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    ANALYSIS OF INTER-AREA OSCILLATIONS VIANON-LINEAR TIME SERIES ANALYSIS TECHNIQUES Daniel Ruiz-Vega*   Arturo R. Messina**   Gilberto Enríquez-Harper*,***   *Programas de Posgradoen Ingeniería EléctricaSEPI-ESIME-Zacatenco. IPN.Mexico City, Mexico.e-mail: **Graduate Programin Electrical EngineeringCinvestav, IPN. PO Box 31-438,Guadalajara Jal. 45090, Mexico.e-mail:***Unidad de Ingeniería EspecializadaComisión Federal de ElectricidadRío Ródano 14, col. CuauhtémocMexico City, Mexico.e-mail: Abstract – This paper compares the characteristics andinformation provided by different modal identificationtools, in the analysis of a very complex forced inter-areaoscillation problem recorded in the Mexican intercon-nected system.These oscillations involved severe frequency and powerchanges throughout the system and resulted in load shed-ding and the disconnection of major equipment. This pa-per reports on the early analytical studies conducted toexamine the onset of the dynamic phenomena.Instances of variations in the amplitude and frequencyof the excited inter-area modes are investigated and per-spectives are provided regarding the nature of studiesrequired to identify and characterize the underlyingnonlinear process.It is shown that nonlinear analysis tools are able toidentify aspects of the dynamic behavior of the system thatare needed in the validation and characterization of theobserved phenomena, even in cases where power systemdynamic characteristics change several times due to loadshedding and generation tripping operations.  Keywords: Power system dynamic behavior, Inter- area oscillations, Modal identifications tools, Non-linear modal identification tools. 1   INTRODUCTION This document details the analytical studies con-ducted to examine the onset of major inter-area oscilla-tions in the Mexican system during the winter of 2004.The study focuses on the use of time-frequency repre-sentations to extract the key features of interest directlyfrom the actual system response.Of primary interest here is the analysis of the timeevolution of recorded signals, since this allows replicat-ing the events leading to the onset of the observed oscil-lations, and analyzing the influence of particular operat-ing conditions on system behavior.The non-stationarity of the data following the trig-gering event makes reliable estimate of the frequencyand damping characteristics of the observed oscillationsdifficult. Traditional methods of time series analysis donot address the problem of non-stationarity in power system signals, and often assume linearity of the process[1]. To circumvent these problems, time-frequencyrepresentations are used to give a quantitative measureof changes in modal behavior on different time scales.Two main analytical approaches have been investi-gated to extract the underlying mechanism from theobserved system oscillations. The first approach is based on the use of time-frequency representations of time series. These models are capable of explaining thenonlinear nature of the observed oscillations and permitthe tracking of evolutionary characteristics in the sig-nals and the development of measures like instantane-ous characteristics to capture mode interaction. Thesecond approach uses conventional analysis techniquescurrently used by the electric industry. Particular atten-tion is paid to the suitability of these techniques as adetector of nonlinear modal interaction.Analyses of observed measurement data via nonlin-ear spectral analysis techniques reveal the presence of complex dynamic characteristics in which the dynamiccharacteristics of the dominant modes of oscillationexcited by the contingency change with time. Themechanism of interaction characterizing the transitionof these modes involves strong nonlinear behavior aris-ing from self –and mutual interaction of the systemmodes. This is a problem that has received limited at-tention in the power system community.A challenging problem in studying this transitionconcerns the identification of the primary modes in-volved in the oscillation and the study of the nature of the coupling among interacting components giving riseto nonlinear, and non-stationary dynamics. The implica-tions of such complex spatio-temporal behavior canthrow much light on the dynamic patterns of the systemand information about the local behavior in both thetime and frequency domains can be extracted. Numerical simulations with nonlinear spectral analy-sis techniques show good correlation with observedsystem behavior and also point to the importance of nonlinear effects arising from changing operating con-ditions. These predictions are the basis for additionalstudies currently being undertaken involving small-signal and large signal performance and are expected toimprove modeling and analysis techniques used inpower system dynamic analysis studies. 