IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011 2125

PMU-Based Monitoring of Rotor Angle DynamicsJie Yan, Member, IEEE, Chen-Ching Liu, Fellow, IEEE, and Umesh Vaidya, Member, IEEE

AbstractOnline monitoring of rotor angle stability in wide areapower systems is an important task to avoid out-of-step instabilityconditions. In recent years, the installation of phasor measuremntunits (PMUs) on the power grids has increased significantly and,therefore, a large amount of real-time data is available for onlinemonitoring of system dynamics. This paper proposes a PMU-basedapplication for online monitoring of rotor angle stability. A tech-nique based on Lyapunov exponents is used to determine if a powerswing leads to system instability. The relationship between rotorangle stability and maximal Lyapunov exponent (MLE) is estab-lished. A computational algorithm is developed for the calculationof MLE in an operational environment. The effectiveness of themonitoring scheme is illustrated with a three-machine system anda 200-bus system model.

Index TermsLyapunov exponent, phasor measuremnt unit(PMU), rotor angle stability.

NOMENCLATURE

Relative rotor angle of generator .

Relative angular speed of generator .

Nonlinear function of and that equals to.

Vector of nonlinear functions of the dynamicalsystem that represents power system in apre-disturbance steady state.

Vector of nonlinear functions of the dynamicalsystem that represents power system duringa contingency period.

Transposition of .

Eigenvalues of .

Time-varying Jacobian matrix of alongwith trajectory .

Predefined time step.

Manuscript received July 01, 2010; revised November 01, 2010; acceptedJanuary 10, 2011. Date of publication March 07, 2011; date of current versionOctober 21, 2011. This work was supported by the Electric Power ResearchCenter (EPRC) at Iowa State University. Paper no. TPWRS-00529-2010.

J. Yan and U. Vaidya are with the Department of Electrical and ComputerEngineering, Iowa State University, Ames, IA 50010 USA (e-mail: jieyan@ias-tate.edu; ugvaidya@iastate.edu).

C.-C. Liu is with the School of Electrical, Electronic and Mechanical Engi-neering, University College Dublin, Dublin, Ireland (e-mail: liu@ucd.ie).

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.2011.2111465

I. INTRODUCTION

O NLINE monitoring of rotor angle stability of powersystem is a challenging task as the dynamics ofpower systems become more and more complex. Traditionalout-of-step relays predict out-of-step in the first swing afterclearance of disturbances. Nevertheless, severe disturbancessuch as double faults along both circuits of double-circuitlines might cause out-of-step in swings after the first swing.As a result, power swing stability cannot be easily predicted.However, the advent of phasor measurement units (PMUs),together with advances in computational and communicationsfacilities, provides an opportunity to perform online monitoringof system dynamics. This paper is focused on the applicationof PMU data to online monitoring of rotor angle stability.

A number of applications of PMU data have been proposed.In [1], a state estimation algorithm using synchronized phasormeasurements is proposed for a multi-area framework. In [2],the conceptual design of a super calibrator is described, usingPMU data to coordinate among individual state estimators. In[3], a method for estimating power system transfer path dy-namic parameters is developed based on the PMU data. Ref-erence [4] proposes an approach for online detection of the startof frequency perturbation in power systems using PMU data. Inthe United States, the North American SynchroPhasor Initiative(NASPI) is working to create a synchronized data network soas to improve power system reliability through wide-area mea-surement, monitoring, and control. The Western Electricity Co-ordinating Council (WECC) has launched the Western Intercon-nection Synchrophasor Program (WISP); 250300 PMUs willbe deployed to enable smart grid functionality [5].

This paper proposes a PMU-based monitoring scheme thatcan be used to recognize patterns of loss of synchronism aspart of the overall vulnerability assessment. Based on our anal-ysis of blackout scenarios, a number of patterns of cascadingevents have been identified: line tripping due to loss of synchro-nism, line tripping due to overloading, generator tripping due toover-excitation, generator tripping due to abnormal voltage andfrequency conditions and under-frequency/voltage load shed-ding [6]. Vulnerability assessment of the power grid against cas-cading events is a critical element of the power infrastructuredefense system [7].

