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

## TL;DR

This paper introduces a two-stage dual dynamic programming method combining nonlinear and linear models for long-term hydro scheduling, with proven convergence and practical advantages over existing approaches.

## Contribution

It extends dual dynamic programming to nonlinear hydro problems using a Benders decomposition approach with convergence guarantees.

## Key findings

- Near-optimal solutions achieved with short nonlinear initial stage
- Method outperforms conventional dynamic programming and existing McCormick envelope approaches
- Bounded suboptimality related to linear-nonlinear trajectory mapping

## Abstract

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.