Mode Estimation Accuracy for Small-Signal Stability Analysis

Evaluation of Mode Estimation Accuracy forSmall-Signal Stability Analysis

Jim Follum, Student Member, IEEE, Ning Zhou, Senior Member, IEEE, John W. Pierre, Senior Member, IEEE

Abstract—This paper proposes a method for determiningelectromechanical mode estimate accuracy by relating modeestimate error to residual values. Mode frequency and dampingratio were estimated using Prony analysis and residuals werecalculated for a 17-machine model with varying levels of loadnoise. Mode estimate error and residuals were found to increaseproportionally to each other as noise values were increased,revealing a distinctly linear relationship. The use of these resultsto develop appropriate confidence in models is discussed. With therelationship established, a method of predicting mode estimateerror values based on residuals in the western North Americanpower system (wNAPS) was developed. The potential of thismethod to evaluate the confidence level of mode estimates isexamined.

Index Terms—Electromechanical modes, Power system mea-surement, Power system monitoring, Power system parameterestimation, Prony analysis.

I. INTRODUCTION

TO provide reliable electric power to consumers, a powergrid must be kept stable. Specifically, the question of

small-signal stability is this: when a system disturbance resultsin the oscillation of an inter-area electromechanical mode,will the oscillation decay harmlessly, or grow in magnitudeuntil system breakup occurs? Answering this question withconfidence is a fundamental step towards ensuring the stabilityof a power system.

Inter-area oscillations refer to generators in one area swing-ing against generators in another area. This phenomenon isconstantly occurring, but the oscillations are of great concernwhen they are not well damped. If the damping torque is insuf-ficient, such a disturbance can cause the oscillations to growin amplitude, leading to system breakup into “islands” [1], [2],[3]. System breakup may lead to a large-scale blackout, suchas the western interconnection power outage that occurred onAugust 10, 1996 [1], [2], [4]. The lightly damped inter-areaoscillations are associated with transmitting large amounts ofpower over long distances [3].

Accurate and timely mode frequency and damping infor-mation is critical in maintaining system stability and applying

This paper was prepared as a result of work sponsored by the U. S.Department of Energy through its Transmission Reliability Program. ThePacific Northwest National Laboratory is operated by Battelle for the U.S.Department of Energy under contract DE-AC05-76RL01830.Jim Follum is with the University of Wyoming, Laramie, WY 82071, USA(e-mail: [email protected]).Ning Zhou is with the Pacific Northwest National Laboratory, Richland, WA99352, USA (e-mail: [email protected]).John Pierre is with the University of Wyoming, Laramie, WY 82071, USA(e-mail: [email protected]).

solutions. For example, when light damping is observed, tie-line flow can be reduced through load shedding or generationre-dispatch [2], [5], [6]. Modal analysis is important becauseit provides the frequency and damping information needed toguide such decisions.

Modal analysis can be performed with either a model-basedor a measurement-based method. In the model-based method,the non-linear differential equations describing the dynamicsof the power system are linearized about an operating point.Mode frequency and damping can then be calculated usingeigenvalue analysis [1], [7]. The accuracy of the calculatedmode is determined by the quality of the power system model.With a measurement-based method, a linear model is estimatedfrom data collected from several phasor measurement units(PMU) distributed throughout the system. By extracting modeinformation from data, this method avoids reliance on theaccuracy of the power system model and thus can serve asa good compliment in monitoring inter-area modes [8], [9],[10], [11], [12].

One major question that a measurement-based method mustaddress is how accurate the estimated linear model and es-timated modes are. Initial efforts were made by [13], [14]to quantify the mode estimation errors from ambient dataanalysis. To complement the study, this paper proposes aMonte Carlo approach to validate an estimated model obtainedthrough ringdown analysis. To accomplish this, the methodseeks to relate mode estimation errors within the model toresiduals.