In [8][11], the first version of an adaptive PMU-basedout-of-step relay was applied to the Florida-Georgia system.The power system is modeled as two interconnected equivalentgenerators. Then two PMU sets are used to monitor the angulardifference between these two equivalent generators. Basedon the monitored angular difference, the equal area criterion(EAC) is utilized to distinguish between an unstable powerswing and a stable one. In [12], the energy function analysisis used to monitor angle stability. Reference [13] reports anout-of-step prediction logic based on the autoregressive model

U.S. Government work not protected by U.S. copyright.

2126 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011

(AR model). A method for monitoring an interarea oscillationmode by the spectrum analysis is presented in [14]. The resultsof [13] and [14] represent forecasting procedures based ontime-series analysis of PMU data during power swings. In[15], synchronized phasor measurements are used as inputsfor computation of the differential/algebraic equation (DAE)model of the post-fault power system. Then, numerical resultsare used for prediction. A fuzzy neural network based on PMUsis applied to power swing stability prediction [16]. In [17], aself-adaptive decision tree approach for online dynamic securityassessment is presented. Offline contingency analysis data suchas PMU data is used to identify critical attributes of the decisiontree nodes and to train the decision tree. In [18], large-sizedecision trees (DTs) extracting selected decision features fromPMU measurements are used for rapid stability assessment.The technique involves offline study. In [19], neural networksare used to estimate rotor angles for monitoring of transientstability in real-time based on PMU measurements.

Neural networks and DTs are able to detect system insta-bility with great accuracy. However, extensive offline study isinvolved. In this paper, an analytical method is proposed formonitoring wide-area rotor angle stability in the operational en-vironment using PMU data. The power system is modeled as amulti-bus system. The Lyapunov exponent is employed to de-termine whether or not a power swing indicates a loss of syn-chronism. The method provides a high level of accuracy with alow computational burden.

The application of Lyapunov exponent was proposed in [20].Their work indicates that the Lyapunov exponent can be usedto predict out-of-step conditions. The result is supported bysimulation results with a small power system. The proposedmethod in this paper extends the work of [20] by articulating themathematical relationship between Lyapunov exponent and theout-of-step instability condition. Furthermore, the connectionbetween the maximal Lyapunov exponent (MLE) and the rotorangle stability of power systems is established. An efficientcomputational algorithm is proposed for calculation of MLEbased on a finite time window of PMU data. The effectivenessof the proposed method is validated by simulation results witha three-bus system and a 200-bus system.

The remainder of this paper is organized as follows. Section IIdescribes the problem of rotor angle stability monitoring.Section III provides the methodology of Lyapunov exponentsand the relationship between rotor angle stability and MLE.In Section IV, a monitoring scheme is provided. Section Vshows the computational method for MLE. In Section VI,simulation results are presented. Section VII is a discussionof the robustness and the computational performance of themonitoring scheme. Finally, Section VIII gives the conclusion.

II. PROBLEM FORMULATION

Rotor angle stability is concerned with the ability of the powersystem to maintain synchronism when subjected to a small orlarge disturbance. Power system faults and line switching resultin a sudden change on the generator outputs. However, mechan-ical power inputs to generators do not change instantaneously.These major disturbances can cause severe oscillations in ma-chine rotor angles and severe swings in power flows. Loss ofsynchronism can occur between one generator and the rest of

the system, or between groups of generators, leading to insta-bility conditions. This research is concerned with the predictionof rotor angle instability following disturbances on the powersystem. The goal of this research is to develop a practical appli-cation of PMU data together with the mathematical and compu-tational foundations.

A power system with buses and generators representedby a dynamical model is considered. Generators are modeled byclassical equations, while loads are represented by ZIP models.Assume that the mechanical power inputs to generators are con-stant during the contingency period. The system model can berepresented by

(1)

where , , and. Prior to a disturbance, a power system is

represented by a -dimensional continuous-time dynamicalsystem denoted by

(2)

where . represents the rotorangles and angular speeds of the generators.

After a disturbance, the parameters of the model are updatedas necessary in order to establish a nonlinear dynamical systemto represent the power system during the contingency period:

(3)

The nonlinear system trajectory in the state space is thepower swing curve following the disturbance. The nonlinearsystem is said to be stable if and only if approaches anasymptotically stable equilibrium point, which meansarrives at and stays within the attractor of an asymptoticallystable equilibrium point. The generators maintain synchronismif the nonlinear system is asymptotically stable. The proposedmethod in this paper is used to assess the nonlinear system sta-bility and hence the rotor angle stability. Assessing the localstability by linearizing around an equilibrium point is a specialcase of the method.

