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A Dynamic Harmonic Regression Approach to PowerSystem Modal Identification and PredictionArmando Jiménez Zavalaa & Arturo R. Messinaa

a Department of Electrical Engineering, The Center for Research and Advanced Studies(Cinvestav) of the IPN, Zapopan, Jalisco, MexicoPublished online: 09 Sep 2014.

To cite this article: Armando Jiménez Zavala & Arturo R. Messina (2014) A Dynamic Harmonic Regression Approach toPower System Modal Identification and Prediction, Electric Power Components and Systems, 42:13, 1474-1483, DOI:10.1080/15325008.2014.934932

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Electric Power Components and Systems, 42(13):1474–1483, 2014Copyright C© Taylor & Francis Group, LLCISSN: 1532-5008 print / 1532-5016 onlineDOI: 10.1080/15325008.2014.934932

A Dynamic Harmonic Regression Approach to PowerSystem Modal Identification and PredictionArmando Jimenez Zavala and Arturo R. MessinaDepartment of Electrical Engineering, The Center for Research and Advanced Studies (Cinvestav) of the IPN, Zapopan, Jalisco,Mexico

CONTENTS

1. Introduction

2. Unobserved Components Time Series Models

3. The Kalman Filter and Smoothing Algorithms

4. Application

5. Conclusion

Funding

References

Keywords: dynamic harmonic regression, Kalman filter, Prony analysis,trend identification, unobserved component model

Received 23 February 2014; accepted 7 June 2014

Address correspondence to Prof. Arturo Messina, Graduate Studies Programin Electrical Engineering, The Center for Research and Advanced Studies(Cinvestav) of the IPN, Zapopan, Jalisco 45019, Mexico. E-mail:aroman@gdl.cinvestav.mxColor versions of one or more of the figures in the article can be found onlineat www.tandfonline.com/uemp.

Abstract—Time series models provide a powerful tool to extractnonstationary features from measured data. In this article, a statis-tical framework based upon a dynamic harmonic regression modelfor examining modal behavior is provided. In this model, temporalpatterns in measured data are modeled within a stochastic state spacesetting. Estimates of the states or time-varying parameters are thenobtained using an optimal estimation method based on the Kalman fil-ter. Techniques to estimate future values of the unobserved signal arealso analyzed. The widely applicable technique is illustrated on bothsimulated and measured data. Factors that affect the performance ofthe method are discussed, including the effects of non-linear trends,data quality, and sampling design. Connections with other modalidentification methods are also investigated.

1. INTRODUCTION

Interest in the use of non-parametric, data-based power systemmodels to characterize system dynamic behavior has increasedin the last few years [1]. These models allow more flexibilityin the modeling of complex, noisy data and can be used to ap-proximate local behavior. References [2–4] summarize recentwork on this topic.

Compared to more traditional approaches, unobservedcomponents time series models have the potential to includefrequency information, local trends, and oscillatory and irreg-ular components. Trend identification is an important issuein load forecasting modeling [5, 6], data analysis and inter-pretation, filtering, and modal identification [7]. Detrendingis also a crucial first step in the application of modal identi-fication methods. The adaptive nature of this model, on theother hand, makes it particularly useful for adaptive model es-timation, forecasting, and extraction of periodic phenomena inmeasured data [8, 9].

There is a large and growing literature on modal identi-fication, including several books and review papers [10–13].Approaches to the characterization of the temporal variabilityof measured data, in particular, are of ongoing interest [14–16].Non-stationary time-series models based on both parametric

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Zavala and Messina: A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction 1475

and non-parametric methods have been used with varied suc-cess for feature extraction and modal identification of time-varying systems [7, 17].

Measured data are often noisy and exhibit non-stationaritiesthat can affect the estimation and interpretation of modal prop-erties. Stochastic time series models attempt to capture thetemporal features of a series by assuming that they followstochastic processes that, when combined together, yield theobservations. Their ability to characterize both modal featuresand local rather than global trends makes it especially usefulfor the analysis of power system data.

In recent work by Messina and Vittal, the use of time seriesmethods to extract local trends from measured data was inves-tigated [4]. The idea behind this approach is that the parametersof the process generating a time series do not remain constantover long periods of time due to observed and unobservedchanges in the environment.

