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Publication# Characterization of Non-Stationary Signals in Electric Grids: a Functional Dictionary Approach

Abstract

With the expanding role of converter-interfaced distributed energy resources, modern power grids are evolving towards low-inertia networks that are increasingly vulnerable to extreme dynamics. Consequently, advanced signal processing techniques are needed to accurately characterize measured signals in power systems during non-stationary conditions. However, as advocated by recent literature, state-of-the-art phasor estimation methods are unable to sufficiently capture the broadband nature of these signal dynamics since they rely on a quasi-steady state, single tone model. Inspired by previous work by the authors, this paper proposes a signal processing method that uses a dictionary of kernels, modeling common signal dynamics, to compress time-domain information into a few coefficients. The identified signal model and the extracted coefficients capture the broadband spectrum of typical power system signal dynamics and allow for an improved reconstruction of the measured signal.

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Electrical grid

An electrical grid is an interconnected network for electricity delivery from producers to consumers. Electrical grids vary in size and can cover whole countries or continents. It consists of:

- po

Dictionary

A dictionary is a listing of lexemes from the lexicon of one or more specific languages, often arranged alphabetically (or by consonantal root for Semitic languages or radical and stroke for logo

Signal processing

Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, , potential fields, seismic signals, altimetry processing, and

The concept of stationarity is central to signal processing; it indeed guarantees that the deterministic spectral properties of linear time-invariant systems are also applicable to realizations of stationary random processes. In almost all practical settings, however, the signal at hand is just a finite-size vector whose underlying generating process, if we are willing to admit one, is unknown; In this case, invoking stationarity is tantamount to stating that a single linear system (however complex) suffices to model the data effectively, be it for analysis, processing, or compression purposes. It is intuitively clear that if the complexity of the model can grow unchecked, its accuracy can increase arbitrarily (short of computational limitations); this defines a tradeoff in which, for a given data vector, a set of complexity/accuracy pairs are defined for each possible model. In general one is interested in parsimonious modeling; by identifying complexity with "rate" and model misadjustment with "distortion", the situation becomes akin to an operational rate-distortion (R/D) problem in which, for each possible "rate", the goal is to find the model yielding t lie minimum distortion. In practice, however, only a finite palette of models is available, the complexity of which is limited by computational reasons: therefore, the entire data vector often proves too "non-stationary" for any single model. If the process is just slowly drifting, adaptive systems are probably the best choice; on the other hand, a wide class of signals exhibits a series of rather abrupt transition between locally regular portions (e.g. speech, images). In this case a common solution is to partition the data uniformly so that the resulting pieces are small enough to appear stationary with respect to the available models. However, if the goal is again to obtain an overall modeling which is optimal in the above R/D sense, it is necessary that the segmentation be a free variable in the modelization process; this is however not the case if a fixed-size time windowing is used. Note that now the reachable points in the R/D plane are in fact indexed not just by a model but by a segmentation/model-sequence pair; their number therefore grows exponentially with the size of the data vector. This thesis is concerned with the development of efficient techniques to explore this R/D set and to determine its operational lower bound for specific signal processing problems. It will be shown that, under very mild hypotheses, many practical applications dealing with nonstationary data sets can be cast as a R/D optimization problem involving segmentation, which can in turn be solved using polynomial-time dynamic programming techniques. The flexibility of the approach will be demonstrated by a very diverse set of examples: after a general overview of the various facets of the dynamic segmentation problem in Chapter 2, Chapter 3 will use the framework to determine an operational R/D bound for the approximation of piecewise polynomial function with respect to wavelet-based approximation; Chapter 4 will show its relevant to compression problems, and in particular to speech coding based on linear prediction and to arithmetic coding for binary sources; finally, in Chapter 5, an efficient data hiding scheme for PCM audio signals will be described, in which the optimal power allocation for the hidden data is determined with respect to the time-varying characteristics of the host signal.

The signal processing community is increasingly interested in using information theoretic concepts to build signal processing algorithms for a variety of applications. A general theory on how to apply the mathematical concepts of information theory to the field of signal processing would therefore be of great interest. This is one of the main goals of this thesis, namely to introduce a mathematical framework for information theoretic signal and image processing. The framework is based on stochastic processes for information transmission and on the error probabilities associated to these transmissions. Within the developed model, the stochastic processes account for the signal processing tasks within probability space, and the error probabilities are the optimization functions that drive the algorithms towards the signal processing objectives. The resulting conceptual framework allows us to directly apply a large number of information theoretic concepts and formulae to signal processing, including lower error bounds for the error probabilities or concepts from rate-distortion theory. In order to illustrate the theoretic framework, we show that several existing information theoretic signal processing algorithms implicitly fit our general model. This allows us to study interesting relationships between several algorithms. More importantly, we also apply the theory to three important target applications, namely multi-modal medical image registration, audio-video joint processing, and non-parametric, non-supervised classification. The first two applications are particular examples of the general concept of multi-modal feature extraction. Multi-modal feature extraction aims to determine those features in a pair of multi-modal signals that carry maximal mutual redundancy. This means that from the feature space representation of one signal we can predict the feature space representation of the second signal with low probability of error. After describing the mathematical basis, we illustrate the algorithm with examples of multi-modal medical image registration, where the algorithm adaptively extracts those features in the initial datasets which best perform the registration task. Again, this is done by determining those features which carry maximal mutual redundancy and therefore define optimally spatial registration. We also apply the model to audio-video signals to predict the localization of a speaker in a video scene from its corresponding speech signal. The resulting algorithms illustrate that the existence of features with large mutual redundancy in multi-modal signals can be used to improve multi-modal signal processing. Furthermore the general theory enables the construction of a wide range of completely new applications. Another illustrative example of the general information theoretic signal processing framework consists of information theoretic classification. Even though the basic model for multi-modal feature extraction and classification is identical, the final mathematical expressions are different and complementary. This allows us to make very interesting analogies between these two distinct applications. In particular, it is interesting to see that in analogy to registration, also classification algorithms aim to minimize error probabilities. The entirely probabilistic nature of the classification framework allows us to add a hidden Markov random field to the algorithms, resulting in the promising concept of non-parametric hidden Markov models. The classification algorithms are validated on synthetic and natural data. For instance, we apply the non-parametric hidden Markov model to the segmentation of medical images and obtain promising results in comparison to the state-of-the-art in this field. In conclusion, the experimental results show that the introduced mathematical framework leads to interesting generalizations of existing signal processing tasks and to promising results for several newly derived signal processing algorithms.

Asja Derviskadic, Guglielmo Frigo, Alexandra Cameron Karpilow, Mario Paolone

As power grids transition towards low-inertia net-works based on converter-interfaced renewable energy resources, they become increasingly vulnerable to extreme dynamics. Currently, the most advanced methods for signal processing in power systems are embedded in Phasor Measurement Units (PMUs), which rely on a stationary phasor model with a single fundamental tone. However, the signal dynamics measured during grid disturbances may have broadband spectra that cannot be sufficiently captured by a narrowband phasor model. Inspired by previous work done by the authors, this paper introduces a signal processing method based on a dictionary containing models of common signal dynamics. The dictionary can be used to identify the signal model and parameters that best capture the dynamics. The method is evaluated and compared to a standard phasor-estimation method for two documented, real-world grid disturbances.