Note that represents the post-fault system trajectory.The brief fault-on period from the occurrence of the fault to relaytripping, breaker opening (and possible breaker reclosure) is notexplicitly modeled. The fault-on period will affect the systemstates and hence the initial conditions of the post-fault period.The proposed model is intended for monitoring of the electro-mechanical dynamics of the power system from the initial con-dition of the post-fault period.

III. METHODOLOGY: LYAPUNOV EXPONENT

The MLE of is calculated to monitor rotor angle sta-bility after a disturbance. If MLE is negative, it is concludedthat will approach an asymptotically stable equilibriumpoint and hence the power swing is (asymptotically) stable. Forthe proposed method, if MLE is higher than or equal to zero, itis judged that will not approach an asymptotically stableequilibrium point and the power swing is considered unstable,which, strictly speaking, means not asymptotically stable inthis context of this paper.

YAN et al.: PMU-BASED MONITORING OF ROTOR ANGLE DYNAMICS 2127

Fig. 1. Nearby trajectories in state space.

The Lyapunov exponent and its relationship with system sta-bility are discussed in the following. In ergodic theory of dy-namical systems, Lyapunov exponents are used to characterizethe exponential divergence or convergence of nearby trajecto-ries as shown in Fig. 1. For an -dimensional continuous-timedynamical system , let be the solutionat time starting from initial condition . The Lyapunovexponents for are defined as the eigenvalues ofthe following limiting:

(4)

where .The limit in (4) is known to exist by Oseledec multiplicative

ergodic theorem [21]. For the purpose of the proposed applica-tion, the following index is defined:

(5)

where is a randomly chosen vector in small scale, asshown in Fig. 1. in (5) is the MLE. A negative (positive) valueof implies exponential convergence (divergence, respectively)of nearby system trajectories. This is true due to the approxima-tion of

(6)

In this research, a relationship between the MLE and the asymp-totic behavior of the dynamical system is established based onthe results of [22].

MLE Stability Criterion: Consider a continuous-time dy-namical system and assume that all Lyapunov exponents arenonzero. Then the steady state behavior of the system consistsof a fixed point. Furthermore, if the MLE is negative, then thesteady state behavior is an attracting fixed point. The proof ofthe MLE stability criterion is outlined in the Appendix.

The MLE stability criterion provides a sufficient condition forsystem stability when MLE is negative. If is positive, nearbysystem trajectories will diverge according to (6).

IV. MONITORING SCHEME

The concept for online monitoring of rotor angle stability isillustrated in Fig. 2. It is intended to be an application in thecontrol center of a power system. The proposed algorithm ob-tains the updated power network configuration from the state

Fig. 2. Concept for monitoring of rotor angle stability.

estimator (SE), say, every 5 min. In order to avoid the penetra-tion of bad data into state estimation, PMU measurements canbe incorporated into the SE process. Reference [23] proposeda reference-free rectangular state estimation method to identifybad data in the SE process by using PMU data.

As discussed in Section II, following a disturbance suchas a fault and the corresponding relay and breaker responses,the sudden change of power network configuration is re-ported through the Supervisory Control and Data Acquisition(SCADA) system in real time. There is a time delay before theSCADA data arrives at the control center. Typically, the controlcenter will receive the SCADA data about the configurationchange in a matter of seconds. For example, in California ISO,the transmission delay is less than 4 s. The updated post-faultdynamic model is then obtained by modifyingparameters according to the SCADA data. After that, thealgorithm extracts synchronized phasor measurements fromthe PMU data concentrator, which obtains real-time PMU datafrom substations equipped with PMUs. Hence, a number of thevariables of are observed from PMU data. Based onthe updated dynamical system and PMU measurements afterthe disturbance, the MLE index of is calculated by thealgorithm which will be discussed in detail in Section V. Ifthe MLE has a negative value, the prediction is that the powerswing is stable; otherwise, it is unstable. Appropriate controlactions may be needed.

If a disturbance occurs during the computation of the MLEand it changes the system configuration, the power system willbe represented by an updated dynamical system consequently

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and the system stability will be re-assessed. The MLE will bere-calculated based on the updated dynamical system and thePMU measurements after the event, in order to determine thesystem stability.