Residuals, also known as prediction errors, are the dif-ferences between measurement and model prediction values.They account for the portion of data that cannot be explainedby the model. A model that describes system features well pro-duces very small residuals. As such, it was hypothesized thatresiduals could provide an indication of how much confidencecould be placed in a model and the mode estimations thatit produced. The results presented in this paper indicate thatresidual values can provide an indication of the error presentin mode estimates.

The purpose of this paper is to explain the method forquantifying the relationship between mode error and residualvalues and discuss applications of the results. The paperis organized as follows. Section II provides an overviewof the analysis techniques that were employed. Section IIIdescribes development of the method, and Section IV presentsthe results from this development. Section V discusses themethod’s application to field measurement data. Conclusionsand expectations for future work are provided in Section VI.

II. ANALYSIS TOOLS

To obtain mode estimates, Prony analysis [15] was used.The error in these estimates was established using Monte Carlotrials. Residual analysis was then applied to determine therelationship between mode estimate error and residual values.Following is an examination of each of these methods and themodel they were employed on (a 17-machine model).

A. Prony Analysis

Prony analysis is a method of estimating damped sinusoidsin noise [16]. In 1990, [15] showed that Prony analysis couldbe successfully applied to ringdowns occurring on the powergrid to estimate modes. This paper also employed the exten-sion of Prony analysis described in [17], which allows multiplechannels to be analyzed concurrently to improve accuracy.In this study, mode frequency, damping ratio (DR), and thereconstructed ringdown approximation were of interest, thelatter being used in residual analysis. The equations used tocalculate these values are

λFreq =Imag

{λi

}2π

(1)

λDR = cos

tan−1

Imag{λi

}Real

{λi

} ∗ 100 (2)

y(k) =

P∑i=1

Bi ∗ zki (3)

where zi is the ith root of the characteristic polynomial and λiis the ith complex eigenvalue (mode) estimate, Bi is the ith

mode magnitude estimate, P is the model order, and k is thesample number, which varies from 0 to K−1. For a completedescription of how these values are obtained, please refer to[15], [16].

B. Residual Analysis

Residuals, as described in [18], were calculated as thedifference between the ringdown being analyzed by Pronyanalysis, y[k], and the ringdown constructed from the modeapproximations obtained during the analysis, y[k] of (3). Aslight adjustment in notation to an equation presented in [18]describes this relationship:

ε[k] = y[k]− y[k] (4)

After obtaining these residuals, two metrics were used forevaluation. These were the maximum residual and the rootmean squared (RMS) residual:

SMAX = max|ε[k]| (5)

SRMS =

√√√√ 1

K

K−1∑k=0

ε2[k] (6)

These metrics provided the necessary information to relatemode estimate error to residuals.

C. Monte Carlo Simulation

It was shown in [19] that if the relationship between theestimated parameters and measurements is linear and noise canbe modeled as Gaussian white noise, the standard deviationsof the estimated parameters is proportional to SRMS and canbe calculated using analytical approaches. For a nonlinearrelationship such as the ringdown and modes, a linearizationprocedure can be applied, and the standard deviation of themodes can be calculated based on the Jacobian matrix. Yet,due to the nonlinear relationship between the modes and thecoefficients of the characteristic polynomial [20], the lineariza-tion procedure may introduce significant errors. Therefore, aMonte Carlo method is used in this paper.

Monte Carlo simulation is a widely used method of deter-mining statistical performance by repeating trials with randomcomponents. Depending on the case, statistical measures suchas the mean, root mean squared error (RMSE), and standarddeviation are calculated after the simulation. All of the statis-tics discussed hereafter were calculated after completing 500-trial Monte Carlo simulations. The value of 500 was chosenfor its balance between computational burden and accuracyimprovement. For this study, the values of interest were themaximum residual, RMS residual, frequency RMSE, and DRRMSE.

