Overview

Engineering design problems are multiobjective in nature. These problems usually optimize two or more conflicting objectives – simultaneously. An approach to multiobjective problem formulation combines the multiple objectives into a single objective function, also known as the Aggregate Objective Function (AOF). This AOF is solved to obtain one Pareto solution. One of several challenges in the area of multiobjective optimization is to judiciously construct an AOF that satisfactorily models the designer's preferences. This chapter provides a concise presentation of the Physical Programming method, which defines a framework to effectively incorporate the designer's preferences into the AOF (Ref. [1]).

Several methods to solve multiobjective optimization problems have been discussed in Chapter 6, such as the weighted sum method, compromise programming, and goal programming. These weight-based approaches require the designer to specify numerical weights in defining the AOF. This process can be ambiguous. For example, consider the following: (1) How does the designer specify weights in weight-based approaches? (2) Do the weights reflect the designer's preferences accurately? If the designer chooses to increase the importance of a particular objective, by how much should he/she increase the weight? Is 25% adequate? Or is 200% adequate? (3) Does the AOF denote a true mathematical representation of the designer's preferences?

The above questions begin to explain that the problem of determining “good weights” can be difficult and dubious. Because of this ambiguity, the weight selection process is often a computational bottleneck in large scale multiobjective design optimization problems. The above discussion paves the way for a multiobjective problem formulation framework that alleviates these ambiguities: Physical Programming (PP) developed by Messac [2].

Physical Programming systematically develops an AOF that effectively reflects the designer's wishes. This approach eliminates the need for iterative weight setting, which alleviates the above discussed ambiguities. Instead of choosing weights, the designer chooses ranges of desirability for each objective. The PP method formulates the AOF from these ranges of desirability, while yielding interesting and useful properties for the AOF.

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.

This chapter zooms out to consider the design of power systems. For example, if a city's power consumption has grown in support of new industry or increased population, should it become the home of a new generator or the destination of a new transmission line? What if there are 100 such choices, all within the same interconnected power system? Additionally, each generation project will take five years to complete and each transmission project ten, over which the energy consumption landscape may change significantly. Finally, each element of the design must respect environmental restrictions, harmonize with a potentially poorly designed legacy system, and be separately financed. It's no wonder that the evolution of power systems looks as much like the product of organic synthesis as deliberate design. It is hence somewhat more reasonable to plan power systems and then hope that plans stay consistent with reality to yield efficient and reliable infrastructures.

Even in the absence of the factors listed above, power system planning remains a source of intractable problems because it almost always entails the use of integer variables. Unfortunately, any plan that builds a one-micron-thick transmission line or a two-watt nuclear power plant is not very useful. This makes many power system planning problems NP-hard, and thus very hard to solve. Fortunately, although these problems cannot be solved in polynomial time, powerful mixed-integer programming heuristics like cutting planes and branch-and-bound make moderate-sized problems tractable.

This chapter ignores many of the above real-world details of power system planning and focuses on its underlying mathematical skeleton. This approach is to respect discrete constraints and seek formulations that have convex continuous relaxations, as discussed in Section 2.2.4. More precisely, if the problem we'd like to solve is

minimize f(x, y)

subject to gi(x, y) ≤ 0

yi ∈ ℤ,

then we want f(x, y) and each gi(x, y) to be convex.

This chapter summarizes the basic technical concepts used throughout this book. As stated in the Introduction, this book focuses on modeling, so most algorithmic aspects are left “under the hood.” Because this book is intended to appeal to anyone familiar with power systems or optimization, background material on both topics is covered, albeit at the minimum depth necessary to access the later material.

Convexity and computational complexity

We begin with a few core concepts. A point x0 ∈ X is a global minimum of the function f(x) over the set X ⊆ ℝn if f(x0) ≤ f(x) for all x ∈ X. If f(x) is continuous and X is compact, which is to say closed and bounded, such a point is guaranteed to exist. x0 is a local minimum of f(x) if there exists an ε > 0 for which f(x0) ≤ f(x) for all x ∈ X satisfying ∥x − x0∥ ≤ ε. All minima of a convex function achieve the same function value and are therefore global. In general, a function may have multiple local and global minima.

To find a global minimum of a convex function, choose a descending algorithm, let it run free, and in a perfect world it will eventually end up there. (In the real world, large problem sizes or bad numerical conditioning can derail any algorithm.) The intuitive simplicity of convexity translates to a genuine computational advantage, which is evinced by the powerful algorithms that exist for convex optimization and the extreme difficulty of nonconvex optimization. This section gives some basic characterizations of convexity and describes the varieties of convex optimization problems encountered in this book. For more comprehensive coverage, the reader is referred to endnotes [1–5], and to endnotes [6, 7] for theoretical treatments of convex functions and sets.

