Risk-Limiting Dynamic Contracts for Direct Load Control
  This paper proposes a novel continuous-time dynamic contract framework that has a risk-limiting capability. If a principal and an agent enter into such a contract, the principal can optimally manage its performance and risk with a guarantee that the agent's risk is less than or equal to a pre-specified level and that the agent's expected payoff is greater than or equal to another pre-specified threshold. We achieve such risk-management capabilities by formulating the contract design problem as mean-variance constrained risk-sensitive control. A dynamic programming-based method is developed to solve the problem. The key idea of our proposed solution method is to reformulate the inequality constraints on the mean and the variance of the agent's payoff as dynamical system constraints by introducing new state and control variables. The reformulations use the martingale representation theorem. The proposed contract method enables us to develop a new direct load control method that provides the load-serving entity with financial risk management solutions in real-time electricity markets. We also propose an approximate decomposition of the optimal contract design problem for multiple customers into multiple low-dimensional contract problems for one customer. This allows the direct load control program to work with a large number of customers without any scalability issues. Furthermore, the contract design procedure can be completely parallelized. The performance and usefulness of the proposed contract method and its application to direct load control are demonstrated using data on the electric energy consumption of customers in Austin, Texas as well as the Electricity Reliability Council of Texas' locational marginal price data. 
Insoon Yang, Duncan S. Callaway, and Claire J. Tomlin
math.OC