Communities living far from a grid network are increasingly being supplied by off-grid renewable energy systems. However, a consensus of the best approach to their design has yet to emerge. A number of options are being trialled and these are described. Small dc photovoltaic systems with batteries are well established but can only supply limited amounts of power. An example of the supply of power to a remote health facility is shown. Microgrids combine a number of energy sources connected using either ac or dc. These connection architectures are demonstrated. An example of an operating ac microgrid is shown, together with the initial design calculations of the scheme. The concept of community energy is explained with a demonstration of the potential benefits of peer-to-peer energy trading. The chapter is supported by 4 examples, 7 questions with answers and full solutions in the accompanying online material. Further reading and online resources are identified.

This chapter is dedicated to examining strategies and technologies for improving power grids’ resilience. The first part of this chapter focuses on traditional power grids by presenting technologies and management approaches for improved resilience at the power generation, transmission, and distribution levels and by discussing strategies for enhanced withstanding capability or reduced restoration speed. The second part of this chapter explores the effect that the evolution of power grids into “smart” grids may likely have in the future. Advanced technologies that have already been implemented at all levels of power grids are discussed. Alternative power distribution approaches implemented at the load level, such as microgrids, able to significantly improve resilience with respect to traditional power grids, are also described in this chapter.

This chapter analyzes linear and nonlinear discrete-time systems described by a discrete-time state-space model whose inputs are uncertain but known to belong to an ellipsoid. For the linear case, even if the input set is an ellipsoid, the set containing all possible values that the state can take is not an ellipsoid in general, but it can be upper bounded by an ellipsoid. We develop techniques for recursively computing a family of such upper-bounding ellipsoids. Within this family, we then show how to choose ellipsoids that are optimal in some sense, e.g., they have minimum volume. For the nonlinear case, we will again resort to linearization techniques to approximately characterize the set containing all possible values that the state can take. The application of the techniques presented is illustrated using the same inertia-less AC microgrid model used in Chapter 5.

This chapter studies continuous-time dynamical systems described by a continuous-time state-space model whose input is subject to probabilistic uncertainty. The first part of the chapter is devoted to the analysis of linear systems and provides techniques for computing the first and second moments of the state vector when the evolution of the input vector is governed by a "white noise" process with known mean and covariance functions. Then, by additionally imposing this white noise process to be Gaussian, we provide a partial differential equation whose solution yields the pdf of the state vector. The second part of the chapter extends these techniques to the analysis of nonlinear systems, with a special focus on the case when the white noise governing the evolution of the input vector is Gaussian. The third part of the chapter illustrates the application of the techniques developed to the analysis of inertia-less AC microgrids when the measurements utilized by the frequency control system are corrupted by additive disturbances.

In this chapter, we first provide some motivation for the type of modeling problems we address in this book. Then we provide an overview of the type of mathematical models used to describe the behavior of the classes of systems of interest. We also describe the types of uncertainty models adopted and how they fit into the mathematical models describing system behavior. In addition, we provide a preview of the applications discussed throughout the book, mostly centered around electric power systems. We conclude the chapter by providing a brief summary of the content of subsequent chapters.

This chapter provides techniques for analyzing discrete-time dynamical systems under probabilistic input uncertainty. Here, the relation between the input and the state is described by a discrete-time state-space model. The input vector is modeled as a vector-valued stochastic process with known first and second moments (or known pdf). The first part of the chapter is devoted to the analysis of linear systems and provides techniques for characterizing the first and second moments and the pdf of the state vector. The second part deals with the analysis of nonlinear systems, where we use the techniques developed in Chapter 4 to exactly characterize the distribution of the state vector when the pdf of the input vector is given. In addition, we rely on linearization techniques to obtain expressions that approximately characterize the first and second moments and the pdf of the state vector. The third part of the chapter illustrates the application of the techniques developed to the analysis of inertia-less AC microgrids under random active power injections.

This chapter introduces the concepts of microgrid and networked microgrids, describes challenges in building networked microgrids, and provides an overview of the topics of this book.