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Complex Empirical Orthogonal Function Analysis of Power System Oscillatory Dynamics

  • P. Esquivel
  • E. Barocio
  • M.A. Andrade
  • F. Lezama
Chapter
Part of the Power Electronics and Power Systems book series (PEPS)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

The proper orthogonal decomposition (POD) method is an optimal technique of finding a basis that spans an ensemble of data, collected from an experiment or numerical simulation [1, 2, 3, 4]. More precisely, assume that \(u(x_j ,t_k )\), 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
$$u_{\rm{m}} ({\bf{x}}) = \left\langle {u\left( {{\bf{x}},t_k } \right)} \right\rangle = \frac{1}{N}\sum\limits_{k = 1}^N {u\left( {{\bf{x}},t_k } \right)}$$
is assumed to be zero [3]. Generalizations to this approach are discussed below.
The POD procedure determines EOFs, \(\varphi _i (x)\), \(i = 1, \ldots ,\infty\), such that the projection onto the first \(p\) EOFs (a low-order representation)
$$\hat u(x_j ,t_k ) = \sum\limits_{i = 1}^p {a_i (t)\varphi _i (x)} ,\;j = 1, \ldots ,n,\;k = 1, \ldots ,N$$
(6.1)
is optimal in the sense that the average least-squares truncation error, \(\varepsilon _j\)
$$\varepsilon _j = \left\langle {\left\| {u(x_j ,t_k ) - \sum\limits_{i = 1}^p {a_i (t)\varphi _i (x)} } \right\|^2 } \right\rangle {\rm{,}}\;{\rm{ }}p \le N$$
(6.2)
is minimized for any \(p \le N\), where \(\left\langle . \right\rangle\) denotes the ensemble average, \(\left\| \,f\, \right\| = \left\langle\, {f,f} \,\right\rangle ^{1/2}\), and \(\left\| . \right\|\) denotes the \(L^2\)norm over \(\Omega\). The \(a_i^{}\)’s are time-dependent coefficients of the decomposition to be determined so that (6.1) results in a maximum for (6.2). These special orthogonal functions are called the proper orthogonal modes (POMs) of the reduced basis for \(u(x_j ,t_k )\).
Following [1], assume that the field is decomposed into a mean value \(u_{\rm{m}} \left( {x_j ,t} \right)\) and a fluctuating part \(\bar u(x,t_j )\)
$$u(x_j ,t_k ) = u_{\rm{m}} (x_j ,t_k ) + \bar u(x_j ,t_k )$$
(6.3)
More formally, let \(L^2\) denote the space of square integrable functions. It then follows that, a normalized basis function \(\varphi\) is optimal if the average projection of \(u\)onto \(\varphi\) is maximized, i.e., [2]
$$\mathop {\max }\limits_{\varphi \in L^2 \left({[0,1]} \right)} \left\langle {|(\bar u(x_j ,t_k ),\varphi )|^2 }\right\rangle \;\hbox{subject to}\;\left\| \varphi \right\|^2 =1{\rm{ }}$$
(6.4)
where the inner product is defined as \(\left\langle {U,V} \right\rangle = \ _{k = 0}^p U_k V_k^\ast = V^H U\), and
$$\left\| \varphi \right\|^2 = \left\langle {\varphi ,\varphi } \right\rangle = \varphi ^{\rm{T}} \varphi = \sum\limits_{j = 1}^n {\varphi _j^2 }$$
The optimization problem can be recast in the form of a functional for the constrained variational problem [3]1
$$J[\varphi ] = \left\langle {\left| {(\bar u\left( {x_j ,t_k } \right),\varphi )} \right|^2 } \right\rangle - \lambda \left( {\left\| \varphi \right\|^2 - 1} \right)$$
(6.5)
A necessary condition for the extrema is that the Gateaux derivative vanishes for all variations \(\varphi + \delta \varPsi \in L^2 \left( {[0,1]} \right)\), \(\delta \in \Re\). This can be expressed as
$$\left. {\frac{{{\rm{d}}J}}{{{\rm{d}}\delta }}\left[ {\varphi + \delta \varPsi } \right]} \right|_{\delta = 0} = 0,\quad \forall \varPsi \in L^2 (\Omega )$$
(6.6)
Consider now the Hilbert space of all pairs \(\left\langle {\,f,g} \right\rangle\) where \(f\)and \(g\)are functions of \(L^2 [0,1]\), i.e., square integrable functions of the space variable \(x\) on the interval \([0,1]\), where \(\left\langle {.,.} \right\rangle\) denotes the standard inner product on \(L^2\) defined by
$$\left\langle \,{f,g} \right\rangle = \int_0^1 {f(x)g^\ast (x)} \,{\rm{d}}x$$
(6.7)
and
$$\left\| {\varphi ^2 } \right\| = \left\langle {\varphi ,\varphi } \right\rangle = \int_\Omega ^{} {\varphi ^2 \,{\rm{d}}x}$$
(6.8)
where \(\Omega\) is the domain of interest over which \(u(x)\) are \(\varphi\) are defined and the asterisk * denotes the conjugate transpose.
It follows immediately from (6.6) that
$$\begin{aligned} &{{\frac{{\rm d}J} {{\rm d}\delta}\left. {[\varphi + \delta \varPsi ]} \right|_{\delta = 0} = 0 = \left. {\frac{{\rm d}J} {{\rm d}\delta}\left[ {\left\langle {\left( {\bar u,\varphi + \delta \varPsi } \right)\left( {\varphi + \delta \varPsi ,\bar u} \right) - \lambda \left( {\varphi + \delta \varPsi ,\varphi + \delta \varPsi } \right)} \right\rangle } \right]} \right|_{\delta = 0}}}\\&{\hskip86pt = 2{\mathop{\rm Re}\nolimits} \left[ {\left\langle {\left( {\bar u,\varPsi } \right)\left( {\varphi ,\bar u} \right)} \right\rangle - \lambda \left( {\varphi ,\varPsi } \right)} \right]} \end{aligned}$$
(6.9)
where use has been made of the inner product properties.
Noting that
$$ {\begin{aligned}& \left\langle {\left( {\bar u,\varPsi } \right)\left( {\varphi ,\bar u} \right)} \right\rangle - \lambda \left( {\varphi ,\varPsi } \right) = \left\langle {\int_0^1 {\bar u(x)\varPsi ^\ast (x)\,{\rm{d}}x} \int_0^1 {\varphi (x^{\prime})u^\ast (x^{\prime})\,{\rm{d}}x^{\prime}} } \right\rangle - \lambda \int_0^1 {\varphi (x)\varPsi ^\ast (x)\,{\rm{d}}x} \\& \hskip 100pt = \int_0^1 {\left[ {\int_0^1 {\left\langle {u(x)u^\ast (x^{\prime})} \right\rangle \varphi (x^{\prime})\,{\rm{d}}x^{\prime}} - \lambda \varphi (x)} \right]} \varPsi ^\ast (x)\,{\rm{d}}x = 0 \end{aligned}}$$
the condition for the extrema reduces to
$$\int_\Omega {\left\langle {\bar u(x)\bar u^\ast (x^\prime)} \right\rangle } \varphi (x^{\prime})\,{\rm{d}}x^{\prime} = \lambda \varphi (x)$$
(6.10)

