronics and Power Systems

Series Editors:Series Editors: M. A. PaiUniversity of Illinois atUrbana-Champaign Urbana, Illinois

Alex StankovicNortheastern UniversityBoston, Massachusetts

Continued from page ii

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Inter-area Oscillations in Power Systems A Nonlinear and Nonstationary Perspective

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