power production management with renewables: market modelling and risk analysis

In this project we have focused on the production of Hourly Price Forward Curves (HPFCs), to model the electricity prices with delivery at some point in the future. We focus mainly on the German electricity market, but the results are applicable for other markets as well. The HPFC gives you the price of electricity on an hourly basis as a function of the currently traded Futures products, typically months, quarters and years. We split the project into three parts: First we focus on how the HPFCs have been constructed in the literature, where we have focused on two different methods. We also propose a new method based on cosntrained least squares optimization used on a trigonometric spline curve. Thereafter, we test these curves against each other to see which method is the best, and under which circumstances. We conclude that all the methods have its strengths and weaknesses, and we give some discussion to how we can use these results to construct one optimal curve. In the second part we focus on how the HPFCs change in time as we come closer to delivery of electricity and the input (FUtures prices) used for construction of these curves change. We focus on two changes, either the price of an observed Futures product is changing, or an observed Futures product is split into two or more products with shorter delivery periods. We observe that a price change in the observed products is similary taken care of in all methods, while the splitting of Futures products give rise to so-called arbitrage opportunities in one of the methods. This means we can construct theoretical risk free trading strategies that will give a certain pay-off. As a last part we use the previous results to construct a stochastic model for the HPFC. This gives us a probability distribution for the price of delivery with delivery at a certain point in the future for arbitrary timepoints between now and delivery. To do this, we assume a probability distribution for the traded Futures contracts, and from use the knowledge we found earlier between how these contracts and the HPFC is connected to construct a probability distribution for the HPFC. To be able to take the splitting of Futures contracts into account we only focus on so called independent and identically distributed distributions. We find that only the normally distributed random variables fit all of our criterias. The results from these thesis give insights in how to construct HPFCs for electricity markets, but the results can also be used for other commodities where Futures contracts are traded in a similar mather, such as oil and gass.

The project will focus on computations relevant to HPFCs(Hourly price forward curves), which is the major tool used for computations concerning power markets. This project aims specifically to develop a methodology that takes into the account the effect of renewable energy sources to these curves. Since renewable energy(e.g. wind and photovoltaic energy) is hard to control, this will lead to high volatility, which in turn will lead to highly volatile prices, so a lot of the work will be concerning the r isk this impose. This project will give a better understanding of the risk and problems concerning renewable energy, which in turn will give a better use and production of such energy sources. The problems concerning this will project will be divided i nto modelling and computational problems. In the modelling part, it might be hard to incorporate the renewables in an appropriate way, and we need to identify the different characteristics concerning the different types of energy sources, but all these pr oblems should be minor, as the project will focus on the computations. For computations it will be several problems, we will first need to specify a realistic model, here problems will be to assume the right kind of model. We will also need to fit data t o this model, which will use different types of regression methods. At last we will need to make use of numerical methods, so we can solve these problems in appropriate time. The findings of this project will lead to better understanding of renewables as an energy source, which will make it possible to utilize these better. Also the findings could potentially be used in other parts of mathematics, where stochastic analysis, model fitting and numerical problems are used(like for other financial products l ike stocks, interest rates etc.).