This article extends the above model to the analysis of bothlocal trends and modal components of a general case of anoisy, time series. Building upon the ideas introduced in [4],modal identification and forecasting algorithms for discrete-time measurements in the presence of random disturbancesand measurement noise are presented. The approach takes intoaccount the non-stationary character of the data and is espe-cially well suited to analyzing measured data from transientprocesses.

Measured and simulated data are used to verify the effi-ciency and accuracy of the proposed algorithms. Factors thataffect the performance of the method are discussed, includingsample spatial distribution, data quality, and sampling design.Also, the ability of the technique to predict future behavior isillustrated.

Results are compared with Prony and Fourier analysis interms of the accuracy and precision of the estimated harmoniccomponents in the presence of noise and non-linear trends.

2. UNOBSERVED COMPONENTS TIME SERIESMODELS

Unobserved components time series models have been recentlyused to characterize non-stationary time series [18]. This sec-tion reviews the nature of these approaches and introduces theadopted representation.

2.1. The Modeling Framework

A typical discrete time unobserved component (UC) model foryt has the general form [3]:

yt = Tt + St + Ct + et ; et = N (0, σ 2

e

), (1)

where yt is the observed time series, t denotes time, and Tt ,St ,and Ct represent the trend, quasi-cyclical, and stochastic com-ponents, respectively; et is an irregular component normallydistributed Gaussian sequence with zero mean value and vari-ance σ 2

e . These components can be characterized by stochastic,time variable parameters (TVPs) with each TVP defined as anon-stationary stochastic variable. Models for the innovationsare discussed below.

2.2. Dynamic Harmonic Regression (DHR)

Among the UC models, DHR has recently attracted attention[4]. In this approach, the time series model (1) is assumed toconsist of three components: a periodic component associatedwith system oscillatory behavior, a quasi-cyclical componentrelated to the signal trend, and an irregular component thatcaptures non-systematic movements. As discussed below, thelevel of the series and the slope of the trend are assumed to bestochastic.

In the problem considered here, the DHR model is a non-stationary time series model. Following [8], consider a modelof the form (1). Let now the time-varying components Tt andSt be given by:

xp jt =

n∑j=1

a jt cos(ω j t) + b jt sin(ω j t), (2)

where the ω j = 2π f j , j = 1,. . ., n are the fundamental andharmonic frequencies of the sinusoidal components associatedwith the jth DHR component. This model has an intuitiverepresentation and can be used to extract both the oscillatorycomponent and the local linear trend model.

The complete DHR model can then be written as:

ydhrt =

R∑j=0

sp jt + et ,

=R∑

j=0[a jt cos(ω j t) + b jt sin(ω j t)] + et .

(3)

In this representation, a jt and b jt are assumed to be stochas-tic, time-varying parameters that follow a generalized randomwalk (GRW) process, and et is used to represent noise in theobserved time series; a0t is a slowly varying component or atrend (Tt = s∞

t = a0t ). As a result, non-stationarity is allowedin the various components.

In practice, the R oscillatory components,ω j , j = 1, . . . , Rare estimated separately using and autoregressive model ofthe observed data [8]. More general approaches, however, arepossible. In this work, a second calculation is made by Pronyanalysis and an energy criterion of the components [19, 20].

In what follows, the time evolution of the model (3) isexpressed within a stochastic state space setting.

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2.3. State Space Modeling Framework

Time varying models assume that the slope (and variance)of the time series change over time. These models may beused in conjunction with a variety of structural time seriesmodel specifications. In this research, a random walk plusnoise model in which the evolution of every 2R + 1 parameterin (3) is characterized by two stochastic variables; the level oramplitude, l jt , and the slope or drift, d jt [8], has been adopted.

Let the stochastic state vector be defined as x jt =[l jt amp; d jt

]T. The state space equation of (1) and (2) can be

written as a Gaussian state space model of the form:

x jt = F j x jt−1 + G jη jt , j = 0, 1, . . . , R, (4)

where η jt = [ v jt ξ jt ]T , {v jt } ∼ w.n.N (0, σ 2v j

), and {ξ jt } ∼w.n.N (0, σ 2

ξ j), with:

F j =[α j β j

0 γ j

],G j =

[δ j 0

0 ; 1

].

Particular cases of this model are discussed in [18].Once the dynamic model has been expressed in state-space

form, the Kalman filter can be used to estimate the states ortime-varying parameters as described below.