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.

Defining
$${\bf{R}}\varphi = \int_\Omega {\left\langle {\bar u(x,t)\bar u^\ast (x^{\prime},t)} \right\rangle } \varphi (x^{\prime})\,{\rm{d}}x^{\prime}$$
(6.11)
where
$$R(x,x^{\prime}) = \frac{1}{N}\sum\limits_{k = 1}^N {\bar u(x,t_k )\bar u(x^{\prime},t_k )}$$
(6.12)
the problem of minimizing (6.9) becomes that of finding the largest eigenvalue of the eigenvalue problem \({\bf{R}}\varphi = \lambda \varphi\), subject to \(\left\| \varphi \right\|^2 = 1\).
In practice, the observations that form the data are only available at discrete spatial grid points herein called snapshots. In this case, the kernel \({\bf{R}}(x,x^{\prime})\) can be written as [5, 6]
$${\bf{R}}{\rm{(}}{\bf{x}}{\rm{,}}{\bf{x}}^{\prime}{\rm{)}}={\begin{bmatrix}{R(x_1 ,x_1 )} & \cdots & {R(x_1 ,x_n )} \\\vdots & \ddots & \vdots \\{R(x_n ,x_1 )} & \cdots & {R(x_n ,x_n )} \\\end{bmatrix}}$$
where \(n\) indicates that the number of measurement points, and
$$R(x_i ,x_j ) = \frac{1}{N}\sum\limits_{k = 1}^N{\bar u(x_i ,t_k )\bar u(x_j ,t_k )} ,\;i,j =1,\ldots,n$$
(6.13)

