Inter-area Oscillations in Power Systems pp 159-187 | Cite as

# Complex Empirical Orthogonal Function Analysis of Power System Oscillatory Dynamics

## Abstract

Multivariate statistical data analysis techniques offer a powerful tool for analyzing power system response from measured data. In this chapter, a statistically based, data-driven framework that integrates the use of complex empirical orthogonal function analysis and the method of snapshots is proposed to identify and extract dynamically independent spatiotemporal patterns from time-synchronized data. The procedure allows identification of the dominant spatial and temporal patterns in a complex data set and is particularly well suited for the study of standing and propagating features that can be associated with electromechanical oscillations in power 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. 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 mode shapes, and time instants of intermittent or irregular transient behavior associated with abrupt changes in system topology or operating conditions.

## Keywords

Mode Shape Singular Value Decomposition Proper Orthogonal Decomposition Empirical Orthogonal Function Instantaneous Frequency## 6.1 Empirical Orthogonal Function Analysis

Empirical orthogonal function (EOF) analysis is a statistical method of finding optimal distributions of energy from an ensemble of multidimensional measurements [1]. The essential idea is to generate an optimal basis for the representation of an ensemble of data collected from measurements or numerical simulations of a dynamic system.

Given an ensemble of measured data, the technique yields an orthogonal basis for the representation of the ensemble, as well as a measure of the relative contribution of each basis function to the total energy with no a priori assumption on either spatial or temporal behavior.

The following sections provide a review of some aspects of the qualitative theory of empirical orthogonal functions that are needed in the analysis of low-dimensional models derived from the technique.

We start by introducing the method in the context of statistical correlation theory.

### 6.1.1 Theoretical Development

*j*= 1,…,

*n*,

*k*= 1,…,

*N*, denotes a sequence of observations on some domain \(x \in \Omega\) where \(x\) is a vector of spatial variables and \(t_k [0,T]\) is the time at which the observations are made. Without loss of generality, the time average of the time sequence

^{1}

^{*}denotes the conjugate transpose.

Equation (6.10) has a finite number of orthogonal solutions over \(\varphi _i (x)\) (the POMs) with corresponding real and positive eigenvalues \(\lambda _i\). They are consequently called empirical orthogonal functions.

In other words, the optimal basis is given by the eigenfunctions \(\varphi _i\) of (6.13) whose kernel is the autocorrelation function \({\bf{R}}\left( {x,x^{\prime}} \right) = \left\langle {\bar u\left( {x_j ,t_k } \right)\bar u\left( {x^{\prime}_j ,t_k } \right)} \right\rangle\).

### 6.1.2 Discrete Domain Representation

*n*locations.

*N*×

*n*-dimension ensemble matrix, \({\bf{X}}\) [6]

The eigenvalues computed from (6.18) are real and nonnegative and can be ordered such that \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _n \ge 0\). The eigenvectors of \({\bf{C}}\)are called the POMs and the associated eigenvalues are called the proper orthogonal vectors (POVs).

For practical applications, the number snapshots *N* can be rather large, leading to a very large eigenvalue problem. There are at least two methods to solve the eigenvalue problem (6.16) [6]: the direct method and the method of snapshots.

The direct method attempts to solve the eigenvalue problem involving the *N* × *N* matrix directly using standard numerical techniques. This can be computationally intensive if the number of observations is larger than the number of observing locations or grid points. The method of snapshots, on the other hand, is based on the fact that data vectors \({\bf{u}}_i\) and the POD modes \(\varphi _l\) span the same linear space. The latter is explored here.

The next section describes in more detail the nature of the approximation employed here to construct the statistical representation.

### 6.1.3 The Method of Snapshots

*n*×

*N*matrix, \({\bf{\boldsymbol{\Phi} }}\), known as the modal matrix. In matrix form Eq. (6.19) becomes

*n*

**C**. In words, the first-order necessary optimality condition for \(\varphi\) to provide a maximum in (6.20) is given by (6.16). This completes the construction of the orthogonal set \(\left\{{\varphi _1 }\ {\varphi _2 }\ \cdots \ {\varphi _n }\right\}\).

*p*is the number of dominant modes and \(\varepsilon\) is an error function.