2   DESCRIPTION OF THE EVENT 2.1   General description of the system The Mexican National Electric Power System is com- posed of 9 control areas. Six of these areas (namely 15th PSCC, Liege, 22-26 August 2005Session 32, Paper 2, Page 1     Northern, Northeast, Central, Oriental, Occidental andPeninsular) form an interconnected power system, andthe remaining three areas operate as electrical islands.The event of concern was registered during a tempo-rary interconnection of the Northwest control area to theMexican Interconnected System (MIS) through a 230kV line, in January 2004 (see Figure 1). The equipmentin charge of the automatic synchronization of both sys-tems had a failure (a fusible blow) and the interconnec-tion was performed with the systems out of phase. Figure 1: Pictorial representation of the MIS showing thelocation of the interconnecting line. The PMU is installed inMZD substation. After the tie line was connected, undamped inter-areaoscillations were observed throughout the system.These were on the order of ± 250 MW in the main in-terconnection, and continued for some minutes beforedamping out. As a consequence, protective relays oper-ated, tripping about 140 MW of load and three generat-ing units in order to compensate for the unbalancedcaused by system oscillations. The line was finallydisconnected. 2.2    Actual recorded response The data used in this study was recorded on Phasor Measuring Units (PMUs) at several key locations in thesystem. Of special relevance, Fig. 2 shows the time behavior of the MZD-DGD 230 kV real power flow.This transmission line interconnects the North-Westerncontrol area and the Mexican Interconnected Systemand constitutes a crucial part of the system’s backbone.On detailed examination, the records indicate the pres-ence of nonlinear, non-stationary behavior which makesdifficult the analysis and interpretation of the observedoscillations using conventional techniques.In the following sections, we discuss in detail thetheoretical studies conducted to analyze these oscilla-tions and compare them with those found experimen-tally. 3   NONLINEAR SPECTRALREPRESENTATION OF THE DATA A distinct characteristic of the observed oscillationsis the presence of nonlinear, non-stationary behavior  Figure 2: Active power flow oscillations in the transmissionline interconnecting North-Western control area with theMexican Interconnected System (MZD-DGD). which precludes direct application of conventionalanalysis techniques. To address the shortcomings of conventional techniques and gain insight into the differ-ent time scales present in the oscillations, time windowshave to be determined in which the oscillations arereasonable (locally) stationary with respect to thesewindows. We next examine the use of time-frequency(TF) techniques to extract the key features of the dy-namic behavior of the system as well as to provide acomparison between nonlinear time series analysis andconventional Fourier analysis. 3.1   The wavelet transform Wavelet spectral analysis provides a natural basis toestimate the time-frequency-energy characteristics of the observed data and is used in this work to identifydynamic trends in the observed system behavior.Consider a time series n  x , with equal time spacing, t  ∆ , and 1...0 −=  N n . Let )( η ψ  o be a wavelet function that depends on a non-dimensional time parameter  η  .In the present analysis, a Morlet wavelet mother func-tion has been used as a basis for the wavelet transformin the form [2],[3] 2/24/1 )( η η ω  π η ψ  −− = ee o jo (1)where o ω  is the non-dimensional frequency to satisfythe admissibility condition.Following Farge [2], the wavelet transformation of adiscrete signal )( t  x is