V. PROPOSED COMPUTATION METHOD

A computationally efficient algorithm to calculate the MLEis proposed. The calculation of MLE is based on (5). In practice,there are three difficulties:

1) One can only calculate , the MLE over a finite time in-terval , instead of over an infinite time interval as shownin (5). The length of time interval, should be preselectedproperly so that and have the same sign.

2) The complete observability of the trajectory is re-quired in (5) to calculate its MLE, which means that therotor angles and angular speeds of all generators are to beobserved by PMU data. However, at present, there are onlya limited number of PMUs in the power system.

3) A fast way to calculate MLE is required for online moni-toring, since rotor angle stability after a disturbance mustbe determined quickly to allow for effective control ac-tions.

The above issues are addressed in the following subsections.

A. Spectrum Analysis

A spectrum analysis is used to compute an appropriate timeinterval length . In practice, power swings exhibit strong peri-odic behavior and the frequency lies within a narrow frequencyband under a given scenario. One can simulate all the credibledisturbances on a power system in the offline study and obtainthe power swing curves following the disturbances. The spec-trum analysis is then applied to the power swing curves to iden-tify a frequency band. In this paper, the R language is employedto conduct the spectrum analysis. Based on the result of the spec-trum analysis, is set as the inverse ratio to the lower limit ofthe frequency band, so that it covers at least one cycle of thepower swing.

B. Implicit Integration Method With Trapezoidal Rule

The implicit integration method with trapezoidal rule isused to approximate the unobservable part of state variables in

. The idea is to estimate the unobservable state variablesat instant based on estimated values of the unobservablestate variables at instant and observed values of the observablestate variables at instant and . For example, assumethat state variables are monitored by PMUs,while the remaining variables are not. The state variables of

are either observed or estimated. The unobservablestate variables of are estimated by the integrationresult of the dynamical system , shown as follows:

...

(7)

where is the observed value of at instantby PMU data. The above equation can be formulated as

. The Newton-Raphson method is then appliedto calculate an estimation of . Note that for the nexttime increments, , can be approximated by thesame procedure.

The Newton-Raphson method requires matrix inversion oper-ations. Correspondingly, the computational burden can be highwhen a large number of state variables are not observable in (7).However, it is noticed that the matrices that need to be inversedhere are the Jacobian matrix of ,which equals to . It is diagonally dom-inant when is small. Note that by simplifying the matricesas diagonal, the computational burden is dramatically reducedwithout compromising the accuracy.

C. Gram-Schmidt Reorthonormalization (GSR)

The standard method with GSR [24] is used to calculateafter the estimation of . The method is based on (6), fromwhich one can obtain

(8)

The algorithm proceeds as follows:1) Let , , and .2) Do the following while

a) Assume that there is another trajectory of dy-namical system with a different initialpoint than . The separation of the two trajecto-ries is at . Then

at , as shown in Fig. 3. Compute the sep-aration at :

(9)

b) Compute the rate of separation at that moment:

(10)

c) Compute the time average of the rate of separation by:

(11)

d) Reorthonormalize the separation, so that hasthe same direction that does, while the magnitudeof equals to 1, as shown in Fig. 3:

(12)

e) .

YAN et al.: PMU-BASED MONITORING OF ROTOR ANGLE DYNAMICS 2129

Fig. 3. Standard method with GSR.

Fig. 4. Three-machine system.

3) When , the time average of the rate ofseparation by equals to , so

.

VI. SIMULATION RESULTS

A three-machine system and a 200-bus system are used forvalidation of the proposed method.

A. Three-Machine System

For illustration of the computational techniques, a simplethree-machine system with lossless transmission lines is used,as shown in Fig. 4. The classical swing equations are as follows:

(13)

In the simulation case, it is assumed that ,, ,

, , , and. A

six-dimensional dynamical system is obtained:

TABLE ISIMULATION RESULTS OF THREE-MACHINE SYSTEM

(14)

Assume that . A three-phasefault followed by normal clearing is applied to each transmis-sion line, respectively. The fault is cleared by relay and breakeroperations, which will last for a fraction of 1 s. Then it is as-sumed that the break recloses successfully at 1 s. During thefault clearing period, one parameter of the dynamical systemin (14) is changed since one branch is de-energized. Then theparameter will change back to the original value if the break re-closes and it is successful.

