Preface

The approaches of large-scale system optimization have long been applied to power system planning and operation, and there is extensive literature on such optimization. On the other hand, optimization is also the basic tool for electricity markets, and is often used with microeconomic models. However, people seldom look at physical power systems and economic market systems in microeconomics from a unified system point of view. In fact, both are large-scale distributed systems, and there are intrinsic connections between optimization approaches of power systems and microeconomics (Figure 0.1). In general, a power system (an engineering system composed of generators, loads, and transmission lines) and a microeconomic system (a social system composed of producers, consumers, and markets) have many common characteristics, such as the following:

1. they both consist of subsystems interconnected together,
2. more than one controller or decision-maker is present, resulting in decentralized computations,
3. coordination between the operation of the different controllers is required, resulting in hierarchical structures, and
4. correlated but different information is available to the controllers.

Figure 0.1 Analogy between a power system and a market system.

Many optimization approaches have been developed for power system planning and operation, such as linear programming, nonlinear programming, integer programming, and mixed integer programming. Decomposition and coordinationtechniques such as Dantzig-Wolfe decomposition, Benders' decomposition, and Lagrangian relaxation are often used. On the other hand, mathematical optimization is essential to modern microeconomics, which is the theory about optimal resource allocation, defined as "the study of economics at the level of individual consumers, groups of consumers, or firms . The general concern of microeconomics is the efficient allocation of scarce resources between alternative uses but more specifically it involves the determination of price through the optimizing behavior of economic agents, with consumers maximizing utility and firms maximizing profit" (from the Economist's Dictionary of Economics). Because the market system can also be regarded as a large-scale system containing many subcomponents (buyers and sellers), the decomposition and coordination principle are also adopted. Then a unified view of optimization for power systems/electricity markets can be established from the large-scale complex systems perspective. This is the starting point of this book.

Here, as an example, we take the unit commitment (UC) problem, which is a classic optimization problem in power system operation. Consider a thermal power system with units. It is required to determine the start-up, shut-down, and generation levels of all units over a specified time horizon . The objective is to minimize the total cost subject to system demand and spinning reserve requirements, and other individual unit constraints. The notation to be used in the mathematical model is defined as follows:

The objective function of UC is to minimize the total generation and start-up cost:

The system power balance constraint is

The individual unit constraints include: unit generation limit, minimum up/down-time, ramp rate, unit spinning reserve limit, etc.

Here we only give a simplified model description, and the detailed formulation of UC will be given in the later chapters.

Different solution methods, such as priority list, dynamic programming, mixed integer programming, and Lagrangian relaxation, have been proposed by researchers. We take the Lagrangian relaxation method as an example. The basic idea of Lagrangian relaxation is to relax the systemwide constraints, such as the power balance constraint, by using Lagrange multipliers, and then to decompose the problem into individual unit commitment subproblems, which are much easier to solve. Lagrangian relaxation can overcome the dimensional obstacle and get quite good suboptimal solutions. By using the duality theory, the systemwide constraint (here referring to the power balance constraint) of the primal problem is relaxed by the Lagrangian function (3). Then the two-level maximum-minimum optimization framework shown in Figure 0.2 is formed. The low-level problems (4) solve the optimal commitment of each individual unit. The high-level problem (5) optimizes the vector of Lagrange multipliers, and a subgradient optimization method is often adopted. When is passed to the subproblems, each individual unit will optimize its own production , namely to minimize its cost or maximize its profit. In this procedure, serves as the function of market prices to coordinate the production of all units to reach the requirement of system demand. The optimization of Lagrange multipliers is in fact the price adjustment process in the market.

Figure 0.2 Illustration of Lagrangian relaxation.

The Lagrangian function is

where is the Lagrange multiplier associated with demand at time .

The individual unit subproblems are

subject to all individual unit constraints.

The high-level dual problem is

We can compare this optimization procedure with a market economy. Consider an economy with agents and commodities . A bundle of commodities is a vector . Each agent has an endowment and a utility function . These endowments and utilities are the primitives of the exchange economy, so we write . Agents are assumed to take as given the market prices for the goods. The vector of market prices is ; all prices are nonnegative.

Each agent chooses consumption to maximize his/her utility given his/her budget constraint. Therefore, agent solves

The consumer's "wealth" is , the amount he/she could get if he/she sold his/her entire endowment. We can write the budget set as

A key concept of a market system is equilibrium. Market equilibrium refers to a condition where a market price is established through competition such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers. There are two kinds of equilibrium considered in microeconomics, namely, competitive equilibrium and Nash equilibrium.

A competitive (or Walrasian) equilibrium for the economy is a vector such that the following hold.

1. Agents maximize their utilities: for all , 8
2. Markets clear: for all , 9

The above model (6) and (9) is apparently a decentralized large-scale optimization model, which is similar in form to power system optimization problems such as the above-mentioned unit commitment. Clearly, we can see that the utility maximization problem (6) of each agent corresponds to the individual unit subproblem (4) except for the opposite sign. At the solution of the high-level dual problem (5) or the primal problem (1), the items with Lagrange multiplier

will tend to zero, and this is just the market clearing condition (9).

The Nash equilibrium is widely used in economics as the main alternative to competitive equilibrium. It is used whenever there is a strategic element to the behavior of agents and the "price taking" assumption of competitive equilibrium is inappropriate. Nash equilibrium is a core concept of game theory, which is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. The mathematical approaches of game theory belong to another kind of decentralized optimization, which also has analogs in power system optimization.

In fact, from the perspective of large-scale system optimization, we shall find in later chapters that the solution method of competitive equilibrium is related to the interaction balance method (or nonfeasible method) and the solution method of Nash equilibrium is related to the interaction prediction approach (or feasible method) of large-scale systems.

The authors' work over a decade has focused on the application of large-scale optimization to power system planning and operation, and also on the application of microeconomics and game theory to electricity markets. The authors have made significant achievements in these research areas. Based on previous research, this book will make a more systematic investigation on large-scale complex systems approaches to power system optimization. The authors believe...