Having computed the relevant eigenmodes, the temporal behavior can be evaluated as the inner product of the eigennmode (the POD mode \(\varphi _i\)) and the original data. To ensure uniqueness of the solution, the normalization condition of \(\left\langle {\varphi _i ,\varphi _i } \right\rangle = 1\) is imposed.

It should also be stressed that no conditions are imposed on the data set; the data can be a sample of a stationary process or a sample of a nonstationary process.

Equation (6.24) is called the Karhunen–Loeve decomposition and the set \(\varphi _j\) are called the empirical basis [5].

### 6.1.4 Energy Relationships

*E*, by \(E = {\rm{Trace}}\,[{\bf{R}}({\bf{x}}_i ,{\bf{x}}_m )],\) one obtains

*p*of the reduced basis \(\varphi\) such that the predetermined level of the total energy

*E*of the snapshot ensemble is captured, i.e., 99%. The

*p*-dominant eigenfunctions are then obtained as

*p*where

*E*is an appropriate energy level.

The key advantage of this technique is that allows extracting information from short and often noisy time series without prior knowledge of the dynamics affecting the time series.

## 6.2 Interpretation of EOFs Using Singular Value Decomposition

A useful alternative method for estimating modal characteristics can be developed based on the analysis of the response matrix \({\bf{X}}\) in (6.15).

Before outlining the procedure for singular value decomposition (SVD) analysis, we introduce some background information on singular value analysis.

### 6.2.1 Singular Value Decomposition

*m*×

*n*matrix. The SVD theorem states that \({\bf{A}}\) can be decomposed into the following form [7]:

*m*×

*m*orthonormal matrix \({\bf{V}}^{\rm{T}} = {\bf{U}}^{ - 1}\), \({{\boldsymbol{\Sigma} }}\) is an

*m*×

*n*pseudodiagonal and semi-positive definite matrix with diagonal entries containing the singular values , \({\bf{V}} = {\rm{col}}[{{\bf{v}}_1 }\quad {{\bf{v}}_2 }\quad \cdots \quad {{\bf{v}}_n }]\) and is an

*n*×

*n*orthonormal matrix \({\bf{U}}^{\rm{T}} = {\bf{U}}^{ - 1}\). The columns of \({\bf{U}}\) and \({\bf{V}}\)are called the left and right singular vectors for \(\,{\bf{A}}\).

Throughout this research, we will, consider only the case when \(m>n\). The diagonal entries of \({\boldsymbol{\Sigma}}\), i.e., the \(\Sigma_{ii} = \sigma _i\), can be arranged to be nonnegative and in order of decreasing magnitude \(\sigma _1 \ge \sigma _2 \ge \cdots \ge \sigma _m \ge 0\).

*r*of the

*u*’s and

*v*’s make any contribution to \({\bf{A}}\), and can be expressed as an outer product expansion

### 6.2.2 Relation with the Eigenvalue Decomposition

An interesting interpretation of the POD modes can be obtained from the singular value analysis of the response matrix \({\bf{X}}\).

*N*×

*N*matrix whose columns are the left singular vectors of \({\bf{X}}\), \({{\boldsymbol{\Sigma} }}{}\) is

*N*×

*n*matrix containing the singular values of \({\bf{X}}\) along the main diagonal and zeros elsewhere, and \({\bf{V}}\) is an

*n*×

*n*orthonormal matrix whose columns correspond to the right singular vectors of \({\bf{X}}\). The response matrix, \({\bf{X}}\), is complex and symmetric and possesses a set of orthogonal singular vectors with positive singular values.

The POMs, defined as the eigenvectors of the sample covariance matrix \({\bf{C}}\) are thus equal to the left singular vectors of \({\bf{X}}\). The POVs, defined as the eigenvalues of matrix \({\bf{C}}\) are the squares of the singular values divided by the number of samples *N*.

## 6.3 Numerical Computation of POMs

- 1.
Given an ensemble of measurements of a nonstationary process, compute the response matrix \({\bf{X}}\). Form the complex time series matrix \({\bf{\hat X}} = {\bf{X}} + {\rm{j}}{\bf{X}}_H\), where \({\bf{X}}_H\) is the Hilbert transform of \({\bf{X}}\)

- 2.
Compute the singular vectors \({\bf{U}},{\bf{V}}\) and the corresponding singular values \(\sigma\).