Assume that there is one PMU at each generator bus; it isdesirable to monitor the real-time values of state variables in(14). The time-varying Jacobian matrix of (14) is obtained. Let

, . is calculated by the standard methodwith GSR. and time-domain simulation results are shownin Table I. As it shows, predicts out-of-step correctly.

Note that in the third contingency in Table I, the power swingmay appear to be stable based on observations during the first 3s, but it turned out to be unstable later, as shown in Fig. 5.predicts the instability correctly.

B. A 200-Bus System

A 200-bus system that resembles the structure of the WECCpower grid, as shown in Fig. 6, is used for testing of the mon-itoring algorithm. The model has 31 generators, and eight ofthem are observable by PMU data. The power system is simu-lated with the PSS/E software tool. Each synchronous machineis modeled as a round-rotor generator, and each load is repre-sented by shunt admittance at its bus. The proposed algorithmis implemented in Matlab. It processes the data generated byPSS/E and calculates the MLE.

The NERC disturbance class B given in the NERC PlanningStandards is considered. Here it is condensed to three-phase

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Fig. 5. Time-domain simulation result for the 3rd contingency.

Fig. 6. A 200-bus system.

fault for 0.07 s with clearing of the generator or transmissioncircuit.

Spectrum analysis is performed in an offline study to deter-mine the appropriate time interval for MLE. The time-domainsimulations are carried out after the NERC disturbance classB. Then spectrum analysis is performed on the resultant powerswing curves. It shows that power swing curves oscillate within0.21.0 Hz. Therefore, is set as 5 s. is to be calculatedto determine angle stability after disturbance.

A 62-dimensional dynamical system model is established torepresent the 200-bus system, as shown in Section II. After each

disturbance, the appropriate parameters of the dynamical systemare changed accordingly. Sixteen state variables of the dynam-ical system are observed during the contingency period, sinceonly eight out of 31 generators are observable by PMU data.The other state variables are estimated by the implicit integra-tion method with trapezoidal rule, as described in Section V.The approximation results are very close to the time-domainsimulation results. For example, the relative rotor angle and an-gular speed of generator at bus 79 after different disturbancesare shown in Fig. 7. Bus 79 is at the upper left corner of Fig. 6,indicated by a circle.

After the generator at bus 18 (circled on the right side ofFig. 6) is disconnected due to fault at bus 18, the relative rotorangle and angular speed of generator at bus 79 oscillate dramat-ically, as Fig. 7(a) and (b) shows. The approximation results ofthe angle and angular speed coincide with the time-domain sim-ulation results very well. After line 104102 (marked on the leftside of Fig. 6) is disconnected, the relative rotor angle and an-gular speed of generator at bus 79 remain stable, as indicatedin Fig. 7(c) and (d). The approximation results are reasonable,since the trends of the approximation results coincide well withthe time-domain simulation results.

The index is calculated by the standard method withGSR, with all state variables either observed or estimated.and the time-domain simulation results for each disturbance areshown in Table II. As shown in the table, is positive whenpower swing is unstable after fault clearing; is negativewhen power swing is stable.

It is noted that, when a disturbance is a generator tripping, thepower swing is more likely to be unstable, as Fig. 7(a) and (b)shows. On the other hand, when a disturbance of a line trippingis applied to the power system, the power swing tends to bestable, as illustrated in Fig. 7(c) and (d).

Although the unstable cases shown in Table II are related tothe separation of a generator due to a three-phase bus fault, the200-bus system, which has a simplified configuration that re-sembles the WECC system, can exhibit complex instability phe-nomena beyond what is shown by the simulation cases in thepaper. For example, system oscillations occur after the initiatingtree contact with a 500-kV line and the following generator trip-ping events [6].

VII. DISCUSSION

The sensitivity analysis with respect to the time intervallength is performed, in order to quantify its impact on theaccuracy of the proposed method. Following a disturbance onthe 200-bus system, the MLE is calculated over different timewindows. The computation results are shown in Table III. Thefollowing facts are observed:

1) As the value of increases, the MLE predicts the systemstability with a higher level of accuracy. For example,fails to detect instability after the first and the third distur-bances. The MLE calculated over a longer time windowprovides a correct prediction.