D. 17-Machine Model

The 17-machine model was developed by Dr. Dan Trud-nowski of Montana Tech and has been used extensively inpower system stability studies such as those presented in [8],[9]. It is a modified version of a simple model representingthe western North American power system (wNAPS). A morecomplete description of the model can be found in [21], anda one-line diagram courtesy of [21] is presented in Fig. 1. Forthis research, the model was used to generate ringdowns foranalysis. These ringdowns are produced with per unit (p.u.)values of power flow. They were generated by simulating a1400-MW brake insertion at bus 35 for 0.5 seconds, reflectingtests performed on the wNAPS [22].

III. METHOD DEVELOPMENT

A. Mode Estimate Error - Residual Relationship

To develop a relationship between mode estimate error andresidual values, the amount of load noise in the system wasvaried. The 17-machine model provides a variable for thispurpose, and accepts values as the percentage of the loadto be random. This value was varied between 0.01% and0.07% in 121 increments. The resulting noisy ringdowns wereanalyzed using Prony’s method for the most energetic modewith frequency 0.4220 Hz and DR 3.63%. Table I presents allof the modes used in the 17-machine model.

For each noise level, Prony analysis with model order 24was performed to obtain modal frequency and DR estimatesfor each trial of a Monte Carlo simulation. These estimateswere then compared to the actual values used by the modelto obtain root mean squared error (RMSE) values. Becausethe true mode values were available during this portion of the

Fig. 1. One-line diagram of the 17-machine model courtesy of [21].

TABLE IMODE FREQUENCY AND DAMPING RATIO VALUES OF THE 17-MACHINE

MODEL

Frequency (Hz) Damping Ratio (%)0.3180 10.74390.4220 3.62760.6349 3.93540.6730 7.63360.8111 4.45580.9125 8.1418

development, it will be referred to as the “known model” case.After each Prony analysis, the ringdown used for analysis wasreconstructed using (3). The reconstructed ringdown was usedto calculate residual values.

When the simulation for each noise level was complete,mode estimate error and residual outliers beyond three stan-dard deviations of the mean were removed in an iterativeprocess which was repeated four times. With outliers re-moved, plots relating mode estimate error and residual metricswere constructed and revealed a distinctly linear relationship.Representative graphs exhibiting this linear relationship arepresented in Section IV.

Linear regressions were then applied to the data points usedto construct these plots such that

εf,SMAX= mf,SMAX

∗ SMAX (7)εf,SRMS

= mf,SRMS∗ SRMS (8)

εζ,SMAX= mf,SMAX

∗ SMAX (9)εζ,SRMS

= mζ,SRMS∗ SRMS (10)

where ε(f,∗) is the modal frequency RMSE in Hz, ε(ζ,∗) is theDR RMSE in percent, m∗,∗ is the slope specific to the modeestimate error and residual type denoted by its subscripts,and SMAX and SRMS are as defined in (5) and (6). The

slopes produced by the regressions quantified the relationshipbetween each of the four combinations of mode error type(frequency and damping ratio) and residual metric (maximumand RMS) because all y-intercepts were quite close to zero.These characteristic slopes were also used to develop a methodfor predicting mode estimate error from residual values.

B. Mode Estimate Error Prediction

The end goal in mode estimate error prediction is to providea method for evaluating the reliability of mode frequency andDR estimates produced while monitoring a power system. Tosubstantiate the viability of the method, it was developed forthe 17-machine model before being applied to measurementdata.

In developing the method for the 17-machine model, it wasrequired that the model be used in a way that reflected theconstraints of analyzing a real-world system. As such, the 17-machine model was used as an unknown system. True modevalues were not available to develop the characteristic slope.This portion of the development will therefore be referred toas the “unknown model” case.

Along with lack of access to true mode values, real-world systems provide only one ringdown for analysis anddo not allow for adjustment of load noise. These factorsmade it impossible to determine the characteristic slope inthe manner described in the previous section. As a substitutefor load noise, white Gaussian measurement noise was addedto the initial ringdown (from now on termed the “referenceringdown”) in varying amounts to create a noisy ringdown.This had the effect of increasing mode estimate error andresidual values, allowing estimation of the characteristic slopethat related these parameters.