A function f is convex if, for any two points in its domain, x and y,

f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y) for all α ∈ [0, 1].

The power system of yore was a jungle. In cities like New York and Chicago, dozens of companies haphazardly strung up their own wires, each carrying its own special blend of electricity. In the United States, Samuel Insull, who began working for Edison in the 1880s, recognized the need for standardization. By the 1920s, he owned a significant share of Chicago's electric power industry, which he made extremely profitable through uniformity, scale, and shady financial practices. His company imploded during the Great Depression, and by the 1930s he had fled to Europe with a villainous reputation. He was acquitted of all charges shortly thereafter but died in 1938 with a debt of nearly $14 million.

Progressive or scoundrel, Samuel Insull's actions paved the way for the Public Utility Holding Company Act of 1935, which effectively made power systems government-regulated, vertically integrated monopolies. Over the next forty years, the power system developed into the dependable infrastructure we know today. Concurrently, it became an inefficient, rigid, environmentally hostile dinosaur. In 1978, the U.S. Congress passed the Public Utility Regulatory Policies Act, which forced utilities to purchase power from less expensive independent power producers at their avoided costs of generation, marking the first incursion of competition into the electricity sector in decades [1, 2]. In spurts over the next twenty years, the North American electric power system was deregulated to the applause of economists and chagrin of many engineers.

Then strange things began to happen, the most notable of which was the California Electricity Crisis [3, 4]. At the center of the crisis was the Enron scandal, and at the center of that, Kenneth Lay, perceived by some as a modern, mirrored version of Samuel Insull. Enron and its ilk gamed every aspect of the vulnerable new electricity markets, leading to extreme price volatility and blackouts.

The application of optimization to power systems has become so common that it deserves treatment as a distinct subject. The abundance of optimization problems in power systems can give the impression of diversity, but in truth most are merely layers on a common core: the steady-state description of power flow in a network. In this book, many of the most prominent examples of optimization in power systems are unified under this perspective.

As suggested by the title, this book focuses exclusively on convex frameworks, which by reputation are phenomenally powerful but often too restrictive for realistic, non-convex power system models. In Chapter 3, the application of classical and recent mathematical techniques yields a rich spectrum of convex power flow approximations ranging from high tractability and low accuracy to slightly reduced tractability and high accuracy. The remaining chapters explore problems in power system operation, planning, and economics, each consisting of details layered on top of the convex power flow approximations. Because all formulations can be solved using standard software packages, only models are presented, which is a departure from most books on power systems. It is a major perk of convex optimization that the user often does not need to program an algorithm to proceed.

I should comment that this book is not an up-to-date exposition of power system applications or optimization theory and that, inevitably, many important topics in both fields have been omitted. My intention has rather been to bridge modern convex optimization and power systems in a rigorous manner. While I have attempted to be mathematically self-contained, the pace assumes an advanced undergraduate level of mathematical exposure (linear algebra, calculus, and some probability) as well as familiarity with power systems and optimization. This book could be used in a course on power system optimization or as a mathematical supplement to a course in power system design, operation, or economics. It is my hope that it will also prove useful to researchers in power systems with an interest in optimization and vice versa, and to industry practitioners seeking firm foundations for their optimization applications.

Recent history

Streetlights, subways, the Internet, this book you are reading now – it is difficult to imagine life without such amenities, all enabled by electric power. To support such a vast set of technologies, electric power systems have grown into some of the most complex and expensive machines in existence. While much of this growth resembles an organic process more than deliberate design, the advent of computing is enabling us more and more to direct the evolution of power systems toward greater efficiency, reliability, and versatility.

At the time of writing, the complexity of power systems is poised to take off. This is largely due to shifts toward renewable energy production and the active involvement of power consumers through demand response, as well as our still-developing handle on economic deregulation. To meet these challenges, new computational tools will be developed, and the most ubiquitous computation in power systems is optimization. An objective of this book is to simplify and unify various topics in power system optimization so as to provide a firm foundation for future developments.

At the heart of most power system optimizations are the equations of the steady-state, single-phase approximation to alternating current power flow in a network. Well-known problems like optimal power flow, reconfiguration, and transmission planning all consist of details layered on top of power flow. Nodal prices, a core component of electricity markets, are obtained from the dual of optimal power flow. It is therefore most unfortunate that the power flow equations are nonconvex, making all of these optimizations extremely difficult. We are thus faced with a tradeoff between realistic models that are too hard to solve at practical scales and tractable approximations.

For many years, linear programming (LP) was the most general efficiently solvable optimization class, and so many large-scale power system models were based on linear power flow approximations or even simpler descriptions like network flow or a real power balance. At the other extreme, a number of nonlinear programming (NLP) algorithms were developed for exact, nonconvex models.

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.