defined as t nk i N k k k n e s x sW  ∆−= ∑= ω  ω ψ  10 )(*ˆˆ)((2)where * denotes the complex conjugate, n W  is thetransformation of the signal, and k  ω  , is the angular frequency defined as ⎪⎩⎪⎨⎧>∆−≤∆= 2:22:2  N k t  N k  N k t  N k  k  π π ω  (3) 15th PSCC, Liege, 22-26 August 2005Session 32, Paper 2, Page 2    We can then define the wavelet power spectrum, 2 )(  sW  n , at time point n and scale  s . To ensure thatthat the wavelet transforms in (2) at each scale are di-rectly comparable to each other, and to the transformsof other time series, the wavelet function at each scale  s is normalized to have unit energy, namely [4])(ˆ2)(ˆ 2/1 k ok   st  s s ω ψ π ω ψ  ⎟ ⎠ ⎞⎜⎝ ⎛ ∆= (4)Using the above normalization and referring to (2),the expectation value for  2 )(  sW  n is equal to  N  timesthe expectation value for  2 |ˆ| k   x . For a white-noise timeseries, this expectation value is  N  / 2 σ  where 2 σ  is thevariance; the normalization by 2 1 σ  gives a measure of the power relative to white noise. The details of thisapproach may be found in [4]. 3.2   Wavelet analysis The data used in the analysis is the actual oscillationtime series extending from 0:43:00 through 0.45:00,with a sampling interval of 0.20 seconds. For the pur- pose of clarity, the analysis was restricted to an observa-tion window between 120 second and 180 secondswhich coincides with the period of concern. This al-lowed us as to concentrate on the onset and analysis of lightly damped inter-area modes.The computed wavelet spectrum for the recordedsignal is presented in Fig. 3. Also shown is the averagevariance of the signal and the normalized wavelet power spectrum. The power spectrum estimate in Fig. 3c)clearly reveals the presence of two dominant modes atabout 0.22 Hz and 0.50 Hz. In addition, the analysisshows a higher frequency component at about 1.25 Hz.One significant feature of the spectrum is the pres-ence of time-varying, non-linear characteristics. Visualinspection of the response suggests, initially, the onsetof two main periods of interest in the analysis of systemresponse. In the first region, the analysis indicates the presence of a nearly constant frequency mode at about0.22 Hz. The continuous nature of the spectrum, and thelack of harmonic frequencies, provides an indicationthat the dominant mode is essentially stationary.For the second region, the analysis discloses harmon-ics superimposed on a slowly changing mode with anaverage frequency of 0.66 Hz. This implies that the periodic structure of the data is non-cosinusoidal in-volving frequency modulation; the presence of harmon-ics in the middle part of the wavelet spectrum indicatesnonlinearity.The analysis of this phenomenon, however, is noteasy to interpret since wavelets introduce spurious har-monics to fit the data. It is also of interest to note thatthe energy distribution for the Wavelet spectrum ismuch more concentrated in the middle time period. Thisis the period of greatest interest to this analysis. Figure 3: Wavelet time series analysis for the MZD-DGDpower signal showing varying frequency characteristics From wavelet analysis it is observed that: •   The spectrum can be divided into four distinctregions. In the first region, the behavior is essen-tially stationary and has a dominant modal com- ponent at about 0.22 Hz. •   The second region identified in the power spec-trum (middle part of the plot) shows the transitionof a low-frequency mode to a higher-frequencymode; the presence of higher order harmonicssuggests the existence of nonlinear characteristics.Of primary interest here, the analysis identifiestwo main frequency components at about 0.42 Hzand 0.62 Hz. •   Subsequent to this period, the frequency of themodal component settles to about 0.25 Hz. Themajority of the signal energy can be associatedwith region 2.A comparison of the power spectra for the MZD-DGD power signal in Fig. 4 shows that wavelet analysisaccurately replicates the dynamic performance of thesystem. Wavelet analysis provides a good visual inter- pretation of the phenomenon but lacks the frequencyresolution to capture the detailed time evolution of theobserved oscillations. These observations prompt fur-ther investigation of the srcin of mechanisms generat-ing such nonlinear behavior. MZD-DGD Real Power Actual Case