3. THE KALMAN FILTER AND SMOOTHINGALGORITHMS

To introduce the model adopted, consider the problem of esti-mating a state vector x(k) associated with a stochastic dynamicsystem modeled by the simple Gauss-Markov process [21]:

x j = F j x jt−1 + G jη jt ,

y j = H j x j + ξ t ,(5)

where y j is a p × 1 dimensional vector of observations thatare linearly related to the state vector x j by the matrix H j andη jt and ξ t are zero-mean statistically-independent white-noisedisturbance vectors with possibly time-variable covariance ma-trices Q j and R j , respectively.

Estimation of the state x, relies on the application of theKalman filter. Given a set of measurements, y (t0) , . . . , y (tN )the optimal estimate x (t + 1) of x (t + 1) can be obtained byminimizing the expected value of the magnitude of the error:

minx

{E

[∣∣xt − xt |τ∣∣2

]},

where the subscripts t |τ refers to an estimate at time t giveninformation up to and including time τ .

As stated by Norton [22], the Kalman filter model esti-mates the parameters of the system (5) based on current andpast observations by first running a one-step-ahead forecastingfollowed by a correction step.

Dynamic harmonic regression estimates the time-varyingparameters using a two-step (prediction-correction) Kalmanfilter, followed by a fixed-interval smoothing algorithm. Theprocess for optimal eigenvector assignment can be describedby the following equations for each time period (t = 1, . . . , N )[22]:

a) Prediction:

xt |t−1 = Fxt−1|t−1,

Pt |t−1 = FPt−1|t−1FT + GQGT ,(6)

b) Correction:

�t = yt − Hxt |t−1,

St = HPt |t−1HT + Rt ,

Kt = Pt |t−1HT S−1t ,

xt |t = xt |t−1 + Kt�t ,

Pt |t = (I − Kt H) Pt |t−1,

(7)

where Q = diag[σ 2v σ

2ξ ], with initial conditions x0 and P0,

and the notation xt |t−1 is used to indicate the estimate of x(t)given the observations x(0),. . ., x(t – 1). The recursive algo-rithm requires specifying the initial condition xo and its errorcovariance Po.

In the above formulation, �t is the one-step-aheadprediction-error (innovation), St is its variance, Kt is the opti-mal Kalman gain, and xt |t and Pt |t , represent the updated stateestimate and the updated estimate covariance for the state vec-tor, respectively.

During the prediction state, the optimal estimate of thestate vector at time t becomes xt |t−1 = Fxt−1|t−1. The asso-ciated covariance matrix of the estimation error is Pt |t−1 =FPt−1|t−1FT + GQGT . The updating equations (7) are thenused to calculate a new estimate of the state as new observa-tions arrive. The filter can be visualized as a feedback system,with the forward part driven by the innovations, which are awhite noise sequence [21].

After the filtering stage, a fixed interval smoother is usedto update (correct) the filter estimated state xt |t . In this case,using the output of the Kalman filter, smoothing takes the formof a backward recursion for t = N , . . . , 1, operating from theend of the sample set to the beginning [22]:

xt+1|N = xt+1|t+1 − P∗t+1FT

t+1λt+1,

P∗t |N = P∗

t |t + P∗t FP∗

t+1|t−1[P∗

t+1|N − P∗t+1|t ]P

∗t+1|t

−1FP∗t |t ,

λt = (I − HT

t+1R−1t+1Ht+1Pt+1|t+1

)FT

t+|λt+1

− HTt+1R−1

t+1

(yt+1 − Ht+1xt+1|t+1

). (8)

Smoothing allows estimating the state vector at a giventime point (given all the available data) and, hence, to inter-polate missing observations in the observed time series and

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Zavala and Messina: A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction 1477

the associated mean square error of the interpolated estimates.Different approaches can be used to avoid numerical issues, asproposed in [14] and [15].

Equations (6)–(8) describe a recursive algorithm that runsbackward in time from the last observation. In the first stage,the estimates xt |t−1 and Pt |t−1 are the one-step-ahead stateestimates and the associated covariance matrix, obtained from(6).