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

Time series are usually recorded in discrete form even though the underlying process itself is continuous. For discretely sampled measured data, the integral time average can be approximated by a sum over the set of sampled data points [1]. In this case, the vectors
$${\bf{x}}_j = {\bar {\bf u}}_j (x_j ,t_k ) = \left[ {\bar u(x_j ,t_1 )}\quad {\bar u(x_j ,t_2 )}\quad \cdots \quad {\bar u(x_j ,t_N )} \right]^{\rm{T}} ,\;j = 1,...,n$$
(6.14)
represent a set of snapshots obtained from the observed data at n locations.
The set of data can then be written as the N × n-dimension ensemble matrix, \({\bf{X}}\) [6]
$${\bf{X}} = \left[{{\bf{x}}_{\rm{1}} }\quad \cdots\quad {{\bf{x}}_n }\right] ={\begin{bmatrix}{\bar u(x_1 ,t_1 )} & \cdots & {\bar u(x_n ,t_1 )} \\\vdots & \ddots & \vdots \\{\bar u(x_1 ,t_N )} & \cdots & {\bar u(x_n ,t_N )} \\\end{bmatrix}}$$
(6.15)
where each column corresponds to the response at a specific time.
Typically, \(n \ne N\), so \({\bf{X}}\) is generally rectangular. Under these assumptions, the actual integral (6.10) can be written as \({\bf{C}}\varphi = \lambda \varphi\), where \(C_{ij} = {1 \mathord{\left/ {\vphantom {{N\sum\nolimits_{k = 1}^N {\bar u(x_i ,t_k )\bar u(x_j ,t_k )} }}} \right. \kern-\nulldelimiterspace} }\). Assuming the EOFs to be of the form \(\varphi _i = \sum\nolimits_{l = 1}^N {w_l^i {\bf{x}}_i }\) where \(w_l^i\) is a coefficient to be determined, the problem of minimizing (6.2) can be recast as the problem of finding the largest eigenvalue of the linear equation
$${\bf{C}}\varphi = \lambda \varphi$$
(6.16)
where \({\bf{C}}\) is the autocorrelation (covariance) matrix defined as
$$ {{\bf{C}} = \frac{1}{N}{\bf{X}}^{\rm{T}} {\bf{X}} =\frac{1}{N}{\begin{bmatrix}{x_{\rm{1}}^{\rm{T}} x_1 } & {x_{\rm{1}}^{\rm{T}} x_2 } & \cdots & {x_{\rm{1}}^{\rm{T}} x_n } \\{x_{\rm{2}}^{\rm{T}} x_1 } & {x_{\rm{2}}^{\rm{T}} x_2 } & \cdots & {x_{\rm{2}}^{\rm{T}} x_n } \\\vdots & \vdots & \ddots & \vdots \\{x_n^{\rm{T}} x_1 } & {x_n^{\rm{T}} x_2 } & \cdots & {x_n^{\rm{T}} x_n } \\\end{bmatrix}} = \frac{1}{N}\sum\limits_{i = 1}^n {\left( {x_i - u_m } \right)^{\rm{T}} \left( {x_i - u_m } \right)}}$$
(6.17)
The resulting covariance matrix \({\bf{C}}\) is a real, symmetric \((C_{ij} = C_{ji} )\) positive, and semi-definite matrix. Consequently, it possesses a set of orthogonal eigenvectors \(\varphi _i ,i = 1,...,n\), i.e.,
$$\varphi _i^{\rm{T}} \varphi _j = {\begin{cases}{\delta _{ij},\,i = j} \\{0,\,i \ne j} \\\end{cases}}$$
Using standard linear algebra techniques, the covariance matrix can be expressed in the form
$${\bf{C}} = {\bf{U \Lambda V}}^{{T}}$$
(6.18)
where \({\bf{U}}\) and \({\bf{V}}\) are the matrices of right and left eigenvectors and \({ \Lambda } = {\rm{diag}}[\lambda _1 \quad \lambda _2 \quad \cdots\quad \lambda _n]\).

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

The method of snapshots is based on the fact that the data vectors \({\bf{u}}_i\) and the POD modes span the same linear space [6]. In this approach, we choose the eigenfunctions \(\varphi\) to be a linear combination of the snapshots:
$$\varphi _i = \sum\limits_{l = 1}^N {w_l^i {\rm{x}}_i }$$
(6.19)
where the coefficients \(w_l^i\) are to be determined such that \({\bf{u}}\) maximizes
$$\mathop {\max }\limits_\varphi \;\frac{1}{N}\sum\limits_{j = 1}^n {\frac{{\left| {x_j ,\varphi } \right|}}{{\left\langle {\varphi ,\varphi } \right\rangle }}}$$
(6.20)
These \(l\) functions are assembled into an n × N matrix, \({\bf{\boldsymbol{\Phi} }}\), known as the modal matrix. In matrix form Eq. (6.19) becomes
$${\bf{\boldsymbol{\Phi} }} = {\bf{XW}}$$
(6.21)
where
$${{\boldsymbol{\Phi}} = {\begin{bmatrix}\uparrow & \uparrow & \uparrow & \uparrow \\{\varphi _1 } & {\varphi _2 } & \cdots & {\varphi _n } \\\downarrow & \downarrow & \downarrow & \downarrow \\\end{bmatrix}},\ {\bf{X}} = {\begin{bmatrix}\uparrow & \uparrow & \uparrow & \uparrow \\{x_1 } & {x_2 } & \cdots & {x_n } \\\downarrow & \downarrow & \downarrow & \downarrow \\\end{bmatrix}},{\bf{W}} = {\begin{bmatrix}\uparrow & \uparrow & \uparrow & \uparrow \\{w_1 } & {w_2 } & \cdots & {w_n } \\\downarrow & \downarrow & \downarrow & \downarrow \\\end{bmatrix}}}$$
in which
$${\bf w}_1 = {\begin{bmatrix}{w_1^1 } \\{w_2^1 } \\\vdots \\{w_N^1 } \\\end{bmatrix}},\ {\bf w}_2 = {\begin{bmatrix}{w_1^2 } \\{w_2^2 } \\\vdots \\{w_N^2 } \\\end{bmatrix}} , \ldots ,\ {\bf w}_n = {\begin{bmatrix}{w_1^n } \\{w_2^n } \\\vdots \\{w_N^n } \\\end{bmatrix}}$$
Substituting the expression (6.19) into the eigenvalue problem (6.16) gives
$${\bf{C}}\sum\limits_{l = 1}^N {w_l^i {\rm{x}}_i = } \lambda \sum\limits_{l = 1}^N {w_l^i {\rm{x}}_i }$$
(6.22)
where \(\,\,C_{ij} = (1/N)\,(\bar u_i ,\bar u_j )\). This can be written as the eigenvalue problem of dimension n
$${\bf{CW}} = {\boldsymbol{\Lambda}} {\bf{W}}$$
(6.23)
where
$${\bf{w}} = \left[{w_1 } \quad {w_2 }\quad \cdots \quad {w_n }\right]$$
and \({\bf{\Lambda }}\) is a diagonal matrix storing the eigenvalues \(\lambda _i\) of the covariance matrix 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 \(\left\{{\varphi _1 }\ {\varphi _2 }\ \cdots \ {\varphi _n }\right\}\).
Once the modes are found using these equations, the flow field can be reconstructed using a linear combination of the modes
$$\hat u_k (x_j ,t_k ) = \sum\limits_{i = 1}^\infty {a_i (t)\,\varphi _i (x)}$$
(6.24)
for some \(a_i (t) \in \Re\), where the \(a_i (t)\) are the time-varying amplitudes of the POD modes \(\varphi _i (x)\).
The truncated POD of \({\bf{\bar u}}\)is
$$\hat u_k (x_j ,t_k ) = \sum\limits_{i = 1}^p {a_i (t)\,\varphi _i (x)} + \varepsilon$$
(6.25)
where 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.