- 3.
Determine the time evolution of the temporal modes, \(a_i\). Extract standing and propagating features using the complex SVD (singular value decomposition) formulation.

For different events recorded at the same location, statistical averaging can be employed to take advantage of the statistics of the data. In this case, the snapshots can be thought of realizations of random fields generated by some kind of stochastic process.

The processing steps are detailed in the sections that follow.

## 6.4 Complex Empirical Orthogonal Function Analysis

Empirical orthogonal function analysis of data fields is commonly carried out under the assumption that each field can be represented as a spatially fixed pattern of behavior. This method, however, can not be used to for detection of propagating features because of the lack of phase information. To fully utilize the data, a technique is needed that acknowledges the nonstationarity and behavior of the time-series data.

This represents a filtering operation upon \(u(x_j ,t_k )\) in which the amplitude of each Fourier spectral component remains unchanged while its phase is advanced by π/2. The eigenvectors here are complex and can be expressed alternatively as a magnitude and phase pair.

In what follows we discuss the extension of the above approach to compute standing and propagating features.

### 6.4.1 Complex EOF Analysis

*j*(spatial points) by

*k*(temporal points) ensemble matrix. From the preceding results, it follows that

*H*denotes a Hermitian matrix, and we assume that

*n*singular values on the diagonal of \(\boldsymbol{\Sigma}\) are the square roots of the nonzero eigenvalues of both \({\bf{X}}\left[ {{\bf{X}}^H } \right]^{\rm T}\) and \(\left[ {{\bf{X}}^H } \right]^{\rm{T}} {\bf{X}}\) where

*n*is the rank of \({\bf{X}}\).

Using the complex SVD, it is possible to compute the spatial amplitude and spatial and temporal phase functions as discussed below.

### 6.4.2 Analysis of Propagating Features

*The time-dependent complex coefficients associated with each eigenfunction can be conveniently split into their amplitudes and phases. From*the

*complex EOF analysis in (*

*6.49*

*)*,

*the ensemble of data can be expressed as the complex expansion*

*[*13, 14

*]*

*i*th mode.

This effectively decomposes the data into spatial and temporal modes.

- 1.
Spatial distribution of variability associated with each eigenmode

- 2.
Relative phase of fluctuation

- 3.
Temporal variability in magnitude

- 4.
Variability of the phase of a particular oscillation

The following definitions introduce these concepts.

### Definition 6.1

This function shows the spatial distribution of variability associated with each eigenmode.

### Definition 6.2

This measure, for which an arbitrary reference value must be selected, varies continuously between 0 and 2π. Together, Eqs. (6.52) and (6.53) give a measure of the space distribution of energy and can be used to identify the dominant modes and their phase relationships. Further, for each dominant mode of interest, a mode shape can be computed by using the spatial part of (6.50).

### Definition 6.3

This function gives a measure of the temporal variability in the magnitude of the modal structure of the field \(u\). As described later, the general form of these equations is very amenable to computational analysis.

### Definition 6.4

Equations (6.52), (6.53), (6.54), (6.55), (6.56) provide a complete characterization of any propagating effects and periodicity in the original data field which might be obscured by normal cross-spectral analysis. Finally, it might be remarked that, in the special case of real analysis, these expressions simplify to the normal definitions.

## 6.5 Application to Time Synchronized Measured Data

To test the ability of the method to analyze complex oscillations, we analyze data from time-synchronized measurements. The data used for this study were recorded by phasor measurement units (PMUs) over a 4 s window during a real event in northern Mexico. A brief description of the data follows. More detailed information on system measurements can be found in [4, 15].

At local time 06:27:42 in the early hours of January 1, 2004, undamped oscillations involving frequency, voltage, and power were observed throughout the northern systems of the Mexican Interconnected System (MIS). The main event that originated the oscillations was a failed temporary interconnection of the Northwest regional systems to the MIS through a 230 kV line between MZD (Mazatlar DOS) and TTE (Tres Estrellas) substations. It is noted that, prior to this oscillation incident, the northwestern system operated as an electrical island.

