The increasing penetration of renewable resources has changed the characteristics of power system and market operations, from one relying primarily on deterministic and static planning to one involving highly stochastic and dynamic operations. In such new operation regimes, the ability of adapting changing environments and managing risks arising from complex scenarios of contingencies is essential. To this end, an operation tool that provides probabilistic forecasting that characterizes the underlying probability distribution of variables of interest can be extremely valuable. A fundamental challenge in probabilistic forecasting for system and market operations is the scalability. As the size of system and the complexity of stochasticity increase, standard techniques based on direct Monte Carlo and machine learning techniques become intractable. This chapter outlines an alternative approach based on an online learning to overcome barriers of computation complexity.

Information theory plays an indispensable role in the development of algorithm-independent impossibility results, both for communication problems and for seemingly distinct areas such as statistics and machine learning. While numerous information-theoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread: Fano’s inequality. In this chapter, we provide a survey of Fano’s inequality and its variants in the context of statistical estimation, adopting a versatile framework that covers a wide range of specific problems. We present a variety of key tools and techniques used for establishing impossibility results via this approach, and provide representative examples covering group testing, graphical model selection, sparse linear regression, density estimation, and convex optimization.

This chapter introduces basic ideas of information-theoretic models for distributed statistical inference problems with compressed data, and discusses current and future research directions and challenges in applying these models to various statistical learning problems. In these applications, data are distributed in multiple terminals, which can communicate with each other via limited-capacity channels. Instead of recovering data at a centralized location first and then performing inference, this chapter describes schemes that can perform statistical inference without recovering the underlying data. Information-theoretic tools are borrowed to characterize the fundamental limits of the classical statistical inference problems using compressed data directly. In this chapter, distributed statistical learning problems are first introduced. Then, models and results of distributed inference are discussed. Finally, new directions that generalize and improve the basic scenarios are described.

This chapter provides an overview of the theory of controlled sensing, and its application to the sequential design of data-acquisition and decision-making processes. Based on the theory, it provides an overview of the applications to the quickest detection and localization of anomalies in power systems. This application is motivated by the fact that agile localization of anomalous events plays a pivotal role in enhancing the overall reliability of the grid and avoiding cascading failures. This is especially of paramount significance in the large-scale grids due to their geographical expansions and the large volume of data generated. Built on the theory of controlled sensing, the chapter discusses a stochastic graphical framework for localizing the anomalies with the minimum amount of data. This framework capitalizes on the strong correlation structures observed among the measurements collected from different buses. This framework, at its core, collects the measurements sequentially and progressively updates its decision about the location of the anomaly.

This chapter reviews the methods used to estimate the state of a power system and its network model based on the measurements provided by supervisory control and data acquisition (SCADA) systems and/or by phasor measurement units (PMU). Initially, it provides an overview of the commonly implemented SCADA-based state estimators. Network observability and bad data processing functions are briefly described. This is followed by a description of the changes in the problem formulation and solution introduced by the incorporation of PMU measurements as well as the associated opportunities and challenges. Finally, detection, identification, and correction of network model errors and impact of such errors on system reliability as well as market operations are presented.

This chapter focuses on distributed control and learning for electric vehicle charging. After a brief survey, it covers three related sets of algorithms: (i) distributed control for electric vehicle charging based on a basic formulation; (ii) distributed control for an extension of the basic setting to include network capacity constraints; and (iii) distributed learning for an extension of the basic setting with limitations in the information flow. The chapter ends with a brief summary of open problems.

This chapter introduces mean field games to capture the mutual interaction between a population and its individuals.Within this context, a new equilibrium concept called mean field equilibrium replaces the classical Nash equilibrium in game theory. In a mean field equilibrium each individual responds optimally to the population behavior. In other words, no individuals have incentives to deviate from their current strategies. This new way of modeling the interactions among members of large populations is used to study dynamic demand response management in electricity grids. Moreover, some generalizations of the classical idea of mean field games are introduced to embrace the situations in which the whole population can be divided into classes of members.

The problem of tracking the system frequency is ubiquitous in power systems. However, despite numerous empirical comparative studies of various algorithms, the underlying links and commonalities between frequency tracking methods are often overlooked. To this end, we show that the treatment of the two best known frequency estimation methodologies: (i) tracking the rate of change of the voltage phasor angles, and (ii) fixed frequency demodulation, can be unified, whereby the former can be interpreted as a special case of the latter. Furthermore, we show that the frequency estimator derived from the difference in the phase angle is the maximum likelihood frequency estimator of a nonstationary sinusoid. Drawing upon the data analytics interpretation of the Clarke and related transforms in power system analysis as practical Principal Component Analyzers (PCA), we then set out to explore commonalities between classic frequency estimation techniques and widely linear modeling. The so-obtained additional degrees of freedom allow us to arrive at the adaptive Smart Clarke and Smart Park transforms (SCT and SPT), which are shown to operate in an unbiased and statistically consistent way for both standard and dynamically unbalanced smart grids. Overall, this work suggest avenues for next generation solutions for the analysis of modern grids that are not accessible from the Circuit Theory perspective.

Data injection attacks serve as the hallmark example of the security concerns posed by the incorporation of advanced sensing and communication capabilities in power systems. Data injection attacks arise when one or several malicious attackers compromise a subset of the meters used by the state estimation procedure with the aim of manipulating the estimate obtained by the network operator. This chapter surveys the main data injection attacks that are formulated under the assumption that the state variables do not posses a probabilistic description and, therefore, the network operator implements unbiased state estimation procedures. Data injection attacks without this assumption are also studied. In particular, when the network operator perform minimum mean square error (MMSE) estimation, a fundamental trade-off is established between the distortion induced by the attacker and the achievable probability of attack detection. Within this setting, optimal attack strategies are described. The chapter also describes stealth attack constructions that simultaneously minimize the amount of information obtained by the network operator and the probability of attack detection.

Machine-learning algorithms can be viewed as stochastic transformations that map training data to hypotheses. Following Bousquet and Elisseeff, we say such an algorithm is stable if its output does not depend too much on any individual training example. Since stability is closely connected to generalization capabilities of learning algorithms, it is of interest to obtain sharp quantitative estimates on the generalization bias of machine-learning algorithms in terms of their stability properties. We describe several information-theoretic measures of algorithmic stability and illustrate their use for upper-bounding the generalization bias of learning algorithms. Specifically, we relate the expected generalization error of a learning algorithm to several information-theoretic quantities that capture the statistical dependence between the training data and the hypothesis. These include mutual information and erasure mutual information, and their counterparts induced by the total variation distance. We illustrate the general theory through examples, including the Gibbs algorithm and differentially private algorithms, and discuss strategies for controlling the generalization error.

A grand challenge in representation learning is the development of computational algorithms that learn the explanatory factors of variation behind high-dimensional data. Representation models (encoders) are often determined for optimizing performance on training data when the real objective is to generalize well to other (unseen) data. This chapter provides an overview of fundamental concepts in statistical learning theory and the information-bottleneck principle. This serves as a mathematical basis for the technical results, in which an upper bound to the generalization gap corresponding to the cross-entropy risk is given. When this penalty term times a suitable multiplier and the cross-entropy empirical risk are minimized jointly, the problem is equivalent to optimizing the information-bottleneck objective with respect to the empirical data distribution. This result provides an interesting connection between mutual information and generalization, and helps to explain why noise injection during the training phase can improve the generalization ability of encoder models and enforce invariances in the resulting representations.