The temporal coefficients are then expressed as
$$a_i = {{\left\langle {{\bf{x}},\varphi _i } \right\rangle } \mathord{\left/ \right. \kern-\nulldelimiterspace} {\left\langle {\varphi _i ,\varphi _i } \right\rangle }}$$
(6.26)
Note that the temporal modes are uncorrelated in time, i.e., \((a_i (t),a_j (t)) = \delta _{ij} \lambda _j ,\) where \(\delta _{ij} = 1\;{\rm{for}}\;i = j,\;0\;{\rm{else}}\), and that the system (6.25) is optimal in the sense that minimizes the error functions
$$\varepsilon (\varphi ) = \sum\limits_{l = 1}^p {\left\| {\bar u(x_j ,t_k ) - \sum\limits_{i = 1}^p {a_i (t)\varphi _i (x)} } \right\|}$$
(6.27)

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

The use of the POD method leads naturally to a discussion of truncation criteria. Several techniques to derive truncated expressions have been proposed in the literature. Here, we choose to reduce the residual terms, \(R = \varepsilon\), such that the mean square value
$$R = O\,\,\left( {\frac{{\lambda _{p + 1} }}{{\sum\nolimits_{i = 1}^n {\lambda _i } }}} \right)$$
(6.28)
be as small as possible.
Among all linear decompositions, the most kinetic energy possible for a projection onto a given number of modes. Defining the total energy E, by \(E = {\rm{Trace}}\,[{\bf{R}}({\bf{x}}_i ,{\bf{x}}_m )],\) one obtains
$$E = \sum\limits_{i = 1}^p {\lambda _i }$$
(6.29)
The associated percentage of total energy contributed by each mode can then be expressed as
$$E_i = \frac{{\lambda _i }}{{\sum\limits_{i = 1}^n {\lambda _i } }}$$
(6.30)
Typically, the order 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
$$\frac{{\sum\nolimits_{i = 1}^p {\lambda _i } }}{{\sum\nolimits_{j = 1}^n {\lambda _j } }} = 99\%$$
(6.31)
for the smallest integer 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

Let \({\bf{A}}\) be a real m × n matrix. The SVD theorem states that \({\bf{A}}\) can be decomposed into the following form [7]:
$${\bf{A}} = {\bf{U}} \boldsymbol{\Sigma} {\bf{V}}^{\rm{T}}$$
(6.32)
where \({\bf{U}} = {\rm{col}}[{{\bf{u}}_1 }\quad {{\bf{u}}_2 } \quad \cdots \quad {{\bf{u}}_m }]\) is an 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 \(\,{\bf{A}}\).
Matrix \(\boldsymbol{\Sigma}\) has the form
$${\boldsymbol{\Sigma}} = {\begin{bmatrix}{\sigma _1 } & 0 & \cdots & 0 & 0 & \cdots & 0 \\0 & {\sigma _2 } & {} & {} & 0 & \cdots & 0 \\\vdots & \vdots & \ddots & {} & \vdots & \ddots & \vdots \\0 & 0 & \cdots & {\sigma _m } & 0 & \cdots & 0 \\\end{bmatrix}} \quad {\rm{for}}\;m<n$$
or
$${\boldsymbol{\Sigma}} = {\begin{bmatrix}{\sigma _1 } & 0 & \cdots & 0 \\0 & {\sigma _2 } & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & {\sigma _m } \\0 & 0 & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \\\end{bmatrix}} \quad {\rm{for}}\;m>n$$

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 \(\sigma _1 \ge \sigma _2 \ge \cdots \ge \sigma _m \ge 0\).