3.1. Trend Extraction

Estimation of the noise covariance matrices Qt , R and the asso-ciated distributions σ 2

εtare central for practical implementation

of this method.Following [3], let the signal-noise ratio (SNR) in Qt be

defined as:

SN R = σ 2ξ

σ 2ε

. (9)

To initiate the algorithm, an estimate of the initial trend andthe SNR matrix are needed, and the SNR must be estimatedseparately.

In the developed procedures, the SNRs are estimated in thefrequency domain. The method is based on the fact that thepseudo-spectrum of the DHR model of the R + 1 differentfrequency components included in the model can be writtenas [3]:

fy

(ω, σ 2

) = 1

8π

R∑j=0

[σ 2ω j(

1 − cos(ω + ω j

))2

+σ 2ω j(

1 − cos(ω − ω j

))2

]+ σ 2

2π, (10)

where σ 2 = [σ 2σ 2ω0σ 2ω1. . . σ 2

ωR]; σ 2

ω0is the variance associated

with the zero frequency term, i.e., with the trend, and the σ 2ωi

are the variances associated with the harmonic components,σ 2 is the variance of et.

Assume in order to introduce the method, that the ω’s in(3) are known. Under the assumption that each element a j , b j

follows an autoregressive (AR) process, the time varying pa-rameters can be estimated by minimizing the functional:

J =R∑

i=0

[fy (ωi ) − f y

(ωi , σ

2)]2

, (11)

where fy (ωi ) is the spectrum of yt .First, the dominant harmonic frequencies ωi ,∀i =

1, 2, . . . , R are extracted from the AR spectrum of the ob-served time series. Then, the unknown parameters are esti-mated from least squares optimization of the functional (11).Once these parameters are estimated, the time-varying trends

and harmonic components can be obtained using the Kalmanfilter approach discussed earlier.

3.2. Accuracy Evaluation

Two parameters are used to assess the accuracy of the method:the normalized mean squared error (NMSE) and the coefficientof determination (CD) R2, which is a number between 0 and1, used to describe how well a regression line fits a set of data.The CD, R2, is defined as [3]:

R2 = 1 − σ 2

σy, (12)

with σ 2 = 1N

∑Nk=1(ek − eN )2, σy = 1

N

∑Nk=1(yk − yN )2, ek is

the difference between the measured and estimated data, andeN is the overall mean value from these differences. A valueof the CD near to 1.0 indicates that a regression line fits thedata well.

By expanding on these ideas, a framework for trend identi-fication and extraction has been implemented. This frameworkis illustrated in Figure 1. Numerical issues associated with thisapproach are discussed next.

4. APPLICATION

In this section, dynamic harmonic regression is applied tothe problem of extracting power system’s electromechanicalmodes from measured data. Emphasis is placed on assess-ing the predictive capability of DHR models to extract time-varying trends and harmonic components, rather than in ana-lyzing the impact of trends on modal estimates.

4.1. Synthetic Data

A synthetic signal was generated to replicate simulated tie-linepower flow response data from a large transient stability modelof the Mexican interconnected system [4]. Figure 2 shows thetime evolution of the simulated data. The signal represents amultimodal unstable oscillation in which two critical modes atabout 0.249 and 0.491Hz become unstable.

In order to assess the potential of the methodology, DHRresults are compared to Prony analysis. For purposes of analy-sis, the original time series in Figure 2 was fitted with a Pronymodel and different types of trends were superimposed on theresulting modal representation. A sequence of white noise wasadded to the signal in order to show the capabilities of thismethod.

The synthetic model takes the general form yt = μt + ψt +et , or

yt = μt + et +4∑

j=1

A j e−σ j t cos(ω j t + φ j ), (13)

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1478 Electric Power Components and Systems, Vol. 42 (2014), No. 13

FIGURE 1. Flow chart of DHR model estimation.

where μt is the synthetic time varying trend, the A j , σ j , ω j ,φ j , with j = 1, . . . , 4, are the modal parameters, and et is anormally distributed white noise with variance σ 2

e = 40 addedto the simulation. Table 1 lists the extracted Prony parametersthat are used for subsequent simulations.

Two case studies involving two different trends were con-sidered:

(a) Case I. In this case, a second order non-linear trend ofthe form μt = a1t + a2t2 was added to the model (13).

FIGURE 2. Simulated tie-line power flow time series.

