Investigation of Game-Theoretic Mechanisms for the Valuation of Energy Resources

Electricity systems are facing the pressure to change in response to the effects of new technology, particularly the proliferation of renewable technologies (such as solar PV systems and wind generation) leading to the retirement of traditional generation technologies that provide stabilising inertia. These changes create an imperative to consider potential future market structures to facilitate the participation of distributed energy resources (DERs; such as EVs and batteries) in grid operation. However, this gives rise to general questions surrounding the ethics of market structures and how they could be fairly applied in future electricity systems. Particularly the most basic question "how should electricity be valued and traded" is fundamentally a moral question without any easy answer. We give a survey of philosophical attitudes around such a question, before presenting a series of ways that these intuitions have been cast into mathematics, including: the Vickrey-Clarke-Groves mechanism, Locational Marginal Pricing, the Shapley Value, and Nash bargaining solution concepts. We compared these different methods, and attempted a new synthesis that brought together the best features of each of them; called the 'Generalised Neyman and Kohlberg Value' or the GNK-value for short. The GNK value was developed as a novel bargaining solution concept for many player non-cooperative transferable utility generalised games, and thus it was intrinsically flexible in its application to various aspects of powersystems. We demonstrated the features of the GNK-value against the other mathematical solutions in the context of trading the immediate consumption/generation of power on small sized networks under linear-DC approximation, before extending the computation to larger networks. The GNK value proved to be difficult to compute for large networks but was shown to be approximable for larger networks with a series of sampling techniques and a proxy method. The GNK value was ethically compared to other mechanisms with the unfortunate discovery that it allowed for participants to be left worse-off for participating, violating the ethical notion of 'euvoluntary exchange' and 'individual rationality'; but was offered as an interesting innovation in the space of transferable utility generalised games notwithstanding. For sampling the GNK value, there was a range of new and different techniques developed for stratified random sampling which iteratively minimise newly derived concentration inequalities on the error of the sampling. These techniques were developed to assist in the computation of the GNK value to larger networks, and they were evaluated in the context of sampling synthetic data, and in computation of the Shapley Value of cooperative game theory. These new sampling techniques were demonstrated to be comparable to the more orthodox Neyman sampling method despite not having access to stratum variances

An engineered empirical Bernstein bound

We derive a tightened empirical Bernstein bound (EBB) on the variation of the sample mean from the population mean, and show that it improves the performance of upper confidence bound (UCB) methods in multi-armed bandit problems. Like other EBBs, our EBB is a concentration inequality for the variation of the sample mean in terms of the sample variance. Its derivation uses a combination of probability unions and Chernoff bounds for the mean of samples and mean of sample squares. Analysis reveals that our approach can tighten the best existing EBBs by about a third, and thereby halves the distance to a bound constructed with perfect variance information. We illustrate the practical usefulness of our novel EBB by applying it to a multi-armed bandit problem as a component of a UCB method. Our method outperforms existing approaches by producing lower expected regret than variants of UCB employing several other bounds, including state-of-the-art EBBs