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1408
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
Wide-Area Detection of Voltage Instability From Synchronized Phasor Measurements. Part I: Principle
Mevludin Glavic, Senior Member, IEEE, and Thierry Van Cutsem, Fellow, IEEE
Abstract—This two-part paper deals with the early detection of an impending voltage instability from the system states provided by synchronized phasor measurements. Recognizing that voltage instability detection requires assessing a multidimensional syst

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1408 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
Wide-Area Detection of Voltage Instability FromSynchronized Phasor Measurements. Part I: Principle
Mevludin Glavic
, Senior Member, IEEE
, and Thierry Van Cutsem
, Fellow, IEEE
Abstract—
This two-part paper deals with the early detection of an impending voltage instability from the system states providedby synchronized phasor measurements. Recognizing that voltageinstability detection requires assessing a multidimensional system,the method ﬁts a set of algebraic equations to the sampled states,and performs an efﬁcient sensitivity computation in order toidentify when a combination of load powers has passed througha maximum. The important effects of overexcitation limiters areaccounted for. The approach does not require any load model.This ﬁrst part of the paper is devoted to theoretical foundationsof sensitivity calculation along the system trajectory, derivation of the algebraic model, and illustration on a simple ﬁve-bus systeminvolving the long-term dynamics of a load tap changer and a ﬁeldcurrent limiter.
Index Terms—
Instability detection, long-term voltage sta-bility, phasor measurement units, sensitivity analysis, wide-areamonitoring.
I. I
NTRODUCTION
T
HE phasor measurement technology [1], [2], developedsince the end of the 1980s, together with advances incomputational facilities, networking infrastructure and com-munications, have opened new perspectives for designingwide-area monitoring, detection, protection and control sys-tems. The phasor measurement unit (PMU) hardware is nowbased on proven technology and is considered as the mostaccurate and advanced time-synchronized technology availableto power engineers [3].As documented in several comprehensive surveys [3]–[5],present and potential applications of synchronized phasor mea-surements range from mere monitoring to tracking system dy-namics in real-time. The study reported in this paper belongsto the last category, as we assume that the monitored regionis equipped with PMUs ensuring full observability of bus volt-ages within that region. Admittedly, present-day power systemsare still far from having such a rich measurement conﬁguration.However, it is likely that in some future, all measurement de-viceswillbeprovidedwithhighprecisiontimetags[3].Further-more, incentive to invest in such a rich measurement conﬁgura-
Manuscript received December 04, 2008. First published June 23, 2009; cur-rent version published July 22, 2009. The work of M. Glavic was supported byan FNRS (Fund for Scientiﬁc Research) Grant. Paper no. TPWRS-00901-2008.M. Glavic is with the Department of Electrical Engineering and Com-puter Science, University of Liège, B-4000 Liège, Belgium (e-mail:glavic@monteﬁore.ulg.ac.be).T. Van Cutsem is with the Fund for Scientiﬁc Research and also with the De-partment of Electrical Engineering and Computer Science, University of Liège,B-4000 Liège, Belgium (e-mail: t.vancutsem@ulg.ac.be).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TPWRS.2009.2023271
tion will be driven by the preliminary demonstration of its po-tential beneﬁts in monitoring, protection and control functions.Voltage stability has been identiﬁed as one area where thosePMU-enhanced functions could prove useful to prevent systemblackouts and the associated social and economical losses.There are two complementary lines of defence againstvoltage instability [6], [7]: preventive and corrective. As re-gards preventive aspects, PMUs can help improving the qualityof present-day state estimation so that better initial operatingpoints are available for real-time voltage security assessmentapplications [7], [8]. They can also improve modeling accuracy[3]. However, the corrective (or emergency) line of defence iswhere the PMU technology is likely to help most signiﬁcantly.In this respect, this paper explores how it could help earlydetecting an impending long-term voltage instability, driventypically by load tap changers (LTCs), overexcitation limiters(OELs), and restorative loads [6], [7].Present-day system integrity protection schemes (SIPS)against voltage instability mostly rely on the detection of low voltage conditions, possibly complemented by signalssuch as excessive ﬁeld currents in neighboring generators[9]. The undervoltage criterion allows simple and possiblydistributed SIPS, for instance for load shedding [10]. However,it essentially relies on the observation of already degradedoperating conditions. The challenge is thus to demonstrate thatPMU-based approaches can offer better anticipation capabil-ities by detecting the inception of