A moment and sum-of-squares extension of dual dynamic programming with application to nonlinear energy storage problems

We present a finite-horizon optimization algorithm that extends the established concept of Dual Dynamic Programming (DDP) in two ways. First, in contrast to the linear costs, dynamics, and constraints of standard DDP, we consider problems in which all of these can be polynomial functions. Second, we allow the state trajectory to be described by probability distributions rather than point values, and return approximate value functions fitted to these. The algorithm is in part an adaptation of sum-of-squares techniques used in the approximate dynamic programming literature. It alternates between a forward simulation through the horizon, in which the moments of the state distribution are propagated through a succession of single-stage problems, and a backward recursion, in which a new polynomial function is derived for each stage using the moments of the state as fixed data. The value function approximation returned for a given stage is the point-wise maximum of all polynomials derived for that stage. This contrasts with the piecewise affine functions derived in conventional DDP. We prove key convergence properties of the new algorithm, and validate it in simulation on two case studies related to the optimal operation of energy storage devices with nonlinear characteristics. The first is a small borehole storage problem, for which multiple value function approximations can be compared. The second is a larger problem, for which conventional discretized dynamic programming is intractable.