Oscillations in the northern systems with periods about 0.61, 0.50, and 0.27 Hz persisted for approximately 1.2 s before the northwestern system was disconnected from the MIS. During the time interval 06:27:42–06:28:54 the system experienced severe fluctuations in frequency, power, and voltage resulting in the operation of protective equipment with the subsequent disconnection of load, independent generation, and major transmission resources.

As discussed later, during this period the system experiences changes in frequency (amplitude) content and mode shapes.

*t*is the time, and

*N*is the number of data points in the time series. For our simulations

*,*2021 snapshots are available, representing equally spaced measurements at three different geographical locations

*.*Each time series is then augmented with an imaginary component to provide phase information and the EOF method is employed to approximate the original data by a general, nonstationary, and correlated frequency model.

The following subsections describe the application of complex empirical orthogonal function analysis to examine the temporal and spatial variability of measured data.

### 6.5.1 Construction of POD Modes via the Method of Snapshots

The method of snapshots was applied to derive a spatiotemporal model of the oscillations. 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. By separating the data into their mean and fluctuating components, EOF analysis is able to selectively determine the temporal behavior of interest. The method may also help reduce the detrimental effects of crossing events, spatial aliasing, and random contamination.

In the application of the proposed method to measured data, we assume that each signal can be construed as a superposition of fast oscillations on top of a slow oscillation (the time-varying instantaneous mean) [16]. The slow oscillation essentially captures the nonlinear (and possibly time-varying) trend while the slow oscillations are the fluctuating parts.

In the second stage, complex EOF analysis is applied to the fluctuating field to decompose the spatiotemporal data into orthogonal temporal modes.

Following Donoho [17], the two-stage approach can be used to reconstruct an unknown function *f* from noisy data \(d_i = f(t_i ) + \sigma z_i ,\quad i{\rm{ = 0,}} \ldots {\rm{,}}n - {\rm{1,}}\) where \(d_i\) is the observed data point (the noise-contaminated measurement point), \(z_i\) is a standard Gaussian white noise, \(t_i\) are equispaced data points, and \(\sigma\)is a noise level. The first stage is to find the estimates \(\hat f(.) = T(y,dy)\,(.)\), where \(T(y,dy)\,(.)\) is a suitable reconstruction formula with spatial smoothing parameters \(\delta\) and \(dy\) is data-adaptive choice of the spatial smoothing parameter. The separating procedure is carried out using the wavelet shrinkage method because of its ability to capture fast changes in the data.

- 1.
Expand the data \(u(x_j ,t_k )\) into wavelet series. Using the wavelet decomposition structure, estimate level-dependent thresholds for signal compression using suitable thresholding approach

- 2.
Obtain a denoised compressed function, \(u_{\rm{m}}\), representing the instantaneous mean, via the wavelet shrinkage, and

- 3.
Compute the fluctuation of the signal by calculating \(\bar u(x_j ,t_k ) =u(x_j ,t_k ) - u_{\rm{m}} (x_j ,t_k ),\) where \(u_{\rm{m}}\) is the time-varying mean of the signal. Among the existing techniques, we use the Birge–Massart strategy in [18].

The algorithm is simple to implement and the computational requirements are small.

### 6.5.2 Spatiotemporal Analysis of Measured Data

Based on the analytical procedure outlined in Section 6.5.1, the complex EOF method was used to determined dynamic trends and to analyze phase relationships. In this procedure, the measured data are augmented with an imaginary component defined by its Hilbert transform, and the temporal patterns are extracted by using the procedure in Section 6.4.

Complex EOF analysis was performed on the original time series, and the nonlinearly detrended (wavelet shrinkage model) time series. The analysis presented here uses the Daubechies wavelet with a fixed decomposition level. A level-dependent threshold is then obtained using a wavelet coefficients selection rule based on the Birge–Massart strategy [18].

Complex EOF analysis of the synchronized measurements of frequency in Fig. 6.6 shows that wide-area system dynamics is well represented by three modes; the two leading modes together account for 96.5% of the total energy. Individually, these modes account for 72, 24.5, and 3.5% of the energy (see Fig. 6.6 caption for details).

On the basis of these results, we conducted detailed analysis aiming at disclosing hidden information in the data. For clarity of exposition, the analysis of temporal and spatial patterns will be presented separately.