Of course, mode estimate error cannot be calculated withoutknowledge of the true modes. Instead, pseudo mode errorwas calculated as the difference between the mode estimatesat a given measurement noise level and the reference modeestimates obtained by applying Prony analysis to the referenceringdown. The model order for the Prony analysis was again24. Residuals were calculated in the same manner describedin the previous section and (4).

The reference ringdown was constructed with load noiseequal to 0.02% of the load power. From this point, pseudomode estimate errors and residuals were calculated for 50measurement noise levels. Measurement noise levels werechosen to correspond to signal-to-noise ratios (SNR) evenlydispersed between 15 and 40 dB. Using such large amountsof noise required an adjustment to the method.

When the method was applied to the ringdown in the knownmodel case, approximately 30 seconds or ringdown data wasanalyzed. In the unknown model case, only the first 15 secondswas used. This ad hoc method was implemented to compensatefor the large amounts of measurement noise being addedto the ringdown. The large noise component had the effectof skewing residual values in the low-amplitude section ofthe ringdown. This procedure greatly improved results, but afurther investigation into its practice is needed.

For each noise level, a 500 trial Monte Carlo simulation wasperformed to obtain mean values for pseudo mode estimateerror and residual metrics. The random component for each ofthe trials was the measurement noise. Outliers were removedand linear regressions were applied to the resulting data foreach of these trials. The results revealed a linear relationshipbetween pseudo mode estimate error and residual metrics.However, the y-intercepts were now far from zero.

Mode estimate error and residuals should approach zerotogether because mode estimates are used to reconstruct theringdown. Thus, the y-intercepts observed when true modevalues were available were near zero. As would be expected,using mode estimates instead of true values for referenceintroduced a vertical shift. Therefore, the ordered pairs ofpseudo mode error and residuals represented a segment ofthe actual linear relationship with a useful slope and a y-intercept that accounted for initial mode estimate errors. Theseslopes were used to predict mode estimate RMSE through thegeneralization of (7) through (10).

ε = m ∗ S (11)

where subscripts denoting mode estimate error type and resid-ual type have been removed for simplicity.

To evaluate the ability of the method to predict frequencyand DR RMSE for a real-world system, the actual RMSEvalues were compared with predictions from the known modeland unknown model cases. The encouraging results obtainedfrom this comparison are presented in Section IV.

IV. METHOD DEVELOPMENT RESULTS

In establishing the relationship between mode estimate errorand residuals, plots such as Figs. 2 and 3 were constructed.These plots present the data points collected as the load noisewithin the 17-machine model was varied. They also showthe trend line that was fit to these data points to obtain acharacteristic slope. Similar results were obtained regardlessof which of the metrics for residuals described by (5) and (6)was used.

The correlation coefficients describing the linearity of therelationships between all four combinations of mode errortype (frequency and DR) and residual metric (maximum andRMS) are presented in Table II. These values are all quiteclose to unity and indicate that the relationship between modeestimate error and residuals is linear. In comparing correlationcoefficients obtained using maximum and RMS residual, it isclear that there is no advantage to using one over the other.The difference between maximum and RMS residual plots issimply scaling.

The data presented in Table III show that, as noted earlier,y-intercepts for trend lines relating mode estimate error andresiduals were found to be close to zero. It was this finding thatled to their exclusion when estimating error in mode estimates.

Experiments on the 17-machine model that examined theplausibility of estimating mode errors in a real-world scenarioresulted in Figs. 4 and 5. These figures compare actual

Fig. 2. Frequency error relationship for the 0.4220 Hz mode. Less than 1%of residual values and less than 5% of mode error values were removed asoutliers before creating the plot and fitting the trend line.

Fig. 3. DR error relationship for the 0.4220 Hz mode. Less than 1% ofresidual values and less than 5% of mode error values were removed as outliersbefore creating the plot and fitting the trend line.