Mode Amplitude f (Hz) σ (1/s) φ (Deg)

1 7.187 0.2494 −0.04128 157.762 5.479 0.4914 −0.08273 −175.203 14.35 0.5275 0.04583 −172.584 17.73 0.7678 0.05717 11.73

TABLE 1. Prony analysis results of the synthetic signal in Figure 2(base case with no trends or noise)

(b) Case II. An oscillatory trend μt = A1e−σ t cos(2π f t)is superimposed to the model (13).

Table 2 gives the time-varying trend models applied and itsparameters. These cases are similar to those presented in [4],only slight modifications in the parameters were introduced.In this study, white noise is also added to the final observeddata.

In the authors’ analysis, the data was fitted with two mod-els: the dynamic harmonic regression model (3), and a Pronymodel [19]. Using (13), the measured data for each case isdecomposed into modal (periodic) components from whichmodal damping parameters can be extracted.

Case I. For the subsequent analyses, the comparison be-tween the simulated and extracted DHR model componentsfor Case I is shown in Figures 3 and 4. Figure 3 shows the ex-tracted trend as well as the actual signal in which white noiseis added. At a glance, DHR seems to have estimated the actualtrend remarkably well with little error along the data.

Figure 4 compares the harmonic components identified inthe DHR model with the time response of the Prony modes inTable 1. As shown, in this plot, the harmonic components 1and 4 correspond to modes 1 and 4, in Table 1, respectively;Allowing a jt and b jt to change in (3) over time, results in achange in amplitude and phase of the equivalent component,as shown in Figure 4.

Modal estimates are in good agreement with the originalmodes 1, 2, and 4; some differences are noted with the 0.527 Hzmode 3 (the least energetic mode) due to the low-dimensionalAR model in (11) used in the simulations. No efforts to improvethe characterization of this mode are conducted since this isan unstable case.

Case Trend Parameters

I μt = a1t + a2t2 a1 = 0.8, a2 = 0.25II μt = A1e−σ t cos(2π f t) A1 = 3.53,

σ = 0.007295, f = 0.1

TABLE 2. Trend characteristics in the examples

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Zavala and Messina: A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction 1479

FIGURE 3. Extracted trend for synthetic signal. Case I inTable 2.

The synthesized model fits very well the signal. More pre-cisely, the NMSE for the identified model is 4.45 × 10−6 andthe R2 = 0.999. In other words, the DHR model explains99.9% of the data. Studies were conducted to extract modalparameters for Cases I and II above.

Prony results for the signal (13) are shown in Table 3. Modes1 and 2 are the Prony’s approximation to the non-linear trends,while modes 3 through 6 correspond to modes 1–4 in Table 1.

Case II. Figure 5 compares the extracted trend with theactual trend in Table 2. Again, the DHR model produces agood fit for the trend model. Small deviations are seen nearthe ends of the signal.

Comparison analysis between both signals, i.e., originalwithout noise and fitted signal, shows an N M SE = 4.4 ×10−6 and R2 = 0.9972.

Figure 6 compares the DHR model with the Prony modes.As shown, the DHR components approximate well the Pronyanalysis results for the noiseless signal.

FIGURE 4. Harmonic components. Case I in Table 2.

Mode Amplitude f (Hz) σ (1/s) φ (Deg)

1 763.34 0 0.0203 02 593 0 −0.0297 03 7.6073 0.2514 −0.0358 141.434 6.2920 0.4907 −0.0773 −170.045 13.0658 0.5288 0.0417 −172.026 16.4025 0.7666 0.0505 16.28

TABLE 3. Prony analysis fit (Case I in Table 2)

For reference and comparison, Table 4 shows Prony resultsfor Case II in Table 2. As expected, simulation results show thatnonlinear trends may affect the accuracy of modal estimates.

4.2. Forecasting Using Time Series Models

The state space estimation based on optimal Kalman filtertogether with fixed interval smoothing is well suited for han-dling missing observations, forecasting, and outliers. Further,

FIGURE 5. Extracted trend for synthetic signal. Case II inTable 2.

FIGURE 6. Harmonic components. Case II in Table 2.

