The present textbook grew out of lecture notes for courses on the methods and applications of dynamic economic analysis that I have been teaching to graduate students over the years. The book is not meant to cover the whole state of the art in this area but to provide a compact presentation of the most essential concepts and results and to illustrate them by selected applications from various fields of economic research. The target readership consists of students and researchers who have little or no experience with solution techniques for dynamic economic models but who have a decent background knowledge in economics, calculus, and linear algebra. I hope that the book helps its readers to get acquainted with the basic issues and the most popular modelling frameworks of dynamic economic analysis and that it raises the appetite of its audience for a more complete and detailed study of this area.

Dynamic economic analysis is a vast area and the relevant literature is extensive. In order to achieve my goal of a compact presentation, I deliberately make two important restrictions that are also reflected in the title of the book. First, I only consider models, that are formulated in discrete time and, second, I do not deal with stochastic dynamics. The main justification of the first restriction is that I want to provide an overview of the most important concepts and methods of dynamic economic analysis without getting lost in technicalities. For some dynamic economic models, the choice between a discrete-time formulation and a continuous-time formulation is simply a matter of taste, whereas for others this choice is driven by the quest for analytical tractability. Moreover, there exist dynamic economic models that generate quite different predictions depending on which of the two formulations of time is applied. In my opinion, however, the basic issues arising in dynamic economic analysis can be illustrated with less technical effort in a discrete-time setting than in a continuous-time setting, which is why I restrict the presentation to the former case. By no means do I consider the discrete-time formulation as more relevant or more realistic than its continuous-time counterpart.

The day-to-day and long-term operation of power systems is made up of a sprawling collection of engineering and economic tasks that subsumes most of the contents of this book. This chapter does not address the full scope of power system operation but rather a handful of relevant problems that are amenable to convex optimization. As will be seen, many of these problems consist of details layered on top of the optimal power flow formulations of Chapter 3. The interested reader is referred to Wood and Wollenberg [1] for broad coverage of power system operation.

Of particular relevance to this chapter is Theorem 3.1 of Section 3.3.1, which guarantees that under certain conditions, the SOC optimal power flow relaxation is exact in radial networks. It is therefore a great boon that distribution systems, the portions of power systems that convey low-voltage electricity from substations to end users, are almost always operated radially. Until recently, distribution systems were essentially passive, predictable energy consumers. The shift from fuel-based centralized power plants to distributed and renewable generation and the new, active role of loads like electric vehicles and smart buildings are transforming distribution systems into highly actuated, potentially volatile consumers and producers of electric power. Consequently, many of the formulations in this chapter are both highly tractable and relevant when specialized to radial networks. This is especially true in Section 4.4, which deals exclusively with power flow in radial networks.

Multi-period optimal power flow

Optimal power flow routines are run every few minutes to update device and resource settings in response to the constantly changing conditions of power systems. The dynamic couplings present over these time scales justify linking these routines over successive time periods. This section formulates the multi-period optimal power flow and elucidates its application through generator ramp constraints and energy storage.

Essentially, multi-period optimal power flow is just a sequence of ordinary optimal power flow routines strung together by dynamic costs and constraints.

When someone referred to “Greek Art” in the nineteenth century, the meaning was clear: the Classical period of Greek art, c. 480–323 bc, usually objects or buildings from Athens. The image that sprang to mind was of marble sculptures and temples, preferably in ruins to convey the romantic notion of a lost past. The passion for antiquity, particularly ancient Greece, inspired innumerable examples of Neoclassical architecture and Neoclassical ornamentation on nearly every continent, Neoclassical painting and sculpture in Europe and North America, enthusiastic borrowing of Greek myth in every artistic medium, including literature, drama, music, and so on. Ancient Greek or classical styles (as opposed to the Classical period) were not only aesthetically favored but also bestowed intellectual cachet (e.g., the façade of the British Museum) or advanced political ideals (e.g., civic buildings and monuments in Washington, DC).

But times have changed, and so have our definitions of Greek art, specifically ancient Greek art (for there is modern and contemporary Greek art, as well). The idea of ancient Greece and its cultural outpouring is very much still with us, but in our multicultural world of global economies, the internet, social networking, and global travel made easily accessible, we have, unwittingly, redefined ancient Greek culture as stretching throughout the region where the ancient Greeks once trod, from Spain to the Hindu Kush, from the earliest Paleolithic “Greeks” to the Byzantine Greeks or even later. What was once clear has become obfuscated because of a plethora of information, as well as the professionalization of the academic fields of art and archaeology and the consequent push to justify budgetary expenditure through publication. What was once of supreme importance to the education of young gentlemen has become the province of anyone and everyone and, ironically, has become obscure and apparently disconnected from the lives of even the well educated.