TABLE IICORRELATION COEFFICIENTS FOR LINEAR REGRESSIONS APPLIED TO

ERROR RELATIONSHIPS FROM THE 17-MACHINE MODEL

Relationship Correlation CoefficientλFreq − SMAX 0.996λFreq − SRMS 0.996λDamp − SMAX 0.994λDamp − SRMS 0.994

TABLE IIIY-INTERCEPTS FROM LINEAR REGRESSIONS APPLIED TO ERROR

RELATIONSHIPS FROM THE 17-MACHINE MODEL

Relationship Y-InterceptλFreq − SMAX −0.098 ∗ 10−3 HzλFreq − SRMS −0.126 ∗ 10−3 HzλDamp − SMAX −0.186 ∗ 10−3%λDamp − SRMS −0.243 ∗ 10−3%

Fig. 4. Actual and estimated frequency RMSE values for a range of loadnoise levels. Less than 1% of residual values and less than 5% of mode errorvalues were removed as outliers before creating the plot.

Fig. 5. Actual and estimated DR RMSE values for a range of load noiselevels. Less than 1% of residual values and less than 5% of mode error valueswere removed as outliers before creating the plot.

mode error with estimates generated in the known model andunknown model cases. Again, the unknown model case rep-resents real-world limitations. Characteristic slope estimateswere first obtained using the method described for each case.These slopes were then used to estimate mode error forseveral different levels of load noise. Figs. 4 and 5 revealthat the estimates obtained under real-world circumstancesprovide valuable approximations of the actual RMSE. TableIV presents the slopes obtained for each case and allows fordirect comparison of the results. As can be seen, correspond-ing slopes are comparable and consequently produce similarRMSE estimates.

V. APPLICATION TO MEASUREMENT DATA

With the results indicating that mode error can be suc-cessfully estimated using a characteristic slope developed

TABLE IVERROR RELATIONSHIP SLOPES FOR THE 17-MACHINE MODEL

Relationship Known Model Unknown Modelper p.u.(MW ) per p.u.(MW )

λFreq − SMAX 0.011 ∗ 10−3Hz 0.009 ∗ 10−3Hz

λFreq − SRMS 0.032 ∗ 10−3Hz 0.025 ∗ 10−3Hz

λDamp − SMAX 2.165 ∗ 10−3% 1.910 ∗ 10−3%

λDamp − SRMS 6.446 ∗ 10−3% 5.211 ∗ 10−3%

Fig. 6. Measurement data from the August 10, 1996 breakup of the wNAPS.This dataset is courtesy of Bonneville Power Administration.

from an initially noisy ringdown, the method was appliedto measurement data from the wNAPS. One channel of MWmeasurements from the breakup of the wNAPS on August 10,1996, courtesy of Bonneville Power Administration (BPA), ispresented in Fig. 6. The average component has been removedto protect the data. The unstable oscillation growth presentedin Fig. 6 has been used to identify that a mode around 0.25Hz caused the system breakup [23].

Three channels of phasor measurement unit (PMU) datawere analyzed using Prony analysis with model order 24. Thetime duration was 18.5 seconds, and the starting point was786.7 seconds in the timescale presented in Fig. 6. Recall thatonly the large-amplitude portion of the ringdown was usedto compensate for the large amounts of measurement noisebeing added to the ringdown. This analysis produced a modeestimate with a 0.220 Hz frequency and -5.106% damping.These values were used for reference as measurement noisewas added to the data. Linear regressions with the correlationcoefficients presented in Table V were fit to the resulting datato establish the relationship between mode error and residuals.The error in the initial estimates was then estimated usingthe described method. The calculated slopes, residual metricvalues, and resulting error estimates are presented in TableVI. For example, the last row of Table VI indicates that whenusing the RMS residual metric, the estimated RMSE in the-5.106% damping estimate is 0.317%.