instability rather than itsconsequences.PMU-based voltage instability monitoring can be classiﬁedinto two broad categories:1) methods based on local measurements, with few or noinformation exchange between the monitoring locations.Most of them rely on Thvenin impedance matching con-dition [11], [12] or its extensions [13]–[15]. As long asthe various buses are checked independently of each other,thesemethodsaccommodatethetimeskewofSCADAdataand no time synchronization is needed;2) methods requiring the observability of the whole regionprone to voltage instability [16]–[19]. They offer the po-tential advantages of wide-area monitoring. The measure-mentsshouldbetime-synchronizedinsofarasanaccurate,dynamic tracking of system states is sought. They shouldpreferably be ﬁltered by a (linear) state estimator.As already mentioned, the approach of this paper belongs tothe second category. Comparisons with Thvenin impedancematching are offered in the companion paper [20].Long-term voltage instability can be triggered by transmis-sion and/or generation outages or by severe load increases. Al-though most of the incidents experienced so far were triggeredby outages, a large part of the existing literature concentrates
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GLAVIC AND VAN CUTSEM: WIDE-AREA DETECTION OF VOLTAGE INSTABILITY 1409
on smooth load increase scenarios. On the contrary, this work focuses on detecting the onset of voltage instability triggeredby a large disturbance. This requires accounting for the “noise”introduced on phasor measurements by short-dynamics not di-rectly linked to long-term voltage instability.Simply stated, voltage instability is linked to the inabilityof the combined generation-transmission system to provide thepower requested by loads, as a result of equipment outages andlimitations of reactive power generation [7]. In this perspec-tive, we propose to compute sensitivities around the “snapshot”system states computed from PMUs, with the purpose of de-tecting that some combination of load powers has gone througha maximum.The use of sensitivities in voltage stability analysis is not new[8], [21]–[24]. Eigenvalue or singular value analysis of variousJacobian matrices has been also proposed for quite some time(e.g., [25]–[27]). Initially intended to provide preventive secu-rity indices[28],theselinearization-basedtechniqueshavebeensuperseded by methods that account for system nonlinearities(such as load power margin computation). Nevertheless, theymay prove useful in scrutinizing the unstable system behaviortriggered by a large disturbance. These aspects have been com-paratively less investigated, with the exception of [27] whereeigenvalues are computed at selected snapshots of the unstablesystem evolution and [23] and [24] where sensitivity analysis iscoupled to a simpliﬁed time-domain simulation. Sensitivity of dynamic system responses have been studied through trajectorysensitivities (e.g., [29]). The latter are computationally muchmore demandingthanthesensitivityof (pseudo-)equilibria con-sidered in this work. Also, it is not clear to the authors whetherthe many discrete controls and delays present for instance in tapchangers, switched capacitors, random load switching, etc. canbe easily taken into account.In this work, it was chosen not to rely on a dynamic modelto predict the system response (e.g., [14], [30]). Indeed, this re-quires a reliable model, especially for loads in emergency con-ditions. Also, it is not clear how to reconcile the model withthe measured system evolution, in case of discrepancies caused,for instance, by events not accounted for in the model. The pro-posed approach does not try to anticipate the load response, butit anticipates generator limitations.In principle the computations presented in this paper couldrely on bus voltages provided by a standard state estimatorprocessing SCADA measurements. However, SCADA dataare not collected and state estimators are not run at the rateconsidered here (and hence some proposed ﬁltering would notbe possible). Furthermore, standard (nonlinear) state estimatorsmay encounter convergence problems in degraded systemconditions. Last but not least, the data collected by RTUs sufferfrom time skew that could make the proposed computationsunreliable in the presence of signiﬁcant transients.This paper does not consider the important problems of PMU placement, communication infrastructure, measurementpre-processing by a state estimator, etc. Instead, we simplyassume that PMUs provide synchronized bus voltage phasors.Inoursimulations, theyareobtainedfromdetailed time-domainsimulation. Circuit breaker statuses are supposed to be providedby the same equipment or by the SCADA system.The paper is organized as follows. In Section II, the theo-retical background of sensitivity analysis is reviewed and ex-
Fig. 1. Two-dimensional illustration of system trajectory.
tended to tracking eigenvalue movement around a maximumload power point. The system equilibrium model ﬁtted to eachsnapshot is detailed in Section III. The various assumptions andtechniques are illustrated on a simple system in Section IV,while concluding remarks are offered in Section V.