2) The value of MLE varies after different disturbances. Itis reasonable because different disturbances change thepower system configuration in different ways, as shown inTable III. Consequently, the MLE is calculated based ondifferent dynamical system configurations.

YAN et al.: PMU-BASED MONITORING OF ROTOR ANGLE DYNAMICS 2131

Fig. 7. Time-domain simulation results and approximation results for compar-ison.

3) When the system is unstable, the value of MLE changeswith respect to the time interval length in a pattern thatexhibits varying characteristics. This is the case because

TABLE IISIMULATION RESULTS OF 200-BUS SYSTEM

TABLE IIISENSITIVITY ANALYSIS RESULTS

the MLE is calculated based on a nonlinear system trajec-tory within one time window. When the nonlinear systemis unstable, small changes in the time window length cancause the value of MLE to vary significantly.

4) When the system is stable, the value of MLE is negative buttends to increase as the window size increases, as shown inTable III. The observation seems to indicate that the systemis considered less stable when a larger set of observationsare obtained by a wider time window. For a power systemundergoing a contingency, it is reasonable that more vari-ations are observed within a larger time frame.

The total computational burden of the monitoring scheme isaffordable for online application. Consider a power system with

generators, of which are not observed by PMU data. If, and the Newton-Raphson method iter-

ates 20 times at most, the main computational burden would bemultiplication operations and

addition operations. Taking the200-bus system in Section VI as an example, it takes 0.0044 s tocalculate for a 3-GHz Pentium 4 CPU. Moreover, the pro-posed algorithm is recursive. It does not have to obtain all PMUdata needed before calculating MLE. For example, if isto be calculated and the PMUs transmit 1-s time window dataevery time, the algorithm can start calculating once the first 1-s

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time window data is obtained and wait for the next one fromPMUs at the same time.

VIII. CONCLUSION

A PMU-based method for online monitoring of rotor anglestability of power systems has been developed. The main ideais to calculate MLE in order to predict out-of-step conditions.The proposed MLE method is based on a solid analytical foun-dation and is computationally efficient. Significant progress hasbeen made in online dynamic security assessment, which allowsthe system to evaluate the impact of various contingencies onsystem stability. As the large-scale deployment of PMUs on thepower grids, it is important to develop new applications of thePMU data. It is believed that the proposed method is an impor-tant step toward the goal.

Future research on the proposed scheme would be helpful inimproving the accuracy and robustness of the method:

1) For a large power system with numerous generators, it isdifficult to establish a dynamical system model to representthe power system during a contingency. The power systemmodel may be simplified based on reasonable assumptions.For example, multiple generators that tend to swing to-gether may be represented by an aggregated model.

2) As a power system grows in scale and complexity, it maybe more difficult to decide the time interval length for MLEmerely by the proposed spectrum analysis. An online pat-tern recognition technique coupled with the spectrum anal-ysis may be used to determine the size of . The dynamicpatterns in a power system can be divided into classes,e.g., fast and slow. Each pattern of dynamic variations isassigned a different time interval length based on thespectrum analysis of the power swings following distur-bances. When a disturbance occurs, the pattern recognitiontechnique is used to recognize the type of perturbation. Afast perturbation can be detected by a short time window,while a slow pattern will require a longer time window.The MLE is then calculated over the corresponding timeinterval length .

3) Many new PMUs will be installed in the future. The place-ment of new PMUs for complete observability should beinvestigated. An integer quadratic programming approach[25] may be used for the optimal placement of PMUs.Complete observability should be ensured under normal aswell as contingency operating conditions.

4) The subject of voltage dynamics requires an expandedmodel of the power systems. In order to use Lyapunovexponent for monitoring voltage dynamics, the powersystem during a dynamic period needs to be representedby a dynamical system whose state variables are busvoltage magnitudes. The work to establish such a dynam-ical system is an important subject for the future work.

5) Further research is needed in order to draw a definitive con-clusion about the relationship between the value of MLEand the size of stability margin.

APPENDIX

Consider an -dimensional continuous-time dynamicalsystem , is differentiable and , where

is a compact set. If the initial point is given, then

Fig. 8. Diminishing separation.

Fig. 9. Colliding trajectories.

there is a corresponding trajectory in the -dimensionalphase space. There also exist Lyapunov exponents foraccording to the definition of Lyapunov exponent.

