[Paper Review] Optimal control of storage incorporating market impact and with energy applications

James R. Cruise, Lisa Flatley+2 more|arXiv (Cornell University)|Jun 13, 2014

Smart Grid Energy Management28 references12 citations

TL;DR

This paper develops a strong Lagrangian framework for optimal control of energy storage systems under market impact, incorporating nonlinear cost functions, capacity and rate constraints, and time-varying inefficiencies. The key contribution is an efficient, horizon-localized algorithm that determines optimal control and Lagrange multipliers—enabling scalable, real-time management of storage over long time horizons with near-optimal performance in stochastic settings.

ABSTRACT

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 inefficiencies or losses in the storage process. We develop an algorithm which is maximally efficient in the sense that it incorporates the result that, at each point in time, the optimal management decision depends only a finite, and typically short, time horizon. We give examples related to the management of a real-world system. Finally we consider a model in which the associated costs evolve stochastically in time. Our results are formulated in a perfectly general setting which permits their application to other commodity storage problems.

Motivation & Objective

• To develop a rigorous mathematical framework for optimal control of energy storage that accounts for market impact, capacity limits, and rate constraints.
• To establish the economic interpretation of Lagrange multipliers associated with capacity and rate constraints for proper storage dimensioning.
• To design a computationally efficient algorithm that determines optimal control decisions based on a short, dynamically identified future time horizon.
• To extend the deterministic framework to a pragmatic, near-optimal approach for stochastic cost environments through sequential re-optimization.
• To generalize the model for application beyond electricity storage to other commodity storage and energy management problems.

Proposed method

• Formulates the storage control problem as a convex optimization problem with nonlinear, time-dependent cost functions and market impact.
• Applies strong Lagrangian duality theory to derive necessary and sufficient optimality conditions, with Lagrange multipliers providing economic insights.
• Develops a sequential algorithm that computes optimal control and multipliers at each time step using only a finite, identifiable future horizon.
• Introduces a time-horizon identification mechanism that ensures the shortest possible look-ahead while preserving optimality.
• Adapts the deterministic solution to stochastic settings via a re-optimization strategy using a time-invariant approximation of future costs.
• Uses a state transformation to map the stochastic problem into a deterministic equivalent, enabling application of the original algorithm.

Experimental results

Research questions

• RQ1 How can optimal storage control be formulated when the store’s actions impact market prices, leading to nonlinear cost functions?
• RQ2 What is the role of Lagrange multipliers in determining the economic dimensioning of storage capacity and rate constraints?
• RQ3 Can an efficient algorithm be designed that limits decision-making to a short, dynamically determined time horizon, even over long planning horizons?
• RQ4 How can the deterministic optimal control framework be adapted for use in a stochastic cost environment with practical computational feasibility?
• RQ5 What is the impact of storage inefficiencies and time-varying price dynamics on the structure of optimal control policies?

Key findings

• The Lagrange multipliers for capacity and rate constraints have direct economic interpretations, enabling optimal dimensioning of storage facilities.
• The optimal control policy is local in time: at each decision point, only a finite, identifiable future horizon is required, and this horizon is the shortest possible.
• The algorithm computes optimal control and multipliers sequentially, with computational complexity independent of the total time horizon.
• The framework accounts for both time-dependent and time-independent inefficiencies, as well as leakage and bid-ask spreads in the storage process.
• In stochastic settings, the proposed re-optimization approach based on a time-invariant approximation yields near-optimal performance and is computationally feasible.
• The theoretical results are general and extendable to other commodity storage and energy management problems beyond electricity.