Notation:
Lowercase bold letters indicate column vectors.Uppercaseboldlettersrefertomatrices. denotestransposition.Complex numbers are overlined.II. T
HEORETICAL
B
ACKGROUND
A. System Trajectory
Let bethevectorofactiveandreactivepowersconsumedbytheloads.Thus,inasystemwith loadbuses,thedimensionof is . Let the system be characterized by a state vectorof dimension . In response to a disturbance, both andevolve with time. To this evolution corresponds a trajectory inthe -dimensional space of the vectors.We assume that this trajectory obeys(1)where is assumed to be a smooth func-tion. The next section will be devoted to deriving a practicalset of equations of this type. Let us already stress that the pro-posed method amounts to assessing the above model linearizedaround sampled points of the trajectory obtained from synchro-nized phasor measurements.A two-dimensional picture of such a trajectory is sketched inFig. 1, where . As both and evolve with time,the system operating point moves along the curve as suggestedby the arrows.Our goal is to identify one point of the trajectory where somelinear combination of the load powers passes throughamaximum,where isan -dimensionalnonzerovector.Notethat is not known beforehand. In the simplecase of Fig. 1 with, the function amounts to . The equi- curvesare the dotted lines parallel to the axis. In the general case, theequi- locusisanhyperplaneorthogonaltothevector .The maximum value of is reached at point M, where one of the equi- line is tangent to the trajectory.
B. Property of Point M
LetusshowthattheJacobianof withrespectto issingularat point M.Indeed,thispointisasolutionoftheconstrainedoptimizationproblem:(2)
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1410 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
(3)The Karush–Kuhn–Tucker necessary optimality conditionsof the above problem are obtained by setting to zero the deriva-tives of the Lagrangian:with respect to , and . This yields(4)(5)(6)where (resp. ) denotes the Jacobian of with respect to(resp. ). Since is nonzero, (6) implies that is also nonzero.Hence, from (5), one easily concludes that the Jacobian issingular. Equivalently, has a zero eigenvalue; is the corre-sponding left eigenvector.As is well known, second-order conditions have to be metin order the solution of (4)–(6) to be a (local) maximum. Weassume that those conditions hold in the situations of concernhere, where loads tend to increase their powers over the periodof time that follows a disturbance.Conversely, at a point where is singular, (5) holds witha nonzero vector . If this vector is such that , nocombination of load powers is found to reach an extremum. Inthe more general case where , the point can in prin-ciplebeinterpretedasalocalextremumofacombinationofloadpowers.However,thismaynotcorrespondtovoltageinstability.In practice, singular Jacobians are not expected in normal oper-ating conditions but may be observed in degraded conditions.
C. Eigenvalue Movement Along the Trajectory
We show next that when passing through M, one real eigen-value of changes sign.Let us consider the “speed” vector , where the dotsdenote time derivative. Since this vector is tangent to the trajec-tory, we have(7)from which one easily derives(8)Let us assume that has all distinct eigenvalues
1
so that itsinverse can be decomposed into(9)where and aretherightandlefteigenvectorsrelativetotheeigenvalue ,respectively.IntheneighborhoodofpointM,oneeigenvalue—say —is close to zero. Hence, the term relativeto dominates in (9) and the following approximation holds:(10)
1
We exclude the case of two (or more) zero eigenvalues as it corresponds totwo (or more) distinct combinations ofload powersbeing maximum atthe sametime.
Introducing (10) into (8) yields(11)Furthermore, in the neighborhood of M, is expected to bevery close to the left eigenvector of the zero eigenvalue, whichsatisﬁes (6). Using the latter equation, (11) can be rewritten as(12)In this last expression, represents the time derivative of . Coming back to Fig. 1, consider point A of the tra- jectory, close to M but reached before M. At this point, the timederivative of is positive. Similarly, at point B close to M butreached after M, the time derivative of is negative.Assuming that the trajectory passes smoothly from A to Bthrough M, the speed vector is continuous. Hence, in (12)the change in sign of must be compensated by the oppositechange in sign of . In other words, as the system passes fromA to B, one real eigenvalue changes sign.