Equivalently, we can express the model as
$${\bf{A}} = \left[{u_1 } \quad \cdots \quad {u_k }\ \vert \ {u_{k + 1} } \quad \cdots \quad {u_m}\right]\left[ {\begin{array}{ccccc}{\sigma _1 } & \cdots & 0 &\vline & 0 \\\vdots & \ddots & \vdots &\vline & \vdots \\0 & \cdots & {\sigma _k } &\vline & 0 \\\hline0 & \cdots & \cdots &\vline & 0 \\\end{array}} \right]\left[ {\begin{array}{c}{v_1^{{T}} } \\\vdots \\{v_k^{{T}} } \\\hline{v_{k + 1}^{{T}} } \\\vdots \\{v_n^{{T}} } \\\end{array}} \right]$$
(6.33)
or
$${\bf A} = \left[ {{\begin{array}{c}{\bf U} \vert {^ \bot {\bf U}} \end{array}}} \right]\,\left[ {{\begin{array}{ccc}\frac{{\bf D} \ {\bf 0}} {{\bf 0}\,\,\, {\bf 0}} \\\end{array}}} \right]\hskip -12.5pt\big\vert\quad\left[ {{\begin{array}{c}\frac{\bf V} {{^\bot {\bf V}}} \end{array}}} \right]$$
where \({\bf{D}}\) is the diagonal matrix of nonzero singular values and \({\bf{U}}{}\) and \({\bf{V}}\) are the matrices of left and right singular vectors, respectively, corresponding to the nonzero singular values.
It is clear that only the first r of the u’s and v’s make any contribution to \({\bf{A}}\), and can be expressed as an outer product expansion
$${\bf{A}} = \sum\limits_{i = 1}^r {\sigma _i } \left( {{\bf{u}}_i {\bf{v}}_i^{\rm{T}} } \right)$$
(6.34)
where the vectors \({\bf{u}}_i\) and \({\bf{v}}_i\) are the columns of the orthogonal matrices \({\bf{U}}\) and \({\bf{V}}\), respectively. Techniques to compute the POMs based on SVD are next discussed.

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 \({\bf{X}}\).

Using the notation in Section 6.1.2 let the response matrix be given by
$${\bf{X}} = {\begin{bmatrix}{\bar u(x_1 ,t_1 )} & \cdots & {\bar u(x_n ,t_1 )} \\\vdots & \ddots & \vdots \\{\bar u(x_1 ,t_N )} & \cdots & {\bar u(x_n ,t_N )} \\\end{bmatrix}}$$
(6.35)
The SVD of the response matrix \({\bf{X}}\) may be written in compact form as
$${\textbf{X}} = {\textbf{U}}\boldsymbol{\Sigma}{\textbf{V}}^{\rm{T}}$$
(6.36)
where \({\bf{U}}{}\) is an orthonormal 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.
In terms of the notation above for SVD, it can be seen directly from (6.18) that the correlation matrix defined previously is given by
$${\bf{XX}}^{\rm{T}} = \left( {{\bf{U\boldsymbol{\Sigma} V}}} \right)\left( {{\bf{U\boldsymbol{\Sigma} V}}} \right)^{\rm{T}} = {\bf{U\boldsymbol{\Sigma} }}^{\bf{2}} {\bf{U}}^{\rm{T}}$$
(6.37)
and
$${\bf{X}}^{\rm{T}} {\bf{X}} = \left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}} \right)^{\rm{T}} \left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}} \right) = {\bf{V \boldsymbol{\Sigma}}}^2 {\bf{V}}^{\rm{T}}$$
(6.38)
Hence, (6.32) becomes
$${\bf{XX}}^{\rm{T}} {\bf{U}} = {\bf{U}}{\begin{bmatrix}{\lambda _1 } & {} & {} & {} \\{} & {\lambda _2 } & {} & {} \\{} & {} & \ddots & {} \\{} & {} & {} & {\lambda _n } \\\end{bmatrix}} = {\bf{U}}{\begin{bmatrix}{\sigma _1^2 } & {} & {} & {} \\{} & {\sigma _2^2 } & {} & {} \\{} & {} & \ddots & {} \\{} & {} & {} & {\sigma _n^2 } \\\end{bmatrix}}$$
(6.39)
It follows immediately from (6.32), (6.33), (6.34), the singular values of \({\bf{X}}\) are the square roots of the eigenvalues of \({\bf{XX}}^{\rm{T}}\) or \({\bf{X}}^{\rm{T}} {\bf{X}}\) [2, 8]. In addition, the left and right eigenvectors of \({\bf{X}}\) are the eigenvectors of \({\bf{XX}}^{\rm{T}}\) and \({\bf{X}}^{\rm{T}} {\bf{X}}\), respectively. Also of interest, the trace of (6.39) is given by
$${\bf{X}}^{\rm{T}} {\bf{X}} = \sum\limits_{i = 1}^r {\sigma _i^2 }$$

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

In this section, a step-by-step description of the algorithm used to extract modal information is presented. The procedure adopted to compute the POMs can be summarized as follows:
  1. 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. 2.

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

     
  3. 3.

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

     
Figure 6.1 illustrates the proposed algorithm.
Fig. 6.1

Conceptual view of the proposed algorithm

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.

