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This paper proposes a novel continuous-time dynamic contract framework thathas a risk-limiting capability. If a principal and an agent enter into such acontract, the principal can optimally manage its performance and risk with aguarantee that the agent's risk is less than or equal to a pre-specified leveland that the agent's expected payoff is greater than or equal to anotherpre-specified threshold. We achieve such risk-management capabilities byformulating the contract design problem as mean-variance constrainedrisk-sensitive control. A dynamic programming-based method is developed tosolve the problem. The key idea of our proposed solution method is toreformulate the inequality constraints on the mean and the variance of theagent's payoff as dynamical system constraints by introducing new state andcontrol variables. The reformulations use the martingale representationtheorem. The proposed contract method enables us to develop a new direct loadcontrol method that provides the load-serving entity with financial riskmanagement solutions in real-time electricity markets. We also propose anapproximate decomposition of the optimal contract design problem for multiplecustomers into multiple low-dimensional contract problems for one customer.This allows the direct load control program to work with a large number ofcustomers without any scalability issues. Furthermore, the contract designprocedure can be completely parallelized. The performance and usefulness of theproposed contract method and its application to direct load control aredemonstrated using data on the electric energy consumption of customers inAustin, Texas as well as the Electricity Reliability Council of Texas'locational marginal price data.

Risk-Limiting Dynamic Contracts for Direct Load Control