Nash equilibria for load balancing in networked power systems

LoBaNet explores the dynamic, multiplayer, economic and operational “games” arising when energy storage and demand-side management technologies are applied to power system balancing. The operators of these technologies, as well as the entity responsible for balancing, are treated as agents within one or more markets for electricity.

By studying dynamic games we address two fundamental research questions: firstly, how the operators of such new technologies should optimally act, and secondly how they should be appropriately rewarded in order to produce a suitable dynamic equilibrium in the balancing service they can provide. Further, by appropriately extending these games to networks we explore how the dynamic equilibria change when such technologies are aggregated through third parties.

LoBaNet is linked to research on stochastic modelling in the Dynamical Systems and Statistical Physics Group and Complex Systems and Networks Group at Queen Mary, University of London. Part of this project explores the effect of multiplex and evolving network topology when, for example, participation in load balancing is infuenced by the participation of peers.

Let a power system operator and the owner of an electricity storage device (battery operator) be parties to a bilateral balancing services ontract under which the system operator chooses times and amounts of electricity to be delivered to the grid by the battery operator. The system operator must pay a fee for the amount of energy it requests each time. Between requests the battery operator may charge the store with electricity purchased in a market. The system operator would like to choose appropriate delivery times and energy amounts which achieve an optimal tradeoff between the physical imbalance (between supply and demand) and the financial cost of actions taken to minimise it (both under this contract, and for using other balancing services when the store is depleted). Simultaneously the battery operator would like to minimise the cost of its energy purchases and any penalties incurred for under-delivery at the request times. Since the store cannot deliver energy once depleted, the strategies of the two players are interdependent and so these optimisation problems combine to form a game between the battery operator and the system operator.

A multi-player version of the above game arises with the addition of third-party aggregators. Suppose now that there are n demand response operators and m ≪ n aggregators. The system operator enters bilateral contracts only with the aggregators who then subcontract with demand response and storage operators, so that the contractual relationships have a network topology. While aggregation in principle increases the reliability of the balancing service provided, it also creates a potential additional game between storage operators. In this setting new dynamic equilibria can be expected due, for example, to potential “free-loading” between storage operators contracted to the same aggregator.

By appropriately applying the mathematical theory of dynamic games and then combining it with network theory and other tools from complexity science and generalized statistical mechanics, the LoBaNet research team aims to establish the existence of dynamic equilibria in games such as Example 1 and Example 2. Further we develop the necessary practical technology, namely the numerical approaches needed to both evaluate (or approximate) and implement the resulting dynamic operational strategies. Finally we generalize the simple examples considered to a much wider context of stochastic modelling of load balance and optimization processes described by superstatistical processes and/or multiplex networks, allowing for time evolution of links and nonstationary behavior.

We closely interact with our industrial project partners (Future Decisions Ltd and Upside Energy) to make sure that the models and algorithmic technology developed are realistic, of practical relevance and potentially of commercial interest.