### 6.5.3 Temporal Properties

These functions display a number of interesting features. As discussed below, the analysis identifies two periods of interest; a transient period associated with the interconnection of the systems (06:27:42–06:28:21) and a nearly stationary period in which the frequency of the interconnected system is restored to its normal value (06:28:54–06:29:39).

The first interval manifests particularly strong temporal activity as can be seen in Fig. 6.8a. In interpreting these results, we remark that break or changes in the temporal functions may signal different physical regimes or control actions.

An examination of the temporal phase in Fig. 6.8b, on the other hand, reveals a nonstationary behavior in which the phase (frequency) content changes with time. Here, the slope of the spatial phase function represents the instantaneous frequency. The slowly increasing trends indicate periods of essentially constant frequencies.

### 6.5.4 Frequency Determination from Instantaneous Phases

Additional insight into the frequency variability of the observed oscillations can be obtained from the analysis of instantaneous frequencies. Recognizing that the instantaneous frequency is the time derivative of the temporal phase function, \(\theta\), the instantaneous frequencies can be estimated from (6.55) for each mode of concern.

Nonstationary features are evident in this plot. Analysis of these plots shows two modal components: a 0.27 Hz component associated with the steady-state behavior of the system, and a 0.64 Hz component associated with the transient system fluctuation following the system interconnection. The 0.27 Hz component captures the slow ambient swings previous to the onset of system oscillations and the steady behavior of the system. The results are consistent with those based on nonstationary analysis of the observed oscillations giving validity to the results.

### 6.5.5 Mode Shape Estimation

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 and control actions. Changes in mode shape may indicate changes in topology or changes in load/generation and may be useful for control decisions and the design of special protection systems. This is a problem that has been recently addressed using spectral correlation analysis [19].

Using the spatial phase and amplitude, the phase relationship between key system locations (the mode shapes) can be determined. In this analysis, we display the complex value as a vector with the length of its arrow proportional to eigenvector magnitude and direction equal to the eigenvector phase.

These results are in general agreement with previously published results based on real EOF analysis and Prony results [4]. The new results, however, provide clarification on the exact phase relationships between key system measurements as a function of time.

### 6.5.6 Energy Distribution

In the previous section it was shown that a linear combination of an individual eigenmode can accurately reconstruct the temporal behavior of simultaneous measurements at different geographical (spatial) locations. A key related question of interest is that of finding a small number of measurements that will provide a good estimation of the entire field of interest.

*x*-axis shows spatial sensor location and the

*y*-axis shows the energy value. From Fig. 6.11, it is evident that modes 1 and 2 are quite prominent at the Mazatlan Dos and Hermosillo substations while mode 3 is more strongly evident at the Tres Estrellas substation. This is consistent with conventional analysis (not shown). However, the proposed approach provides an automated way to estimate mode shapes without any prior information of the time intervals of interest.

## 6.6 Concluding Remarks and Directions for Future Research

In this chapter, a new method of temporal representation of nonstationary processes in power systems has been presented. Complex empirical orthogonal function analysis provides an efficient and accurate way of looking at the temporal variability of transient processes while at the same time providing spatial information about each one of the dominant modes with no a priori assumption on either spatial or temporal behavior. The main advantage of this approach is its ability to compress the variability of large data sets into the fewest possible number of temporal modes.

Complex empirical orthogonal function analysis is shown to be a useful method for identifying standing and traveling patterns in wide-area system measurements. Using wide-area frequency information, spatiotemporal analysis of time-synchronized measurements shows that transient oscillations may manifest highly complex phenomena, including nonstationary behavior. Numerical results show that the proposed method can provide 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. The identified system modes from the decomposition may also serve to reveal relevant, but unexpected structure hidden in the data such as that resulting from short-lived transient episodes. Other issues such as the effect of numerical approximations on modal estimates will be investigated in future research.

A generalization of this theory is also needed to treat statistical data from an ensemble of nonstationary oscillations. This is an aspect that warrants further investigation. Finally, the generalization of the proposed technique to determine the most suitable locations for phasor measurement devices and the analysis of modal coherency are topics worthy of further investigation.

## Footnotes

- 1.
Given a function to maximize, \(f(P)\) subject to the constraints \(g(P) = 0\), the Lagrange function can be defined as \(F(P,\lambda ) = f(P) - \lambda g(P)\).

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