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4 Background: Mathematical Research Areas Important for the Grid
Pages 61-83

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From page 61...
... The mathematical sciences provide essential technology for the design and operation of the power grid. Viewed as an enormous electrical network, the grid's purpose is to deliver electrical energy from producers to consumers.
From page 62...
... The material in this chapter is intended to present sufficient background about these mathematical areas to understand important issues raised in their application to the power grid, building on the power grid material presented in Chapters 1 to 3. Chapter 5, Preparing for the Future, discusses challenges that the next-generation grid will present requiring new mathematical analysis, and Chapter 6, Mathematical Research Priorities Arising from the Electric Grid, discusses the new mathematical capabilities required to meet these challenges.
From page 63...
... Differential equations derived from the physical laws of electricity describe the rates of change of the variables. Kirchhoff's laws for currents and voltages in an electrical network, together with models for network devices like generators, transformers, and motors, are the core ingredients for models of the electric grid.
From page 64...
... Models of the electrical grid have a network structure inherited from the physical network. An important theoretical challenge is to determine how the network structure of coupled systems of oscillators (like the power grid)
From page 65...
... General-Purpose Optimization Methods and Software Continuous optimization, including both theoretical analysis and numerical methods, has been an active research area since the late 1940s. During the decades since then, there has been consistent and significant progress, punctuated by bursts of activity when a new, or apparently new, idea becomes known.
From page 66...
... High-quality optimization software invariably uses linear algebraic techniques whose speed depends on the size, structure, and sparsity of relevant matrices, and these properties may change during the course of iteration toward a solution. For example, some optimization methods factorize or update a matrix whose dimension increases as the number of currently active constraints increases, while other methods factorize a matrix whose dimension decreases in the same circumstances.
From page 67...
... In a power engineering context, the unit commitment problem (see Sheble and Fahd, 1994, for background) decides which generators will operate over a certain time window.
From page 68...
... It is easy to argue for the importance of this field in a power engineering context. Both problems discussed above, the unit commitment and line switching problems, arise in an ac power flow setting, in which case the problems have both continuous and binary variables but where the underlying equations are nonlinear.
From page 69...
... For example in the unit commitment setting, one might be concerned that a generator may need to go off-line in the next 24 hours owing to a mechanical condition. In that case the decision may be made to trip the generator now or to postpone that decision until 12 hours from now, when more information will become available and when the relative likelihoods may be well understood -- that is, the probabilities -- of the events that will cause the various realizations of that information.
From page 70...
... However, the loads during the second stage are not precisely known, and instead it is assumed that one of a fixed family S of known scenarios, each specifying a set of loads for the second stage, will be realized starting at time Δ; further, the probability of each scenario is known at t = 0. The actions available to the power grid operator are these: (1)
From page 71...
... It is used to control localized processes. Generation is one element of the power system where primary control is critical to the efficient and reliable operation of the power grid.
From page 72...
... Transient excitation boost was installed on some generators to temporarily raise the rotor field current during local faults to boost terminal voltage and help coordinate with the protection scheme. Other nongenerator examples of primary control include voltage-regulating transformers in distribution sub­ stations or static var compensators to continuously regulate voltage at a transmission substation by adjusting the reactive power output of the device.
From page 73...
... RISK ANALYSIS, RELIABILITY, MACHINE LEARNING, AND STATISTICS Power systems are composed of physical equipment that needs to function reliably. Many different pieces of equipment could fail on the power system: Generators, transmission lines, transformers, medium-/low-voltage cables, connectors, and other pieces of equipment could each fail, leaving customers without power, increasing risk on the rest of the power system, and possibly leading to an increased risk of cascading failure.
From page 74...
... Regression is a classic problem that is pervasive, and much work in modern statistics and machine learning still revolves around variants of linear regression. The most important regression problems related to the power grid are those of estimating demand: • System load forecasting (estimation of demand for a region)
From page 75...
... . Important classification and reliability problems related to the power grid include the following: • Asset failure prediction and condition-based maintenance.
From page 76...
... Since there are no ground truth labels for clustering, there are many different viable methods for measuring the quality of a clustering. One application of clustering for the power grid is energy disaggregation and nonintrusive load monitoring.
From page 77...
... In the case of the power network, each piece of equipment on the power grid (substations, transformers, wind turbines, consumers) and their influence on one another would be modeled.
From page 78...
... From the viewpoint of the data, the model is providing an extrapolation of the state to times and locations where no measurements are available. This balance between the two types of information, that derived from measurements and that from physical laws, distinguishes data assimilation from other statistical methods such as state estimation, machine learning, and data-driven modeling.
From page 79...
... COMPLEXITY AND MODEL REDUCTION IN THE TIME OF BIG DATA Realistic modeling involves large numbers of equations and degrees of freedom improving spatial and temporal resolution of the model, incorporating more physical effects, and describing uncertainty and noise. What is considered "large" in a model has progressed from thousands to tens and even hundreds of millions of variables as computational power has been systematically expanding.
From page 80...
... This type of effective modeling, which crucially relies on machine learning for the detection of the important observables parameterizing the high-dimensional state data, holds promise for many complex but effectively simple systems models, especially for the power grid. A nontrivial twist arises from the fact that some of the data that one wishes to effectively compress are not just real numbers (components of a vector in Rn for very large n)
From page 81...
... There are aspects of uncertainty that have always been present in grid modeling and cannot be prescribed or measured accurately in advance. The uncertainty of grid loads is ever present, but the uncertainty inherent in renewable energy production, especially in solar panels and wind turbine farms owing to weather variability, poses a new set of challenges.
From page 82...
... 2015. "Mixed-Integer Programming: It Works Better Than You May Think." Presentation to the National Research Council work shop Analytical Research Foundations for the Next-Generation Electric Grid on February 11-12, 2015, in Irvine, Calif.
From page 83...
... 2012. A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output.


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