Characterization of spatial and temporal changes in the dynamic patterns of a nonstationary process is a problem of great theoretical and practical importance. On-line monitoring of large-scale power systems by means of time-synchronized Phasor Measurement Units (PMUs) provides the opportunity to analyze and characterize inter-system oscillations. Wide-area measurement sets, however, are often relatively large, and may contain phenomena with differing temporal scales. Extracting from these measurements the relevant dynamics is a difficult problem. As the number of observations of real events continues to increase, statistical techniques are needed to help identify relevant temporal dynamics from noise or random effects in measured data. In this paper, a statistically based, data-driven framework that integrates the use of wavelet-based EOF analysis and a sliding window-based method is proposed to identify and extract, in near-real-time, dynamically independent spatiotemporal patterns from time synchronized data. The method deals with the information in space and time simultaneously, and allows direct tracking and characterization of the nonstationary time-frequency dynamics of oscillatory processes. The efficiency and accuracy of the developed procedures for extracting localized information of power system behavior from time-synchronized phasor measurements of a real event in Mexico is assessed.

Phenomena observed in power system oscillatory dynamics are diverse and complex. Remotely sensed measured data are known to exhibit noisy, nonstationary fluctuations resulting primarily from small magnitude, random load changes in load, driven by low-scale motions or nonlinear trends originating from control actions or other changes in the system. Extracting from these sets indices that capture significant spatial and temporal dynamics is very challenging [

Recent improvements in wide-area monitoring schemes have led to renewed investigation of nonlinear and nonstationary behavior of system oscillations [

Analysis and characterization of time-synchronized system measurements requires mathematical tools that are adaptable to the varying system conditions, accurate and fast, while reducing the complexity of the data to make them comprehensible and useful for real-time decisions. This is particularly true in the study of large datasets of dynamic processes whose energy changes with time or frequency.

In this paper, a statistically based, data-driven framework that integrates the use of a wavelet-based empirical orthogonal function (EOF) analysis and the method of snapshots is proposed to identify and extract dynamically independent spatio-temporal patterns from time-synchronized data. Extensions to current approaches to estimating propagating and standing features in near-real-time that can be associated with observed or measured data are discussed and numerical issues are addressed.

The procedure allows identification of the dominant spatial and temporal patterns in a complex dataset and is particularly well suited for the study the temporal evolution of critical modal parameters. It is shown that, in addition to providing spatial and temporal information, the method improves the ability of conventional correlation analysis to capture temporal events and gives a quantitative result for both the amplitude and phase of motion, and modal content, which are essential in the interpretation and characterization of transient processes in power systems.

The efficiency and accuracy of the developed procedures for capturing the temporal evolution of the modal content of data from time-synchronized phasor measurements of a real event in Mexico is assessed. Results show that the proposed method can provide accurate estimation of nonstationary effects, modal frequency, time-varying modes shapes, and time instants of intermittent or irregular transient behavior associated with abrupt changes in system topology or operating conditions in a near-real-time setting.

Time-synchronized phasor measurements collected by PMUs or other dynamic recorders can be interpreted conveniently in terms of statistical models involving both temporal and spatial variability [

Visualization of PMU data in terms of spatial and temporal information.

Assume, in order to introduce the more general ideas that follow, that measured data is available at

Using this notation, the set of data can be represented by an

In this formulation, each column corresponds to the system response at a specific time, and can represent the system response to a single event or represent an ensemble of time responses to multiple events measured at a single location.

Empirical orthogonal function (EOF) analysis provides a basis for the modal decomposition of an ensemble of data in terms of the smallest possible number of basic modes or proper orthogonal modes (POMs, most energetic global functions) [

The temporal and spatial components are calculated from the eigenvectors and eigenfunctions of the covariance matrix

In standard real-EOF analysis, matrix

Two important properties of EOFs are that the spatial distributions are orthogonal and their time series are uncorrelated. The method accounts for spatial and temporal changes and can be used to extract dynamic patterns from measured data. Measured data, however, may exhibit quite different dynamics at each system location or exhibit abrupt changes in modal quantities that cannot be captured using existing stationary models [

In real-world applications data are multiscale due to events occurring at different locations with different localization in time and frequency. To address the problem of multiscale modeling in data, a wavelet-based EOF approach has been developed. In this approach, the empirical orthogonal functions of the wavelet coefficients are computed at each scale and then combined the results at relevant scales. The approach is referred to as wavelet-based EOF analysis [

computing the wavelet decomposition for each column in the data matrix,

computing the covariance matrix of wavelet coefficients for each scale of interest, and combining the results at the dominant scales of interest,

extracting dynamic features from the selected scales.

Appendix

A complex, near-real-time formulation with the ability to resolve localized information is then proposed.

As it was highlighted in the previous section, conventional EOF analysis assumes stationarity of the underlying dynamic process and is therefore not suitable for an on-line implementation. In addition, such an approach, can only provide information about standing waves and cannot isolate portions of the phenomena in which dynamic changes take place [

In order to overcome the above limitations, a sliding window-based method is used to systematically analyze the observational data in near-real-time. This allows to isolate and extract the portions of data where oscillations are present and improves numerical efficiency and accuracy.

Assume that

A sliding window-based approach has been combined with EOF analysis to resolve localized information. In this approach, a sliding window frame of fixed size, say

Visualization of PMU data in terms of spatial and temporal information.