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1480 Electric Power Components and Systems, Vol. 42 (2014), No. 13

Mode Amplitude f (Hz) σ (1/s) φ (Deg)

1 1334.58 0 −6.7 × 10−6 −5.3 × 10−16

2 5.6356 0.1010 0.0320 6.72103 6.9037 0.2497 −0.0438 155.954 5.9522 0.4916 −0.0795 −177.775 15.1447 0.5283 0.0466 −175.586 16.6483 0.7686 0.0611 6.67

TABLE 4. Prony analysis fit (Case II in Table 2)

the ability to interpolate missing observations avoids the needto preprocess data to better fit theoretical assumptions.

Forecasting is carried out by omitting the updating equa-tions while retaining the prediction equations in the algorithmpresented in (7) and (6), respectively. The prediction can becarried out under the condition that matrices H and F areconstant with time or its value can be estimated properly [7].

In this section, harmonic regression is proposed as a predic-tion method for data collected over space and time. To illustratethe performance of the method, simulated data from a transientstability model of the Mexican interconnected system is used.Figure 7 shows the time evolution of that data. The signal rep-resents an unstable power oscillation on a major interface inwhich five modes are seen to interact. For prediction purposes,only the first 24 seconds (solid line, labeled as observed data)are used to analyze and extract the harmonic components.

As a first step, the AR spectrum of the signal was estimated,and the dominant frequencies were identified. Figure 8 com-pares the actual AR-spectrum with the fitted spectrum usingthe DHR model. Using the first five frequency peaks in Fig-ure 8 (0.24–1.24 Hz), a DHR model is estimated. Figure 9shows the temporal evolution of the five extracted frequencycomponents, with the lower frame presenting the modes M4and M5.

Figure 10 shows the resulting extrapolated series, togetherwith the simulated (measured) series and the extracted trend.As shown, the DHR model is able to approximate well future

FIGURE 7. Observed data.

FIGURE 8. Autoregressive spectrum for the simulated data.

FIGURE 9. Harmonic components.

FIGURE 10. Extracted trend and forecasting of the originalsignal.

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Zavala and Messina: A Dynamic Harmonic Regression Approach to Power System Modal Identification and Prediction 1481

FIGURE 11. Recorded inter-tie power oscillations at a majortransmission corridor.

system behavior based on past observations, in other words, itis able to obtain a true forecast.

Comparing the identified signal with the complete originalsignal, the coefficient of determination indicates that the modelexplains 99.7% of the original data. A separate analysis for thefirst 26 sec shows that the NMSE = 1.9 × 10−12 and R2 =0.99999, and for the last 6 sec (i.e., the forecast horizon), anNMSE = 6.5 × 10−5 and R2 = 0.9933.

4.3. Application to PMU Data

To evaluate the efficacy of the method in estimating modalparameters under noisy conditions, DHR was applied to syn-chronized phasor measurement unit (PMU) data from a realevent in the Mexican Interconnected System (MIS) [23].

The event was triggered by a combination of system eventsinvolving operational changes and weak transmission condi-tions, and resulted in unstable oscillations that lasted for about150 sec until protective equipment disconnected a major gen-erator participating in the oscillation.

The initiating event was the increase from 150 to 200 MWof unit #3 of the hydroelectric plant El Caracol in the south-western network of the MIS. System oscillations developedfor about 2 min until the 230 kV line interconnecting the El

FIGURE 12. AR spectrum. Time interval (14:41:23.000 to14:43:34.650).

FIGURE 13. Harmonic components. Entire oservationinterval.

Caracol plant was tripped off by overcurrent relays. Figure 11shows recordings of real power flow at a major transmissionline captured by the CFE WAMS on July 31, 2008. Measure-ments were obtained over a 250 seconds period at a rate to 20samples per second.

In what follows, attention is focused on assessing the abil-ity of the technique to characterize both the transient oscil-lations following the event and the ambient conditions oncethe critical generator was disconnected. For clarity of illus-tration, the observation interval is divided into two data seg-ments (14:21:23.000 to 14:43:34.650) and (14:43:54.650 to14:44:55.650), which capture transient and ambient operatingconditions.

Figure 12 compares the actual AR spectrum and theidentified spectrum using DHR for the transient interval14:41:23.000 to 14:43:34.650. DHR analysis identifies a dom-inant mode at about 1.0 Hz and two additional modes at about0.37 and 0.55 Hz. These frequencies coincide with major inter-area modes in the MIS.

FIGURE 14. Stochastic trend.