Definition 1 [26]: The equilibrium point is stable if, for each , there is such that

asymptotically stable if it is stable and can be chosen,such that

Now define with an arbitrary initial point .Then there is another trajectory in the state space. Ac-cording to the definition of Lyapunov exponent, if the MLE of

, , , such that

Hence, the separation between trajectory and will di-minish to 0 as time goes on, if and the initial pointsand are close enough, as Fig. 8 shows.

Outline of Proof for MLE Stability Criterion: If , asmentioned before, , such that

Then , such that . Let, then , since the dynamical system is

autonomous, as Fig. 9 shows.Hence

Moreover

YAN et al.: PMU-BASED MONITORING OF ROTOR ANGLE DYNAMICS 2133

Since

Therefore, will approach an equilibrium point . canbe viewed as a special trajectory: it starts at and stays at .The special trajectory of has the same MLE as does.Then , such that

Hence, by Definition 1, is asymptotically stable.

REFERENCES[1] L. Zhao and A. Abur, Multiarea state estimation using synchronized

phasor measurements, IEEE Trans. Power Syst., vol. 20, no. 2, pp.611617, May 2005.

[2] V. Vittal, G. T. Heydt, and A. P. S. Meliopoulos, A Tool for Online Sta-bility Determination and Control for Coordinated Operations BetweenRegional Entities Using PMUs, PSERC, Tempe, AZ, 2008.

[3] J. H. Chow, A. Chakrabortty, L. Vanfretti, and M. Arcak, Estimationof radial power system transfer path dynamic parameters using syn-chronized phasor data, IEEE Trans. Power Syst., vol. 23, no. 2, pp.564571, May 2008.

[4] Q. Zhao, J. Dong, T. Xia, and Y. Liu, Detection of the start of fre-quency excursions in wide-area measurements, in Proc. IEEE PESConversion and Delivery of Electrical Energy in the 21st Century, Jul.2008, pp. 16.

[5] V. VanZandt and M. Bianco, The western interconnection syn-chrophasor program (WISP)smart grid investment grant, in Proc.NASPI Work Group Meeting, Feb. 2425, 2010.

[6] K. Yamashita, S. K. Joo, J. Li, P. Zhang, and C. C. Liu, Analysis,control and economic impact assessment of major blackout events,Eur. Trans. Elect. Power, vol. 18, no. 8, pp. 854871, Nov. 2008.

[7] C. C. Liu, J. Jung, G. T. Heydt, V. Vittal, and A. Phadke, The strategicpower infrastructure defense (SPID) system, IEEE Control Syst. Mag.,pp. 4052, Aug. 2000.

[8] V. Centeno, A. G. Phadke, A. Edris, J. Benton, M. Gaudi, and G.Michel, An adaptive out-of-step relay for power system protection,IEEE Trans. Power Del., vol. 12, no. 1, pp. 6171, Jan. 1997.

[9] V. Centeno, A. G. Phadke, and A. Edris, Adaptive out-of-step relaywith phasor measurement, in Proc. 6th Int. Conf. Developments inPower System Protection (Conf. pub 1. no. 434), Mar. 2527, 1997,pp. 210213.

[10] V. Centeno, A. G. Phadke, A. Edris, J. Benton, and G. Michel, Anadaptive out-of-step relay, IEEE Power Eng. Rev., vol. 17, no. 1, pp.3940, Jan. 1997.

[11] V. Centeno, J. de la Ree, A. G. Phadke, G. Michel, R. J. Murphy, andR. O. Burnett, Jr, Adaptive out-of-step relaying using phasor measure-ment techniques, IEEE Comput. Appl. Power, vol. 6, no. 4, pp. 1217,Oct. 1993.

[12] J. H. Chow, A. Chakrabortty, M. Arcak, B. Bhargava, and A. Salazar,Synchronized phasor data based energy function analysis of dominantpower transfer paths in large power systems, IEEE Trans. Power Syst.,vol. 22, no. 2, pp. 727734, May 2007.

[13] K. Yamashita and H. Kameda, Out-of-step prediction logic for wide-area protection based on an autoregressive model, in Proc. IEEE PESPower Systems Conf. Expo., Oct. 2004, vol. 1, pp. 307312.