Referring to Figure

More formally, given a set of available sample time series, the temporal autocorrelation matrix,

In our formulation, matrix

An interesting and useful interpretation of the covariance matrix can be obtained by rewriting (

Now if we let

This analysis suggests that the covariance matrix at time step

Similar to the previous development, it can be shown that for a fixed window size,

Equation (

In an effort to extend standard real-EOF analysis to deal with propagating phenomena a complex formulation is adopted. Here, the real part is augmented with an imaginary obtained from the Hilbert transform of each time series. Refer to Esquivel and Messina for more details [

Assume then that

Given a model of the form (

Appendix

Many power system phenomena derive from nonlinear interactions between traveling waves of different spatial scales and temporal frequencies. This section extends the developed model to detect propagating phenomena in nonstationary processes.

Once the spatial eigenvectors associated with the real and imaginary part of

From EOF analysis [

Let now the complex field (

Equation (

In the developed models, a criterion for choosing the number of relevant modes is given by the energy percentage contained in the

Equations (

It might be remarked that, in the special case of real analysis, these expressions are simplified to the standard definitions.

A flowchart of the proposed approach is shown in Figure

Algorithm for near-real-time computation of standing and propagating features from measured data.

The following sections describe the application of multivariate statistical techniques to measured data including the estimation of instantaneous parameters.

The data used for this study were recorded by multiple phasor measurement units during a real event in northern Mexico. See [

The main event that originated the oscillations was a failed temporary interconnection of the Northwestern regional system to the Mexican interconnected system through a 230 kV line between Mazatlan Dos and Tres Estrellas substations. As a result of topological changes and load shedding, the observed oscillations exhibit highly complex phenomena including transient motions characterized by changing frequency content and variations in the mode shapes of critical electromechanical modes.

Among the existing PMU locations, frequency measurements at three major substations round the north, northwestern and northeastern systems were selected for study: Hermosillo (H), Mazatlan Dos (MZD) and Tres Estrellas (TTE). Figure

Time traces of recorded bus frequency swings recorded January 1, 2004 and detail of the oscillation buildup.

Measurements were recorded over 400 ms collected at a rate of 0.20 samples per second.

System measurements in this plot demonstrate significant variability suggesting a nonstationary process in both space and time. As observed in these plots, the most prominent variations occur in the interval during which the oscillation starts at 06 : 27 : 42 and the interval in which the operating frequency is restored to the nominal condition (60 Hz) by control actions at about 06 : 28 : 21.

Based on the frequency data collected by PMUs, EOF analysis was applied to reveal spatial and temporal dynamics. As a first step towards the development of an empirical basis, the covariance matrix was formed by the ensembles of the frequency observations at different system locations, that is,

To visualize the complex temporal and spatial dynamics that takes placed in the system following the failed interconnection, each time series is augmented with an imaginary component using Hilbert analysis and the complex EOF method is employed to approximate the original data. Further, in order to improve the ability of the method to capture temporal behavior, the individual time series are separated into their time-varying mean and fluctuating components; complex-EOF analysis is applied to the fluctuating field to decompose the spatio-temporal data into orthogonal modes associated to the standing and traveling wave components.

Using the proposed approach, two statistical modes describing standing and traveling features were identified. Mode 1, which accounts for 98.5% of the total energy describes the main (standing) feature in the records. The second mode contains approximately 1% of the energy and describes propagating features.

Figure

RMS error in the approximation.

Signals | Off-line (rms) | On-line (rms) |
---|---|---|

Hermosillo | 0.3591 | 0.2614 |

Mazatlan Dos | 0.4928 | 0.4008 |

Tres Estrellas | 0.7536 | 0.5477 |

Reconstruction of the original data with the traveling wave mode identified using the off-line and on-line analysis.

Also of interest, Figure

(a) average energy of traveling wave mode using the off-line analysis, and (b) instantaneous energy of traveling wave mode using the on-line analysis.

(a) mode shape of traveling wave mode using off-line analysis, and (b) time-varying mode shape of traveling wave mode using the proposed on-line analysis method

One of the most attractive features of proposed technique is its ability to detect changes in the shape properties of critical modes arising from topology changes [

The proposed approach provides an automated way to estimate mode shapes without any prior information of the time intervals of interest. The computational effort for analysis of large blocks of data is directly related to the number of spatial variables and the size of the observation window considered. In our numerical simulations the data analysis window corresponds to on-going that research is being conducted to estimate the appropriate window size for real-time applications.

Finally, Figure

Spatial pattern of the leading mode showing temporal variability.

Wide-area, real-time monitoring may prove invaluable in power system dynamic studies by giving a quick assessment of the damping and frequency content of dominant system modes after a critical contingency. In this paper, an alternative technique based on time-dependent complex EOF analysis of measured data is proposed to resolve the localized nature of transient processes and to extract dominant temporal and spatial information. A method of spatially decomposing oscillation patterns in near-real-time into their standing and travelling parts is presented.

By combining a sliding window approach with complex EOF, a novel framework for on-line characterization of temporal behavior is proposed that attempts to consider the influence of both, spatial and temporal variability. Numerical results show that the proposed method provides accurate estimation of nonstationary effects, modal frequency, time-varying mode shapes, and time instants of intermittent transient responses. This information may be important in determining strategies for wide-area control and special protection systems.

It is shown that, in addition to providing spatial and temporal information, the method improves the ability of conventional correlation analysis to capture temporal events and gives a quantitative result for both the amplitude and phase of motions, which are essential in the interpretation and characterization of transient processes in power systems.

Extensions to this approach to determine the optimal locations to place PMUs are underway. Applications to real-time system monitoring, protection and control will be addressed in future work.

Consider a function

The continuous wavelet transform (WT) of

The function

This section reviews the singular value decomposition and its features that relevant in the context of the proposed formulation.

Using the SVD decomposition, a complex matrix

Using (

From the decomposition given in (

The imaginary part measures the grade of asymmetries when the sum of both matrixes is different from zero; this is used to define the existence of traveling wave components.