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We present an approximate method for solving nonlinear control problems overlong time horizons, in which the full nonlinear model is preserved over aninitial part of the horizon, while the remainder of the horizon is modeledusing a linear relaxation. As this approximate problem may still be too largeto solve directly, we present a Benders decomposition-based solution algorithmthat iterates between solving the nonlinear and linear parts of the horizon.This extends the Dual Dynamic Programming approach commonly employed foroptimization of linearized hydro power systems. We prove that the proposedalgorithm converges after a finite number of iterations, even when thenonlinear initial stage problems are solved inexactly. We also bound thesuboptimality of the split-horizon method with respect to the originalnonlinear problem, in terms of the properties of a map between the linear andnonlinear state-input trajectories. We then apply this method to a case studyconcerning a multiple reservoir hydro system, approximating the nonlinear headeffects in the second stage using McCormick envelopes. We demonstrate thatnear-optimal solutions can be obtained in a shrinking horizon setting when thefull 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 bothconventional dynamic programming and a multi-cell McCormick envelopeapproximation from literature.

Two-Stage Dual Dynamic Programming with Application to Nonlinear Hydro Scheduling