  We present an approximate method for solving nonlinear control problems over
long time horizons, in which the full nonlinear model is preserved over an
initial part of the horizon, while the remainder of the horizon is modeled
using a linear relaxation. As this approximate problem may still be too large
to solve directly, we present a Benders decomposition-based solution algorithm
that iterates between solving the nonlinear and linear parts of the horizon.
This extends the Dual Dynamic Programming approach commonly employed for
optimization of linearized hydro power systems. We prove that the proposed
algorithm converges after a finite number of iterations, even when the
nonlinear initial stage problems are solved inexactly. We also bound the
suboptimality of the split-horizon method with respect to the original
nonlinear problem, in terms of the properties of a map between the linear and
nonlinear state-input trajectories. We then apply this method to a case study
concerning a multiple reservoir hydro system, approximating the nonlinear head
effects in the second stage using McCormick envelopes. We demonstrate that
near-optimal solutions can be obtained in a shrinking horizon setting when the
full nonlinear model is used for only a short initial section of the horizon.
For this example, the approach is shown to be more practical than both
conventional dynamic programming and a multi-cell McCormick envelope
approximation from literature.
