Optimization Methods for Gas and Power Markets

In the Preface, we mentioned that optimization problems represent a class of mathematical problems that we cannot consider always homogeneous. Different kinds of problems ask for different representation and solution tools. Hence, a bit of classification may be worthwhile for the sake of clarity.

Optimization is the branch of mathematics which faces the problem of selecting a best element (with respect to some criteria) from some set of available alternatives. A general optimization problem can be represented in the following way: where f maps the elements of a set A to the set of real numbers and X ⊆ A is the set of choices. The simplest optimization problem consists in maximizing a real function over an allowed set. By convention, here the standard form of an optimization problem is stated in terms of maximization, but it is always possible to switch from minimization to maximization using the relationship Optimization problems are commonly divided into classes, depending on the mathematical properties of the function f and the geometrical form of X. For every class of problems suitable optimization methods have been developed. When f is a linear function of x, and X can be described using linear (in)equalities, then the problem is said to be a Linear Optimization Problem. This is the subject of Section 2.1. When f is not linear, but constraints satisfy some regularity conditions, then the algorithm to solve the optimization problems differs from the linear case. This is the subject of Section 2.2. Finally, when the quantity we want to optimize is uncertain (for example the profit and loss of a trading position on a stock market), then the optimization problem is called stochastic and the solution methods are slightly different from the deterministic case. This is the subject of Section 2.5.

In the analysis of this business case, we will move from static optimization problems to dynamic ones, still, obviously, characterized by uncertainty.

Virtual Power Plant Contracts (VPP) are energy-structured products built up to replicate the payoff of a real power plant asset. These kinds of structured products are mainly used to hedge the risk exposure of a complex energy portfolio made up of different (for technology, location and efficiency) power generation assets in a more effective way compared with those reachable through standard products like forwards or plain vanilla options.

Traditionally, day-ahead markets have been considered the spot part of electricity markets. The trading and price formation mechanism of day-ahead markets were and are pretty much homogeneous among different power markets around the world. They are auction-based markets with a system marginal price formation mechanism. All the 24 hours of the day following the auction date can be traded, independently or in blocks. However, in recent years the impressive penetration of non-programmable renewable energy sources in many countries has introduced inefficiency into the day-ahead market framework. Wind and solar generation units are not typically able to forecast exactly their production 36–24 hours in advance, hence for them a day-ahead market is not sufficient to avoid dangerous unbalancings. Figure 8.1 displays forecast error reduction for wind generation in Germany as the forecast time horizon reduces.