# Electricity markets: modelling, optimization and simulation (completed)

The project **Electricity markets: modelling, optimization and simulation,** acronymed **EMMOS** is funded by the Norwegian Research Council under the EVITA program. EMMOS is running for 4 years, starting from January 1, 2011, and focuses on development and analysis of electricity markets. The project leader is professor Fred Espen Benth.

Hydro power is important in the Nordic electricity market. Illustration by Colourbox.no

## About the project

Futures contracts in energy markets are used for hedging price risk. Norwegian electricity producers use futures markets at the Nordic power exchange NordPool to insure their production, while electricity retailers and the metal industry use futures contracts to hedge the risk of sudden price spikes. Norway’s gas production generates large revenues from export to the UK and European gas markets. The optimal risk management for these market participants is a fundamental problem which calls for an integrated use of advanced mathematics and numerical analysis.

The traded futures contracts deliver energy over different time periods, like for instance a specified month, quarter or a year. Furthermore, the energy markets are integrated (Nordic electricity with continental markets like the German EEX, and gas markets with electricity, say). This poses demanding modelling questions for describing the stochastic dynamics of price required for managing risk when operating in these markets.

The approach to model (cross-)commodity futures markets is to use the Heath-Jarrow-Morton approach adopted from interest-rate theory. One specifies a (multi-)dimensional stochastic process as the solution of a stochastic differential equation with values in a function space. This space will consist of functions of “time-to-delivery” *x*, and one derives the prices for energy futures by aggregation to get a price dynamics in *x* and “length-of-delivery” *y*.

The classical models are typically based on finite-dimensional Brownian motion noise in the dynamics. Empirical studies points strongly towards non-Gaussian models. But more importantly, there is a high degree of idionsyncratic risk in the market, meaning that different contracts have a significant specific risk not shared with other. Hence, it is natural to consider noise processes being truly infinite dimensional. Some papers have suggested such a dynamics, but very little is known on the extension to energy markets, and the incorporation of many markets. Other characteristics of energy markets are strong seasonality in price variations and the occurrence of spikes in the spot dynamics which affects futures prices to some degree.

The analysis of various risk management problems can typically be formulated as stochastic control problems, where one tries to find an investment strategy in the different available contracts under some criterion for optimality. Such criterions may be given by utility functions measuring the risk aversion of the company, or some risk measure like conditional value-at-risk. Given the complex structure of the market, implying highly non-trivial stochastic dynamics of the futures prices, such stochastic control problems will call for new theory to prove the existence and uniqueness of solutions. In particular, we will have to work with dynamic programming principles and Hamilton-Jacobi-Bellman (HJB) equations in function (Hilbert) spaces, a theory which is not very well-developed.

Although the modelling and mathematical analysis of the optimization problems are interesting n itself, we aim at also understanding the practical issues in this project. This calls for the development of numerical algorithms for implementing the risk management in practice. Rarely, stochastic control problems allow for analytic solutions. In addition, since we will have a formulation of the price processes living in function spaces, the optimal solutions will be general and not implementable in practice. Hence, we search for finite-dimensional approximations which lead to practically implementable strategies. Two main approaches will be investigated. The first will be based on finite-dimensional approximation of the price processes, and Monte Carlo methods for stochastic control. Essentially, this involves, among other things, simulation schemes for first-order stochastic partial differential equations.The second will involve development of numerical schemes for HJB equations. The idea here is to represent the solution in terms of basis functions of the function space, and then use numerical schemes for solving finite-dimensional HJB-equations.

## Objectives

The project aims at developing infinite dimensional stochastic processes for modelling and analysis of the term structures (futures markets) in energy. Investment strategies for optimal management of risk will be derived, based on trading in energy futures and options. The study of these investment decisions can be formalized as stochastic control problems, and we will be concerned with both the theoretical analysis and the practical implementation. This involves development of theory in mathematics and numerical analysis. On the theoretical side, new methods for stochastic control in infinite dimensions, including dynamic programming principles and Hilbert-space valued Hamilton-Jacobi-Bellman (HJB) equations. On the numerical side, we will focus on Monte Carlo based simulation methods of infinite dimensional stochastic processes, including stochastic partial differential equations, and numerical methods for Hilbert-space valued HJB equations. For the latter, we will use finite-dimensional approximations combined with numerical schemes for partial differential equations. The theoretical and numerical developments will have practical problems and solutions in view.

## Outcomes

## Financing

THe EMMOS project is financed by the Norwegian Research Council under the EVITA program.

## Cooperation

Professor Rudiger Kiesel and professor Steen Koekebakker are active research partners on the proejct. Dr. Fridthjof Ollmar of Agder Energy is partner from the industry.