  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.