Wavelet-Domain Reconstruction 120140160180 Time in seconds -300-200-1000100200300400    P  o  w  e  r   i  n   M   W -300-200-1000100200300400   Figure 4: Comparison of the srcinal system oscillations withthe wavelet transform 15th PSCC, Liege, 22-26 August 2005Session 32, Paper 2, Page 3    3.3   Estimation of instantaneous attributes: The Hilbert-approach In order to more accurately describe the event in bothtime and frequency, the Hilbert-Huang transform(HHT) method [5,6] was used to determine the nonlin-ear, non-stationary characteristics of the process. TheHHT is a two-step data-analyzing method. In the firststep, the time series )( t  x is decomposed into a finitenumber  n of intrinsic mode functions (IMFs), whichextract the energy associated with the intrinsic timescales using the empirical mode decomposition (EMD)technique. The srcinal time series )( t  x is finally ex- pressed as the sum of the IMFs and a residue: ∑+= = n jn j r t C t  x 1 )()( ˆ (5)where n r  is the residue that can be the mean trend or aconstant. Each IMF represents a simple oscillatorymode with both amplitude and frequency modulations.Having decomposed the signal into n IMFs, the Hil- bert transform of the kth component of the function k  c  in the interval ∞<<−∞ t  can be written as ∫ ∞∞− −= '')'(1)(ˆ dt t t t c P t c k k  π  (6)From signal theory, a real signal k  c and its Hilberttransform define an analytic signal given by))((exp)()( ˆ)()( t  jt  At c jt ct C  k k k k k  ϕ  =+= (7)Thus, the local amplitude k   A of the analytic functionis )(ˆ)()( 22 t ct ct  A k k k  += (8)and its phase k  ϕ  and instantaneous frequency k  ω  are dt t d t  t ct carctg t  k k k k k  )()(;)()(ˆ)( ϕ ω ϕ  =⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ = (9)It follows that the srcinal signal ()  xt  can then beexpressed as the real part of the complex expansion ⎥⎦⎤⎢⎣⎡= ∑∫ = nk k k  dt t  jt  At  x 1 ))((exp)(Re),( ω ω  (10)where () k   At  and () k  t  ω  are the instantaneous amplitudeand frequency, respectively. Eq (10) represents a gener-alized form of the Fourier expansion with time variableamplitude and frequency; this allows to accommodatenonlinear, nonstationary data. For the sake of economyof space, details of the theory are omitted. A compre-hensive account may be found in [5].Fig. 5 shows the intrinsic mode functions (IMF) for the actual system response. Application of the HHTyields seven IMFs associated with different time scalesof the data. The first IMF captures the higher-frequencymodes whilst the subsequent IMFs give informationabout the lower frequency modes. The residue essen-tially gives the trend of the function. Figure 5: The intrinsic mode function (IMF) components forthe MZD-DGD power signal Once the IMFs are computed, the instantaneous charac-teristics were determined using the approach in [6]. Thisis illustrated in Fig. 6 that shows the instantaneous am- plitude and frequency of each IMF. We con fine our discussion on the analysis of the first four IMFs, sincethey have the largest contribution to system behavior and are of relevance to the inter-area mode phenome-non. It should be noted in analyzing these results thatthe relative significance of each mode is determined byits peak (amplitude) values compared to those of thesrcinal data. Figure 6: Instantaneous amplitude and frequency of the firstfour intrinsic mode functions (IMFs) showing selected timewindows for linear spectral analysis The fluctuating nature of the IMFs and the variationof the amplitude provides and indication of nonlinear,non-stationary effects in the driving mechanism. In particular, the varying frequency characteristics suggestfrequency modulation, and therefore nonlinear behavior especially for IMFs 1 and 2. On the basis of this repre-sentation, the recorded signal was divided into four main observation (time) windows. Each time windowwas then segmented into sub-intervals to investigatespecific characteristics of interests. These are: 15th PSCC, Liege, 22-26 August 2005Session 32, Paper 2, Page 4    Time window 1 . A window in which the system re-sponse is initially dominated by two main modes; anessentially constant amplitude, nearly stationary modeat 0.18 Hz, and a oscillation mode about 0.22 Hz (IMF2). A third IMF is also observed with a frequencyslightly higher that the