Convex Optimization of Power Systems

In the late 1880s, Thomas Edison and George Westinghouse fought the so-called War of the Currents to decide whether the incumbent direct current or Nicola Tesla's alternating current technology would become the standard for future power systems. Endnotes [1, 2] provide excellent historical accounts. The winning argument was that it was much easier using the technology of the time, transformers, to change voltage levels in AC power systems, enabling efficient high-voltage transmission of power and lower voltage generation and end usage (see (2.6) in Section 2.5.1). Beyond pride, part of Edison's opposition to AC transmission was rooted in the higher level of mathematics necessary to understand it.

Today, power electronics have enabled direct current to make a comeback in certain applications like long-distance transmission and microgrids, cf. Section 3.5.1. Some even say that we are now constrained by the mathematical model of AC power flow, which while simple to write down is a quagmire for analysis and computation. Here we tackle this issue head on in one of its purest forms, optimal power flow.

In words, optimal power flow is the problem of minimizing some function of voltage, current, and power, subject to the resulting flow being able to feasibly traverse a transmission or distribution system. Since its introduction by Carpentier [3], virtually every algorithm for continuous optimization has been applied, cf. endnotes [4–12] and the surveys [13–15]. Optimal power flow had similar but separate beginnings in the Russian academic literature [16]. System operators solve optimal power flow routines to do long-term planning, days- to hour-ahead scheduling, real-time dispatch, and pricing (to name a few), making it one of the most frequently employed optimization routines in power systems. As will be seen over the course of this book, many other power system optimizations are essentially optimal power flow models with additional layers of detail.