Let \(u(x_j ,t_k )\) be a space–time scalar field representing a time series, where \(x_j ,\;j = 1, \ldots ,n\) is a set of spatial variables on a space \(\Omega _k\) and \(t_k ,\;k = 1, \ldots ,N\) is the time at which the observations are made. Provided \(u\) is simple and square integrable, it has a Fourier representation of the form [9]
$$u(x_j ,t_k ) = \sum\limits_{m = 1}^\infty {[a_{j(m)} (\omega )\cos (m\omega t_k ) + b_{j(m)} (\omega )\sin (m\omega t_k )]\;}$$
(6.40)
where \(a_{j(m)} (\omega )\) and \(b_{j(m)} (\omega )\)are the Fourier coefficients defined as
$$\begin{aligned}a_{j(m)} & = \frac{1}{{\rm{\pi }}}\int\limits_{ - {\rm{\pi }}}^{\rm{\pi }} {u(x_j ,t_k )\cos \left( {m\omega t_k } \right)\,{\rm{d}}\omega } \\b_{j(m)} &= \frac{1}{{\rm{\pi }}}\int\limits_{ - {\rm{\pi }}}^{\rm{\pi }} {u(x_j ,t_k )\sin \left( {m\omega t_k } \right)\,{\rm{d}}\omega }\end{aligned}$$
This allows description of traveling waves. Equation (6.40) can be rewritten in the form
$$u_c (x_j ,t_k ) = \sum\limits_{m = 1}^\infty {c_{j(m)} (\omega ){\rm{e}}^{ - {\rm{i}}m\omega t_k } }$$
(6.41)
where \(c_{j(m)} (\omega ) = a_{j(m)} (\omega ) + ib_{j(m)} (\omega )\) [1] and \({\rm{i}} = \sqrt { - 1}\). Expanding (6.41) and collecting terms gives
$$\begin{aligned} U_c (x_j ,t_k ) &= \sum\limits_{m = 1}^\infty {\left\{ {\left[ {a_{j(m)} (\omega )\cos (m\omega t_k ) + b_{j(m)} (\omega )\sin (m\omega t_k )} \right]} \right\}} \\& + i\sum\limits_{m = 1}^\infty {\left\{ {\left[ {b_{j(m)} (\omega )\cos (m\omega t_k ) - a_{j(m)} (\omega )\sin (m\omega t_k )} \right]} \right\}} \\&= u(x_j ,t_k ) + {\rm{i}}u_H (x_j ,t_k ) \end{aligned}$$
(6.42)
where the real part of \(U\) is given by (6.40) and the imaginary part is the Hilbert transform of \(u(x_j ,t_k )\)[10]
$$u_H (x_j ,t_k ) = - \frac{1}{{\rm{\pi }}}\int\limits_{ - \infty }^\infty {\frac{{h\left[ {u\left( {x_j ,t_k } \right)} \right]}}{{t_k - x_j }}\,{\rm{d}}x}$$
(6.43)

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 the proposed formulation \(u\) can be estimated more efficiently by performing a filtering operation on \(u\) itself. Equation (6.43) can be rewritten in the form of a convolution as
$$u_H (x_j ,t_k ) = \sum\limits_{\ell = - L}^L {u(x_j ,t_k - \ell )h(\ell ),\quad L = \infty }$$
(6.44)
where \(h\) sis a convolution filter with unit amplitude response and 90° phase shift.
In practice, a simple filter that has the desired properties of approximate unit amplitude response and π/2 phase shift is given by [11]
$$h(l) = \begin{cases} \frac{2}{{{\rm{\pi }}l}}\sin ^2 \left( {\frac{{{\rm{\pi }}l}}{2}} \right),& l \ne 0 \\0,& l = 0 \\\end{cases}$$
(6.45)
where \(- L \le l \le L\). As \(L \to \infty\), Equation (6.45) yields an exact Hilbert transform.

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

6.4.1 Complex EOF Analysis

Drawing on the above approach, an efficient formulation to compute complex time-dependent POMs has been derived. Following Susanto et al. [12] assume that \({\bf{X}}\) is a j (spatial points) by k (temporal points) ensemble matrix. From the preceding results, it follows that
$${\bf{X}} = {\bf{U}}\boldsymbol{\Sigma} {\bf{V}}^H\;$$
(6.46)
where \({\bf{V}}^H\) is the conjugate transpose of \({\bf{V}}\), the superscript H denotes a Hermitian matrix, and we assume that
$$\begin{aligned}{\bf{U}}^H {\bf{U}} &= {\bf{I}} \\{\bf{V}}^H {\bf{V}} &= {\bf{I}} \\\end{aligned}$$
Now, it can be easily verified that
$$\begin{aligned}{\bf{XX}}^H &= \left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}^H} \right)\left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}^H} \right)^{H} = {\bf{U}}\boldsymbol{\Sigma} \boldsymbol{\Sigma}^{{\rm T}} {\bf{U}}^H \\{\bf{X}}^H {\bf{X}} & = \left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}^H} \right)^H \left( {{\bf{U}}\boldsymbol{\Sigma} {\bf{V}}^H} \right) = {\bf{V}}\boldsymbol{\Sigma}^{\rm{T}} \boldsymbol{\Sigma} {\bf{V}}^H \\\end{aligned}$$
(6.47)
where \({{\boldsymbol{\Sigma} }}^{\rm{T}}\) denotes the transpose of \(\boldsymbol{\Sigma}\). As is apparent from Eq. (6.46), the columns of \({\bf{U}}\) are the eigenvectors of \({\bf{X}}\left[ {{\bf{X}}^H } \right]^{\rm{T}}\), and that the columns of \({\bf{V}}\) are the eigenvectors of \(\left[ {{\bf{X}}^H } \right]^{\rm{T}} {\bf{X}}\). The 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 \({\bf{X}}\).
Once the spatial eigenvectors are calculated, their corresponding time evolution is given by the time series \({\bf{A}}_i (t)\) which is obtained by projecting the time series \({\bf{X}}\) onto the proper eigenvector \(\varphi _i\), and summing over all locations:
$${\bf{A}}_i (t) = \sum\limits_{j = 1}^n {{\bf{{\rm X}}}(x_j ,t_k ){\boldsymbol{\upvarphi }}_i (x)}$$
(6.48)
The original complex data field, \({\bf{X}}(x,t)\), can be reconstructed by adding this product over all modes, i.e.,
$${\bf{X}}(x{\rm ,}t) = \sum\limits_{i = 1}^n {{\bf{{\rm A}}}_i (t){\boldsymbol{\upvarphi }}_i^{\, H} (x)}$$
(6.49)

