# Stochastic decomposition applied to large-scale hydro valleys management

## TL;DR

This paper introduces Dual Approximate Dynamic Programming (DADP), a novel method for near-optimal control of large-scale stochastic systems like hydro valleys, overcoming the curse of dimensionality inherent in traditional dynamic programming.

## Contribution

The paper proposes DADP, an original Lagrangian relaxation-based algorithm with statistical approximation, tailored for large-scale energy management problems, especially hydro valley control.

## Key findings

- DADP achieves near-optimal solutions in large hydro valley management.
- The method outperforms the state-of-the-art SDDP in computational efficiency.
- Numerical results demonstrate the effectiveness of DADP on real-world systems.

## Abstract

We are interested in optimally controlling a discrete time dynamical system that can be influenced by exogenous uncertainties. This is generally called a Stochas-tic Optimal Control (SOC) problem and the Dynamic Programming (DP) principle is one of the standard way of solving it. Unfortunately, DP faces the so-called curse of dimensionality: the complexity of solving DP equations grows exponentially with the dimension of the variable that is sufficient to take optimal decisions (the so-called state variable). For a large class of SOC problems, which includes important practical applications in energy management, we propose an original way of obtaining near optimal controls. The algorithm we introduce is based on Lagrangian relaxation, of which the application to decomposition is well-known in the deterministic framework. However, its application to such closed-loop problems is not straightforward and an additional statistical approximation concerning the dual process is needed. The resulting methodology is called Dual Approximate Dynamic Programming (DADP). We briefly present DADP, give interpretations and enlighten the error induced by the approximation. The paper is mainly devoted to applying DADP to the management of large hydro valleys. The modeling of such systems is presented, as well as the practical implementation of the methodology. Numerical results are provided on several valleys, and we compare our approach with the state of the art SDDP method.