D. Using Sensitivities
To detect the change in sign of one real eigenvalue, there isno need to explicitly compute eigenvalues of . Instead, sensi-tivities involving the inverse Jacobian can be used.We consider the sensitivities of the total reactive power gen-eration to individual load reactive powers. Let the load reactivepowers be grouped into . The sought sensi-tivities are obtained from a general sensitivity formula [7] as(13)where denotes the gradient of with respect to andthe is the Jacobian of with respect to . With the modeldetailed in the next section, this matrix includes 0’s and 1’s.In normal operating conditions, the above sensitivities arepositive, and usually larger than one. As point M is approached,one real eigenvalue approaches zero and the sensitivities in-crease. After crossing point M, the sensitivities are negative dueto the change in sign of . As the trajectory leaves point M,moves away from zero and the sensitivities decrease in mag-nitude. Thus, when passing through M, the sensitivities changesign through inﬁnity.Intheory,allsensitivities changesignatthesametime,whatever the bus . In practice, however, this effect is less pro-nounced as one moves away from the region experiencing thelargest voltage drops, because the numerator in (10) becomessmaller [24].In practice, discontinuities and trajectory sampling may pre-vent sensitivities from reaching very high values, as will be il-lustratedlateron.Whatissoughtisasuddenchangeinsign,i.e.,we seek to identify a discrete time such that(14)where and are thresholds to be adjusted.Computing merely requires solving one linear systemwith as matrix of coefﬁcients and as independentterm. The main computational effort lies in the factorization
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GLAVIC AND VAN CUTSEM: WIDE-AREA DETECTION OF VOLTAGE INSTABILITY 1411
of , for which efﬁcient sparsity programming packages areavailable. In turn, the most expensive step is the optimal or-dering of the matrix, which can be done only infrequently (aftermajor changes in topology; otherwise, zero entries can be usedforoutagedequipment).Sparsevectortechniquescouldtakead-vantage of the zero components of . Finally, the Jacobiancanbelimitedtotheregionpronetovoltageinstability,asshownin the companion paper [20].Eigenvalue computation is comparatively more demandingbecause it requires solving iteratively a sequence of linear sys-tems of the same size. Furthermore, tracking the movement of eigenvalues in case of discontinuities due to, e.g., OELs addssome complexity. For detection purposes, the easily computedsensitivities were found to work satisfactorily.III. S
YSTEM
M
ODELING
A. Basic Assumptions
The following basic assumptions are made:ã the network is represented by its standard bus admittancematrix. Real-time breaker status information is used to as-semble this matrix;ã the short-term dynamics of generators, automatic voltageregulators, speed governors, static var compensators, etc.are not tracked but replaced by accurate equilibrium equa-tions. This assumption is reasonable in so far as long-termvoltageinstabilityisofconcern.Provisionismadeforlargetransients that cause the system to deviate from the as-sumed equilibrium;ã the long-term dynamics driven by OELs, LTCs, andrestorative loads are reﬂected through the change in mea-sured voltages from one snapshot to the next;ã whether a generator is voltage controlled or ﬁeld currentlimited is known or detected. Equations are adjustedaccordingly;ã since the method aims at detecting a maximum of a com-bination of load powers, only the consumed powers needto be known; no information about load behavior is neededin the proposed method.
B. Overview of the Model
Based on the above assumptions, the algebraic model (1) isobtained as follows.Decomposing the bus admittance matrix , the vector of busvoltagesandthevector ofnodalcurrentsintotheirrealandimaginary parts:(15)the network relations take on the form(16)(17)We take the th bus as reference by setting the phase angle of its voltage to zero, or equivalently(18)The short-term dynamics model can be written in compactform as(19)(20)where includes state variables such as ﬂux linkages, rotorspeeds, controls, etc. while (20) relates to the stator of syn-chronous and induction machines, static loads, SVCs and otherFACTS devices, etc. Assuming short-term dynamics at equilib-rium as mentioned above, (19) is replaced by(21)The active and reactive powers consumed by the load at theth busrelatetovoltageandcurrentcomponentsthrough(22)(23)Hence, the algebraic model (1) consists of (16)–(18),(20)–(23) involving the vectors of variables(24)Finally, the total reactive power generation used in (13) isgiven by(25)where the sum extends over all generator, synchronous con-denser, and static compensator buses.
C. Modeling the Synchronous Machine and Its Controls
In this section, the synchronous generator model is consid-ered in some detail. Other components are modeled in the samespirit.In theory, (21) are obtained by setting the left-hand side of (19) to zero. In practice, however, the reduced model detailedhereafter, extensively used in quasi steady-state (QSS) simula-tion [7], [23], offers a good compromise between simplicity andaccuracy.Each synchronous machine is characterized by the emf whosemagnitudeisproportionaltotheﬁeldcurrent,andtheemf behind saturated synchronous reactances. Assuming negli-gible armature resistance, the stator equations are written in themachine reference frame as(26)
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