TABLE VCORRELATION COEFFICIENTS FOR LINEAR REGRESSIONS APPLIED TO

ERROR RELATIONSHIPS FROM MEASUREMENT DATA

Relationship Correlation CoefficientλFreq − SMAX 0.998λFreq − SRMS 0.998λDamp − SMAX 0.995λDamp − SRMS 0.996

TABLE VIERROR RELATIONSHIP RESULTS FOR MEASUREMENT DATA

Relationship Slope Residual RMSE(MW) Estimate

λFreq − SMAX 0.024∗10−3 HzMW

28.173 0.664∗10−3Hz

λFreq − SRMS 0.072∗10−3 HzMW

9.939 0.711∗10−3Hz

λDamp − SMAX 0.011 %MW

28.173 0.296%λDamp − SRMS 0.032 %

MW9.939 0.317%

VI. CONCLUSIONS AND FUTURE WORK

In this paper, the linear relationship between mode estimateerror and residual values was established. Knowledge of thisrelationship is valuable in model validation efforts because itallows for an assumption about mode estimate error based onresidual values. Application of the method to field measure-ment data provides valuable information about mode estima-tion accuracy. The ability to establish a distinct relationshipbetween mode error and residual values for each dominantmode in a power system would help place appropriate levelsof confidence in their estimates. In particular, damping ratioRMSE estimates could be used to inform system operatorsof how close a mode’s damping can approach zero beforeconclusions about system stability become unreliable.

Future work will be focused on validating the method’sapplication to field measurement data and extending analysisbeyond the highest energy mode. Much work exists beforethe findings presented in this paper could be applied to actualsystem monitoring applications, but they do indicate that suchan application is plausible.

VII. ACKNOWLEDGMENT

The authors gratefully acknowledge Mr. William Mittel-stadt, Dr. John Hauer, and Dr. Dan Trudnowski for their helpfuldiscussions and support.

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Jim Follum (S’10) received the B.S. degree in electrical engineering from theUniversity of Wyoming, Laramie, WY in 2011. He interned with the PacificNorthwest National Laboratory in Richland, WA as a research assistant fromJune to August 2010. After completing a second internship there from Mayto August 2011, he will return to the University of Wyoming in pursuit of the

Ph.D. in electrical engineering. His research interests include power systemstability and signal processing.

Ning Zhou (S’01- M’05- SM’08) received his B.S. and M.S. degrees inautomatic control from the Beijing Institute of Technology in 1992 and1995, respectively. In 2005, he received his Ph.D. in EE with a minor instatistics from the University of Wyoming. From 1995-2000 Ning workedas an assistant professor in the Automatic Control Department at BeijingInstitute of Technology. He is currently with the Pacific Northwest NationalLaboratory as a power system engineer. Ning is a senior member of the IEEEPower Engineering Society (PES). His research interests include power systemdynamics and statistical signal processing.

John W. Pierre (S’86-M’86-S’87-M’91-SM’99) received the B.S. degreein electrical engineering with a minor in economics from Montana StateUniversity, Bozeman, in 1986 and the M.S. degree in electrical engineeringwith a minor in statistics and the Ph.D. degree in electrical engineering fromthe University of Minnesota, Twin Cities, in 1989 and 1991, respectively.He worked as an Electrical Design Engineer at Tektronix before attending theUniversity of Minnesota. Since 1992, he has been a Professor at the Universityof Wyoming (UW), Laramie, in the Electrical and Computer EngineeringDepartment. He served as Interim Department Head from 2003 to 2004. Forpart of his 2007-2008 sabbatical, he worked at Pacific Northwest NationalLaboratory, Richland, WA, and Montana Tech, Butte. His research interestsinclude statistical signal processing applied to power systems as well as DSPeducation.

Dr. Pierre received UW’s College of Engineering Graduate Teaching andResearch Award in 2005. He is a member of the IEEE Signal Processing,Education, and Power Engineering Societies.