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 ]
$${\bf{X}}(x,t) = \sum\limits_{i = 1}^n {{\bf R}_i(t)\angle \boldsymbol{\uptheta} _i\, {\bf S}_i (x)\angle \boldsymbol{\upphi} _i }$$
(6.50)
where \({{\textit{R}}}_i (t)\) is the complex temporal amplitude function, \({\textbf{\textit{S}}}_i (x_j )\) is the complex spatial mode or eigenvector, and \({\boldsymbol{\theta}}\), \({\boldsymbol{\phi}}\) are the phase functions corresponding to \({\textbf{\textit{R}}}_i (t)\) and \({\textbf{\textit{S}}}_i (x_j )\). These phase functions describe the propagation characteristics of the ith mode.
Equation (6.50) can be rewritten as
$$u(x_j ,t) = \sum\limits_{i = 1}^n {\big| {{\bf R}_i (t)} \big|\big| {{\bf S}_i (x )} \big|{\rm{e}}\,^{{\rm{j}}[ {\boldsymbol{\uptheta} _{_{R_i } }(t) + \boldsymbol{\upphi} _{S_i } (x)} ]} }$$
(6.51)

This effectively decomposes the data into spatial and temporal modes.

Four measures that define possible moving features in \(u({\bf{x}},t)\) can then be defined [9].
  1. 1.

    Spatial distribution of variability associated with each eigenmode

     
  2. 2.

    Relative phase of fluctuation

     
  3. 3.

    Temporal variability in magnitude

     
  4. 4.

    Variability of the phase of a particular oscillation

     

The following definitions introduce these concepts.

Definition 6.1

(Spatial amplitude function, \({\bf{S}}_i (x)\)) The spatial amplitude function, \({\bf{S}}_i (x)\), is defined as
$${\bf{S}}_i (x) = \sqrt {{\boldsymbol{\upvarphi}}_i^H (x){\boldsymbol{\upvarphi}}_i (x)}$$
(6.52)

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

Definition 6.2

(Spatial phase function, \({{\boldsymbol{\upvarphi}}}_i\)) This function shows the relative phase fluctuation among the various spatial locations where \(u\) is defined and is given by
$${{\boldsymbol{\upphi} }}_i (x) = \tan ^{ - 1}\left( {\frac{{{\mathop{\rm Im}\nolimits} \{ {\boldsymbol{\upvarphi}}_i (x)\} }}{{{\mathop{\rm Re}\nolimits} \{ {\boldsymbol{\upvarphi}}_i (x)\} }}} \right)$$
(6.53)

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

(Temporal amplitude function, \({\bf{R}}_i\)) Similar to the description of the spatial amplitude function in (6.52), the temporal amplitude function, \({\bf{R}}_i\) can be defined as
$${\bf{R}}_i (t) = \sqrt {{\bf{A}}_i^H (t){\bf{A}}_i(t)}$$
(6.54)

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

(Temporal phase function, \({\boldsymbol{\uptheta }}_i\)) This temporal variation of phase associated with \(u(x,t)\) is given by
$${\boldsymbol{\uptheta }}_i (t) = \tan ^{ - 1}\left( {\frac{{{\mathop{\rm Im}\nolimits} \{ {\bf{A}}_i (t)\}}}{{{\mathop{\rm Re}\nolimits} \{ {\bf{A}}_i (t)\} }}}\right)$$
(6.55)
For the simple case of a sinusoidal wave with fixed frequency and wave number, \(\theta _i (t) = \omega t\). In the more general (and interesting case), the space derivative of the phase and frequency of the modal components can be calculated from
$$\begin{aligned}{\textbf{\textit{k}}}_i &= {{{\rm{d}}({{\boldsymbol{\upphi} }}_i )}}/{{{\rm{d}}(x)}} \\{\boldsymbol{\upomega }}_i &= {{{\rm{d}}({\boldsymbol{\uptheta }}_i )}}/{{{\rm{d}}(t)}} \\ \textbf{\textit{c}}_i &= {{{\boldsymbol{\upomega }}_i }}/{{\textbf{\textit{k}}_i }} \\\end{aligned}$$
(6.56)
where \(\textbf{\textit{c}}_i\) is the phase velocity of the function.

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.