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1482 Electric Power Components and Systems, Vol. 42 (2014), No. 13

Mode Amplitude f (Hz) σ (1/s) φ (Deg)

Actual 3.5696 0.9962 −0.0323 −42.8991DHR 3.9449 0.9965 −0.0306 −44.8722

TABLE 5. Prony analysis results for 1 Hz modes (time interval:14:41:23.000 to 14:43:30.000)

Figure 13 shows the time evolution of the extracted periodiccomponents at 0.373, 0.553, and 0.995 Hz, while Figure 14shows the extracted trend. Simulation results suggest that themethod accurately characterizes system behavior under bothtransient and ambient conditions.

In order to verify the accuracy of the extracted modes,Prony analysis was applied to both the original signal and thedominant 0.99 Hz component (Mode M3 in Figure 13), for thetime interval 14:41:23.000 to 14:43:30.000. Simulation resultsin Table 5 show that DHR can be efficiently used to identifyoscillatory components from noisy data.

The error analysis shows that the DHR model explains99.99% of the original data. These ideas are framed inFigure 14, where the identified stochastic trend is shown. Atthe end of the analysis, an AR spectrum of the DHR model isobtained and presented as a dashed line in Figure 12.

A key advantage of DHR analysis over other techniquesis its ability to characterize system behavior under noisymeasurements. In order to demonstrate this feature, DHRwas applied to ambient data (time interval 14:43:54.650 to14:44:55:650).

Simulation results in Figure 15 show two modes at about0.37 and 0.51 Hz associated with inter-area oscillations. Com-parison between the Fast Fourier Transform (FFT) of the actualoscillations and the FFT of the DHR components in Figure 16shows that the technique can be used to extract modal behav-ior from ambient operating conditions. No physically meaningresults are obtained for this time interval using Prony analysisand are therefore not shown.

FIGURE 15. Harmonic components. Time interval (14:43:54.650 to 14:44:55.650).

FIGURE 16. FFT spectrum. Time interval (14:43:54.650 to14:44:55.650).

Interestingly, with DHR analysis the effect of slow trendsin the spectra is eliminated as shown in Figure 16. Slow trendsmay make the analysis and interpretation of system behaviordifficult and may lead to incorrect or biased results.

5. CONCLUSION

Structural time series models provide a useful framework forprocessing and modeling time-varying trends in measureddata, integrating the processes of forecasting, interpolation,and seasonal adjustment. Also, competitive results were ob-tained from modal identification of dynamic harmonic regres-sion components. As illustrated, the capability of forecastinghas proved to be a satisfying tool that can compete with otherexisting techniques.

The ability of DHR models to compute time varying param-eters also opens a new possibility in the study of complex, non-linear systems. There are many possible directions in whichthis theory may be expanded. The use of instantaneous infor-mation from other techniques to estimate the dominant fre-quencies and the introduction of different smoothers to allowimplementations in real time is an opportunity area. Furtheranalysis can be made by dividing the data set into numeroussegments, allowing the DHR model to have a time-varyingfrequency approach using instantaneous frequency estimationalgorithms to help in the segmentation procedure.

FUNDING

The authors gratefully acknowledge financial support fromCONACYT, Mexico, under grant 234455.

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BIOGRAPHIES

Armando Jimenez Zavala received his B.Sc. degree inelectrical engineering from Morelia Institute of Technology,Michoacan, Mexico, in 2007, and the M.Sc. degree in electricalengineering from Cinvestav, Mexico, in 2012. He is currentlyworking toward a doctoral degree in electrical engineering atCinvestav. He is a Student Member of the IEEE. His researchinterests include measurement-based methods to the study ofpower system oscillations, and power system identification.

Arturo R. Messina received the M.Sc. degree (Honors) inelectrical engineering from the National Polytechnic Instituteof Mexico, in 1987, and the Ph.D. degree from Imperial Col-lege of Science Technology and Medicine, London, U.K., in1991. Since 1997, he is a professor at the Center for Researchand Advanced Studies (Cinvestav) of the National Polytech-nic Institute of Mexico. He is an Editor of the Electric PowerComponents and Systems Journal. His areas of interest in-clude power system stability analysis and control, and the de-velopment and application of advanced measurement-basedsignal processing techniques to the study and characterizationof inter-area oscillations in power systems.

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