[14] N. Kakimoto, M. Sugumi, T. Makino, and K. Tomiyama, Monitoringof interarea oscillation mode by synchronized phasor measurement,IEEE Trans. Power Syst., vol. 21, no. 1, pp. 260268, Feb. 2006.

[15] C. W. Liu and J. Thorp, New methods for computing power systemdynamic response for real-time transient stability prediction, IEEETrans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 3, pp.324337, Mar. 2000.

[16] C. Liu, M. Su, S. Tsay, and Y. Wang, Application of a novel fuzzyneural network to real-time transient stability swings prediction basedon synchronized phasor measurements, IEEE Trans. Power Syst., vol.14, no. 2, pp. 685692, May 1999.

[17] K. Sun, S. Likhate, V. Vittal, V. S. Kolluri, and S. Mandal, An onlinedynamic security assessment scheme using phasor measurements anddecision trees, IEEE Trans. Power Syst., vol. 22, no. 4, pp. 19351943,Nov. 2007.

[18] I. Kamwa, S. R. Samantaray, and G. Joos, Development of rule-basedclassifiers for rapid stability assessment of wide-area post-disturbancerecords, IEEE Trans. Power Syst., vol. 24, no. 1, pp. 258270, Feb.2009.

[19] A. Del Angel, P. Geurts, D. Ernst, M. Glavic, and L. Wehenkel, Esti-mation of rotor angles of synchronous machines using artificial neuralnetworks and local PMU-based quantities, Neurocomput., Elsevier,vol. 70, no. 1618, pp. 26682678, Oct. 2007.

[20] C. Liu, J. S. Thorp, J. Lu, R. J. Thomas, and H. Chiang, Detectionof transiently chaotic swings in power systems using real-time phasormeasurements, IEEE Trans. Power Syst., vol. 9, no. 3, pp. 12851292,Aug. 1994.

[21] V. I. Osedelec, Multiplicative ergodic theorem: Lyapunov character-istic exponent for dynamical systems, Moscow Math. Soc., vol. 19, pp.539575, 1968.

[22] J. P. Eckman and D. Ruelle, Ergodic theory of chaos and strange at-tractors, Rev. Modern Phys., vol. 57, pp. 617656, 1985.

[23] J. Zhu and A. Abur, Bad data identification when using phasor mea-surements, in Proc. IEEE Power Tech Conf., Lausanne, Switzerland,Jul. 2007, pp. 16761681.

[24] K. Ramasubramanian and M. S. Sriram, A comparative study of com-putation of Lyapunov spectra with different algorithms, Physica D,vol. 139, no. 12, pp. 7286, May 2000.

[25] S. Chakrabarti, E. Kyriakides, and D. Eliades, Placement of synchro-nized measurements for power system observability, IEEE Trans.Power Del., vol. 24, no. 1, pp. 1219, Jan. 2009.

[26] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ:Prentice-Hall, 2002.

Jie Yan (M09) received the B.Eng. degree in electrical engineering from Zhe-jiang University, Hangzhou, China, in 2005 and the M.S. degree in electricalengineering from Huazhong University of Science and Technology, Wuhan,China, in 2007. He is currently pursuing the Ph.D. degree in electrical engi-neering at Iowa State University, Ames.

Chen-Ching Liu (F94) received the Ph.D. degree from the University of Cal-ifornia, Berkeley.

During 20062008, he was Palmer Chair Professor of Electrical and Com-puter Engineering at Iowa State University (ISU), Ames. Prior to joining ISU,he was a Professor of electrical engineering at the University of Washington,Seattle. He is currently a Professor of Power Systems at University CollegeDublin, Dublin, Ireland.

Dr. Liu received the IEEE PES Outstanding Power Engineering EducatorAward in 2004. He served as Chair of the Technical Committee on Power SystemAnalysis, Computing and Economics, IEEE Power and Energy Society, during20052006.

Umesh Vaidya (M07) received the Ph.D. degree in mechanical engineeringfrom the University of California at Santa Barbara in 2004.

He was a Research Engineer at the United Technologies Research Center(UTRC), East Hartford, CT. He is currently an Assistant Professor in the De-partment of Electrical and Computer Engineering, Iowa State University, Ames.His research interests include dynamical systems and control theory, in partic-ular analysis and control of nonequilibrium behavior in nonlinear systems andapplication of ergodic theory methods to control problems.