Figure 6.2 shows a geographical diagram of the MIS showing the PMU locations and the location of the initiating event. For demonstration purposes, three buses spread across the system are selected. To allow comparison with previous work, bus frequency signals are used in the analysis.
Fig. 6.2

Schematic of the MIS system showing the location of the observed oscillations. Measurement locations are indicated by shaded circles

Figure 6.3 is an extract from PMU measurements of this event showing the observed oscillations of selected bus frequencies. For the purpose of further comparison to EOF analysis, the relevant time interval of concern is zoomed in. System measurements in this plot demonstrate significant variability suggesting a nonstationary process in both space and time. The most prominent variations occur in the interval during which the oscillation start at 06:27:42 and the interval in which the operating frequency is restored to the nominal condition (60 Hz) by control actions (06:28:21).
Fig. 6.3

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

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

As a first step toward the development of a POD basis, the observed records are placed in a complex data matrix, \({\bf{X}}(x_j ,t_k )\), as
$${\bf{X}}(x_j ,t_k ) = \left[ {{\bf{f}}_H(t),{\bf{f}}_{{\rm{MZD}}} (t),{\bf{f}}_{{\rm{TTE}}} (t)} \right] ={\begin{bmatrix}{u(x_1 ,t_1 )} & \cdots & {u(x_n ,t_1 )} \\\vdots & \ddots & \vdots \\{u(x_1 ,t_N )} & \cdots & {u(x_n ,t_N )} \\\end{bmatrix}}$$
where \(j = 1,2,3\) is the spatial position index (grid location), tis 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.

A two-stage analysis technique based on wavelet shrinkage is proposed to determine the temporal properties of time-synchronized information. In the first step, the original system time histories are decomposed into their time-varying mean speeds and fluctuating speeds through wavelet shrinkage. More formally, the recorded time series are decomposed into their time-varying mean frequencies, \(u_m (x_j ,t_k ),\) and nonstationary fluctuating components, \(\bar u(x_j ,t_k ),\) as follows:
$$u(x_j ,t_k ) = u_m (x_j ,t_k ) - \bar u(x_j ,t_k)$$
(6.57)

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.

The adopted separation/identification procedure can be summarized as follows:
  1. 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. 2.

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

     
  3. 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].

As seen in Fig. 6.4 , this model very effectively describes the long-term behavior of the data while also capturing transient fluctuations. This, in turn, results in improved characterization of system behavior.
Fig. 6.4

Time-varying means and fluctuating components of the recorded bus frequency signals. The time series are smoothed using wavelet shrinkage

Spectral analysis results for the leading POM in Fig. 6.5 show that the main power is concentrated in oscillations with frequencies about 0.61, 0.50, and 0.27 Hz, which can be associated with major inter-area modes in the system [4]. Small peaks in Fig. 6.5 may indicate nonlinear interactions between frequency components. From Fig. 6.5 it is also seen that the time-varying instantaneous mean (bold gray line) acts as a high-pass filter. This approach can be used to separate the slow from the fast components as well as to improve the numerical accuracy of the method. This is discussed further in the next two subsections.
Fig. 6.5

Comparison of the Fourier transform spectrum of the original signal and the spectra constructed with the instantaneous mean removed

Figure 6.6 presents the corresponding eigenspectrum of these modes computed to capture 99.9% of the signal’s energy. In these plots, the horizontal axis shows the number of modes required to attain 99% of the averaged total energy; the vertical axis shows the energy in (6.13) captured by each POM. The data set being analyzed corresponds to bus frequency measurements observed with the PMUs at various geographical locations.
Fig. 6.6

Energy captured as a function of the number of modes. The percentage of energy located in the jth mode is measured by \(E = {{100\lambda _j } \mathord{\left/ {\vphantom {{100\lambda _j }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{i = 1}^n {\lambda _i } }}\)

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).

Figure 6.7 compares the reduced solution using the POD basis functions to the full solution obtained from the measured data for the Hermosillo signal in Fig. 6.4. As observed in this plot, using only three modes we are able to accurately approximate the measured data over the entire observation period. The agreement between the reduced order model and the observed behavior illustrates the high degree of accuracy that is possible with a simplified model.
Fig. 6.7

Reconstruction of the original data using the three leading POMs. Solid lines represent the original time series and dotted lines represent the composite oscillation obtained by adding the temporal modes

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

In order to reveal the hidden wave signatures in the time series, we examine both amplitude and temporal patterns in system behavior in the light of complex orthogonal function analysis. Figure 6.8 illustrates the temporal evolution (amplitude and phase) of the dominant mode.
Fig. 6.8

Temporal patterns of variability associated with the dominant mode

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.

The study focuses on POM 1 which is the mode that captures most of the variability in the signal. Figure 6.9 gives the instantaneous frequency of POM 1 for the interval of interest in this study. Also plotted, is the instantaneous mean frequency (nonlinear trend) determined using wavelet shrinkage analysis above.
Fig. 6.9

Instantaneous frequency of POM 1. Time interval 06:27:42–06:29:39

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.

Figure 6.10 shows the mode shape for the three intervals of interest above (06:27:42–06:28:06, 06:28:06–06:28:21, and 06:28:21–06:28:54 ) computed using the spatial function (6.50). It is interesting to note that the dominant POM mode shape changes with time. The effect is more pronounced for the time interval 06:28:06–06:28:21 in which several control actions take place in the system. This information may be useful to identify the dominant generators involved in the oscillations, and ultimately devise control mechanisms to damp the observed oscillations.
Fig. 6.10

Mode shape of POM 1 for various time intervals of interest

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.

Based on the decomposed EOFs, complex EOF analysis was used to determine the locations with the most energy. Figure 6.11 shows the participation of each location to the total energy of the record. The 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.
Fig. 6.11

Energy distribution

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. 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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Copyright information

© Springer Science+Business Media,LLC 2009

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringThe Center for Research and Advanced StudiesCinvestavMexico

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