This chapter introduces topics that extend beyond the electricity market, and focuses in particular on oil, natural gas, and biofuels. Short- and long-term equilibriums are analyzed in the context of oil markets. Monopolies, cartels, and the model of the dominant firm are also analyzed in the context of the oil market. The tax incidence problem is formulated as an equivalent optimization problem and analyzed in the context of natural gas markets. One-way substitutability is analyzed in the context of biofuel markets, and the tortilla crisis is illustrated through an optimization model. Hotelling’s rule is stated and proven by considering a dynamic optimization model of a finite nonrenewable resource that is gradually depleted over time while satisfying a price-elastic demand.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.

This chapter introduces bandlimited signals, sampling theory, and the method of reconstruction from samples. Uniform sampling with a Dirac delta train is considered, and the Fourier transform of the sampled signal is derived. The reconstruction from samples is based on the use of a linear filter called an interpolator. When the sampling rate is not sufficiently large, the sampling process leads to a phenomenon called aliasing. This is discussed in detail and several real-world manifestations of aliasing are also discussed. In practice, the sampled signal is typically processed by a digital signal processing device, before it is converted back into a continuous-time signal. The building blocks in such a digital signal processing system are discussed. Extensions of the lowpass sampling theorem to the bandpass case are also presented. Also proved is the pulse sampling theorem, where the sampling pulse is spread out over a short duration, unlike the Dirac delta train. Bandlimited channels are discussed and it is explained how the data rate that can be transmitted over a channel is limited by channel bandwidth.

This chapter introduces the continuous-time Fourier transform (CTFT) and its properties. Many examples are presented to illustrate the properties. The inverse CTFT is derived. As one example of its application, the impulse response of the ideal lowpass filter is obtained. The derivative properties of the CTFT are used to derive many Fourier transform pairs. One result is that the normalized Gaussian signal is its own Fourier transform, and constitutes an eigenfunction of the Fourier transform operator. Many such eigenfunctions are presented. The relation between the smoothness of a signal in the time domain and its decay rate in the frequency domain is studied. Smooth signals have rapidly decaying Fourier transforms. Spline signals are introduced, which have provable smoothness properties in the time domain. For causal signals it is proved that the real and imaginary parts of the CTFT are related to each other. This is called the Hilbert transform, Poisson’’s transform, or the Kramers–Kronig transform. It is also shown that Mother Nature “computes” a Fourier transform when a plane wave is propagating across an aperture and impinging on a distant screen – a well-known result in optics, crystallography, and quantum physics.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.

This chapter presents the Laplace transform, which is as fundamental to continuous-time systems as the z-transform is to discrete-time systems. Several properties and examples are presented. Similar to the z-transform, the Laplace transform can be regarded as a generalization of the appropriate Fourier transform. In continuous time, the Laplace transform is very useful in the study of systems represented by linear constant-coefficient differential equations (i.e., rational LTI systems). Frequency responses, resonances, and oscillations in electric circuits (and in mechanical systems) can be studied using the Laplace transform. The application in electrical circuit analysis is demonstrated with the help of an LCR circuit. The inverse Laplace transformation is also discussed, and it is shown that the inverse is unique only when the region of convergence (ROC) of the Laplace transform is specified. Depending on the ROC, the inverse of a given Laplace transform expression may be causal, noncausal, two-sided, bounded, or unbounded. This is very similar to the theory of inverse z-transformation. Because of these similarities, the discussion of the Laplace transform in this chapter is brief.

This chapter discusses the application of operations research models in the energy industry. The applications that are covered include linear programming in the economic dispatch problem, mixed integer programming in unit commitment, nonlinear programming in the alternating current optimal power flow, stochastic programming in hydrothermal scheduling, and various other classes of mathematical programs. The capacity expansion problem is then analyzed through an analytical approach that relies on load duration curves and screening curves. The model is used to highlight the missing money problem in energy only markets with price caps, introduces the notion of competitive equilibrium, and discusses the distinction between short-term and long-term competitive equilibrium.

This chapter presents the direct current optimal power flow (DCOPF) problem, which is a linear approximation of the alternating current optimal power flow problem. Two equivalent formulations of the DCOPF are provided: one based on power transfer distribution factors (PTDFs), and one based on susceptances. The optimal solution of the DCOPF is characterized using the KKT conditions of the model. The optimal transmission switching and optimal transmission expansion problem are introduced. The nodal pricing or locational marginal pricing (LMP) mechanism is defined, and various properties of LMPs are characterized through examples and KKT conditions. Congestion rent and congestion cost are defined and compared. The equivalence of DCOPF to a competitive market model for transmission and energy is established using KKT conditions. A DCOPF with losses is introduced. Zonal pricing is defined, and the motivations of its origins are discussed. Two models of zonal pricing are analyzed, one based on a transportation network and one based on flow-based market coupling. Various notions of zonal pricing are defined, including available transfer capacities, loop flows, transit flows, critical branches, zone-to-line PTDFs, remaining available margin, and generation shift keys. Redispatch is defined and demonstrated through examples, and the INC-DEC gaming strategy is described.

A number of properties relating to the inverse z-transform are discussed. The partial fraction expansion (PFE) of a rational z-transform plays a role in finding the inverse transform. It is shown that the inverse z-transform solution is not unique and depends on the region of convergence (ROC). Depending on the ROC, the solution may be causal, anticausal, two-sided, stable, or unstable. The condition for existence of a stable inverse transform is also developed. The interplay between causality, stability, and the ROC is established and illustrated with examples. The case of multiple poles is also considered. The theory and implementation of IIR linear-phase filters is discussed in detail. The connection between z-transform theory and analytic functions in complex variable theory is placed in evidence. Based on this connection, many intriguing examples of z-transform pairs are pointed out. In particular, closed-form expressions for radii of convergence of the z-transform can be obtained from complex variable theory. The case of unrealizable digital filters and their connection to complex variable theory is also discussed.

This chapter described the intended audience of the book, summarizes the content of each chapter, describes the use of the material in courses, describes how exercises are used in the book, provides explanations about notation and terminology, and includes acknowledgements.

This chapter covers basic concepts in power system operations and electricity markets. Basic concepts of power generation include variable cost, marginal cost, fixed cost, investment cost, the weighted average cost of capital, and the definition of a natural monopoly. Basic concepts of transmission and distribution include a discussion of Ohm’s law, Kirchhoff’s laws, the power flow equations, and the direct current power flow. Basic concepts of consumption include the notion of valuation/marginal benefit, demand functions, demand elasticity, and the value of lost load. The actors of electricity markets are introduced, including transmission/independent system operators, distribution system operators, utilities, load serving entities, retailers, power exchanges, and transmission companies. Reserves and ancillary services are then introduced, and details about the forward-looking and rolling nature of power system operations are discussed. Exchanges and pools are informally defined, and the debate between uniform and pay-as-bid pricing is detailed. A blueprint of a typical electricity markets, with the participating actors and traded products and services, is introduced. The California and Central Western European markets are compared in order to introduce the debate between zonal and nodal pricing, as well as different approaches in pricing.

This concise and rigorous textbook introduces students to the subject of continuum thermodynamics, providing a complete treatment of the subject with practical applications to material modelling.

• Presents mathematical prerequisites and the foundations of continuum mechanics, taking the student step-by-step through the subject to allow full understanding of the theory.

• Introduces more advanced topics such as theories for the investigation of material models, showing how they relate to real-world practical applications.

• Numerous examples and illustrations, alongside end-of-chapter problems with helpful hints, help describe complex concepts and mathematical derivations.