  This work considers a short-term, continuous time setting characterized by a
coupled power supply system controlled exclusively by a single provider and
comprising a cascade of hydropower systems (dams), fossil fuel power stations,
and a storage capacity modeled by a single large battery. Cascaded hydropower
generators introduce time-delay effects in the state dynamics, which are
modeled with differential equations, making it impossible to use classical
dynamic programming. We address this issue by introducing a novel Lagrangian
relaxation technique over continuous-time constraints, constructing a nearly
optimal policy efficiently. This approach yields a convex, nonsmooth
optimization dual problem to recover the optimal Lagrangian multipliers, which
is numerically solved using a limited memory bundle method. At each step of the
dual optimization, we need to solve an optimization subproblem. Given the
current values of the Lagrangian multipliers, the time delays are no longer
active, and we can solve a corresponding nonlinear Hamilton--Jacobi--Bellman
(HJB) Partial Differential Equation (PDE) for the optimization subproblem. The
HJB PDE solver provides both the current value of the dual function and its
subgradient, and is trivially parallelizable over the state space for each time
step. To handle the infinite-dimensional nature of the Lagrange multipliers, we
design an adaptive refinement strategy to control the duality gap. Furthermore,
we use a penalization technique for the constructed admissible primal solution
to smooth the controls while achieving a sufficiently small duality gap.
Numerical results based on the Uruguayan power system demonstrate the
efficiency of the proposed mathematical models and numerical approach.
