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.

The first part of Chapter 7 deals with oscillating water columns (OWCs). The concepts of radiation conductance and susceptance are introduced. The former is related to the radiated power, whereas the latter represents the reactive power. Expressions for the power absorbed by the OWC are derived, which are analogous to those of the oscillating body WEC. The potential energy of the OWC is also discussed. The last part of Chapter 7 deals with wave energy converters that move in modes other than the six conventional rigid-body modes. The theory of generalised modes are described, and some examples are given to illustrate the utility of the theory.