second IMF whose amplitudedecreases slowly. Time window 2 . A window in which, the analysis of the instantaneous frequency shows that the frequency of the 0.28 Hz mode increases to about 0.42 Hz. It is alsoof interest to observe that the frequency of the secondand third IMFs increase slightly. Time window 3 . A window in which the frequency of the second IMF increases to about 0.65 Hz and strongnonlinear frequency modulation is observed at about1.25 Hz, suggesting the presence of a second harmonic. Time window 4 . A windows in which the frequencyof second IMF decreases to about 0.25 Hz and instancesof nonlinear behavior are barely observed.These results agree very well with wavelet analysis, but in this approach, the time evolution of the observedoscillations is more closely captured.By dividing the time series into several periodswhich are nearly stationary and linear, it is possible toapply conventional analysis techniques to the study of the phenomena of concern and obtain results which aremeaningful. In the sequel, we concentrate on the analy-sis of system behavior in each of these observationwindows in an effort to identify modal characteristics. 4   LINEAR SPECTRAL REPRESENTATION OFTHE DATA On the basis of the pervious results, conventionalanalysis techniques were used to assess system dynamicbehavior over a range of time scales. Fourier spectralanalysis and Prony analysis were performed over theobservation windows selected in previous studies and toensure stationarity, the average value within each win-dow was substracted; this enables to focus on specificfeatures of interest in the data. The selected periods of time for Fourier and Prony analyses are clearly indi-cated in Fig. 7. 4.1   Fourier spectral analysis of data To confirm the non-stationary nature of the observedoscillations, and estimate local characteristics of thesystem response, we computed the Fourier spectra for each of the time windows described above. Fig. 8 showsthe discrete Fourier spectra of the actual power data for each of these windows. From this analysis, severaltrends can be identified: •   For the first observation window, the analysis re-veals a major dominant mode at about 0.22 Hz.As the windows moves through the second period,Fourier spectral analysis identifies the presence of  Figure 7: Selected time windows for linear spectral analysis. three major modes: a mode at about 0.42 Hz, a modeat about at about 1.25 Hz and a third mode at 0.90Hz. These results are in good agreement with the re-sults in Fig. 6. •   In turn, the analysis of the time window 3 showsthat the frequency of the slowest mode increases toabout 0.62 Hz whilst the frequency of the 1.25 Hzremains practically constant. An interesting observa-tion should be noted; the higher frequency mode ap- pears to be harmonically related to the 0.62 Hz (sec-ond harmonic) mode, indicating the presence of nonlinear behavior. These results are in goodagreement with mid-term behavior of the waveletspectrum in Fig. 4. •   Finally, the analysis of time window 4 shows thatthe frequency of the modes decreases to 0.22 Hz and0.40 Hz as suggested from the HHT method. Simulation results are in good agreement with previousfindings, but the combined use of Fourier spectralanalysis and Hilbert spectral analysis enables to charac-terize the spatio-temporal dynamic behavior of the sys-tem. 4.2   Prony analysis Prony analyses are applied to the recorded power of the line for each one of the selected time windows of Figs. 6 and 7. The results of Prony analysis are reportedin Table 1. In order to filter out spurious modes fromthe results, the sliding windows technique and othertechniques like de-trending the signal were used [7]. In Table 1, column 1 indicates the time window of con-cern in seconds; columns 2 and 3 present the frequencyand damping ratio of the identified dominant modes.Finally, column 4 shows the “signal to noise ratio”,which is a measure of the accuracy of Prony analysis.Good accuracy is achieved for SNR values around 40dB [8].It can be seen in the results of Table 1 that the informa-tion provided by Prony analysis also shows that thesignal behavior is non-stationary. The mode frequencies 15th PSCC, Liege, 22-26 August 2005Session 32, Paper 2, Page 5
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