Optimal control of storage incorporating market impact and with energy applications

Large scale electricity storage is set to play an increasingly important role in the management of future energy networks. A major aspect of the economics of such projects is captured in arbitrage, i.e. buying electricity when it is cheap and selling it when it is expensive. We consider a mathematical model which may account for nonlinear|and possibly stochastically evolving|cost functions, market impact, input and output rate constraints and both time-dependent and time-independent ineciencies or losses in the storage process. Our main concern is to develop the associated strong Lagrangian theory. The Lagrange multipliers associated with the capacity constraints in particular have important economic interpretations with regard to the dimensioning of storage|both with respect to its capacity and its rate constraints| and prove key to the ecient control of a store. We also develop an algorithm which determines, sequentially in time, both these Lagrange multipliers and the optimal control. This algorithm further identies, for each point in time, a time horizon beyond which it is not necessary to look in order to identify the optimal control at that point; this horizon is furthermore the shortest such. The algorithm is thus particularly suitable for the management of storage over extended periods of time. We give examples related to the management of real-world systems. Finally we consider a pragmatic approach to the real-time management of storage in a stochastic cost environment, which is computationally feasible, optimal under certain ideal conditions, and which may in general be expected to perform close to optimally. Our results are formulated in a general setting which permits their application to other energy management problems, and to other commodity storage problems.