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Real Options Analysis for physical trading in electricity
[Thesis]. Manchester, UK: The University of Manchester; 2018.
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Abstract
This thesis implements a quantitative study of various reserve contracts in an energy balancing context. Under any of these contracts, the owner of a storage device, such as a battery, helps smooth fluctuations in electricity demand and supply by using the device to increase either electricity consumption or generated electricity in the network. We formulate appropriate problems in the different chapters which require the mathematically modelling of the corresponding variables. This means that throughout the whole thesis we assume that the imbalance process evolves as a stochastic process and the electricity price in the network can be derived from the instantaneous imbalance level. In each chapter we study the problem of calculating the so called "real value" of related optimal trading strategies. We consider both analytical and numerical treatments of the problems. In order to derive analytical results we use some simplifying assumptions; the imbalance process is a Wiener process and the electricity price is a capped linear transformation of the imbalance value. Furthermore the contract considered in this analytical work contains American-style put options with electricity as the underlying. Thus the following problem is formulated for the battery owner: determine the optimal time to enter the contract and, if necessary, the optimal time to discharge electricity before entering the contract. The previously stated problem is formulated as one of optimal stopping, and is solved explicitly in terms of the model parameters and instantaneous values of the power system imbalance. The optimal operational strategies thus obtained ensure that the battery owner has positive expected economic profit from the contract. Furthermore, they provide explicit conditions under which the optimal discharge time is consistent with the overall objective of power system balancing. For the numerical study we relax some tight assumptions and derive a realistic framework with the help of UK electricity market data. The imbalance price is modelled as a more general Ornstein\textendash Uhlenbeck process, furthermore the electricity price is a time and imbalance dependent image of the instantaneous imbalance. The proposed trading strategy for this numerical part includes the offer of American-style put or call options. This means that the storage operator has now the possibility to optimally decide which option to offer, with the restriction of having the storage device in the appropriate mode at any time. The system operator accepts these offers as possible long-term real-time balancing resorts. Using the relation with variational inequalities, we calculate the real value of the optimal trading strategy for the storage operator and at the same time we calculate the balancing cost of this resort compared to a so called "target cost" for the system operator. These results reveal that with proper parameter choices, mutual benefit is available, i.e. a financial profit for the storage operator whilst the balancing cost can also be reduced for the electricity system operator. Our results are illustrated via numerical calculations, which also contain optimal operational strategies. In the subsequent chapters we propose various extensions to the framework with the aim to obtain more realistic models. We consider a situation in which there are multiple options with different parameter values, among which the storage operator can choose the optimal one to offer. We also allow the system operator not to accept unconditionally any of the offered options, instead we establish a likelihood function to mathematically express the preference of the system operator towards the different options. The exercise decision of the system operator has been modelled in the early chapters via a fixed threshold strategy. We propose a more sophisticated strategy than this fixed threshold one by modelling the exercise decision using Poisson processes. We also consider an extension of the underlying imbalance process in which it is modelled by Jump Added Ornstein\textendash Uhlenbeck processes. With the help of statistical tests we are able to we examine whether or not including jumps yields a better modelling of historical UK imbalance process values, and how sensitive the contract results are on the actual imbalance process choice. Using any of the aforementioned model extensions to conclude that the main contribution of the numerical study still holds, meaning that with proper parameter choices, mutual benefit is available.