Méthodes numériques appliquées à la programmation dynamique stochastique pour la gestion d’un système hydroélectrique

RÉSUMÉ : La Programmation Dynamique Stochastique (PDS) est une méthode couramment utilisée pour la gestion de petits systèmes hydroélectriques. La PDS décompose le problème principal en une succession de petits problèmes non linéaires à résoudre. Néanmoins, son utilisation peut être problématique dans un contexte opérationnel en raison du temps de calcul important requis pour résoudre tous ces sous-problèmes. L’objectif du projet de recherche consiste donc à améliorer sa vitesse d’exécution en proposant une nouvelle méthode de résolution des sous-problèmes générés par la PDS. Cette approche combine deux méthodes d’optimisation non linéaire, soit une méthode de Programmation Linéaire Séquentielle (PLS) pour la phase de récursion de la PDS et une méthode de points intérieurs pour la simulation. La méthode proposée sera comparée à trois grandes familles de méthodes d’optimisation non linéaires avec contraintes : les méthodes du lagrangien augmenté, les méthodes séquentielles et les méthodes de points intérieurs. Pour évaluer convenablement l’efficacité de chacune de ces méthodes, les logiciels d’optimisation les plus établis sont utilisés dans cette analyse. Par contre, presque aucune implémentation de la méthode de la PLS n’a été trouvée dans la littérature. Une partie de ce travail sera consacrée à l’implémentation de cet algorithme. Les résultats numériques sont obtenus par l’application de la PDS sur le système hydroélectrique du Saguenay-Lac-Saint-Jean, opéré par la compagnie Rio Tinto. Étant donné la tendance linéaire des fonctions de production sur une majeure partie de leur domaine, il est possible d’approcher efficacement ces dernières par une approximation linéaire par morceaux. Ainsi, une approche hybride combinant la PLS et le logiciel IPOPT s’avère une solution efficace autant pour le temps de calcul que la qualité de la solution.----------ABSTRACT : Stochastic Dynamic Programming (SDP) is a common approach used for hydropower systems management. SDP decomposes the main problem by a sequence of small nonlinear subproblems. Nevertheless, its application can be difficult in an operational context because of the excessive computation time required to solve all its subproblems. The objective of this study consists of increasing SDP speed by introducing a new methodology to solve these subproblems. This approach combines two nonlinear methods, the Sequential Linear Programming (SLP) is used for the computation of policy and an interior points method for the simulation. The method proposed will be evaluated to the main nonlinear constraints methods: augmented Lagrangian, sequential methods, and interior points methods. To adequately compare their efficiency, state-of-the-art solvers are used in the analysis. Few established implémentations of SLP algorithm have been found in the literature. Thus, a part of this work will be an implementation of this method. Numerical results are obtained by SDP application on the Saguenay-Lac-Saint-Jean hydropower system, operated by the company Rio Tinto. Considering that hydropower functions exhibit a linear behavior over large portions of the domain, it is possible to accurately approximate the function by a piecewise linear approximation. In that case, an hybrid approach combining SLP and the software IPOPT is proved to be efficient to reduce the computation time with no lack of quality solution

A fast solution approach to solve the generator maintenance scheduling and hydropower production problems simultaneously

The Generator Maintenance Scheduling Problem (GMSP) is a problem that combines a hydropower optimization problem with a scheduling problem. Both problems are known to be hard to solve and combining them leads to an even more challenging mathematical problem. Since the hydropower production functions are nonlinear, hyperplane curve fitting is used to linearize each power production function. The goal of the GMSP is to find an optimal schedule plan to decide when to shut down generators for maintenance. Therefore, one production function needs to be formulated per generator combinations leading to a rather large number of constraints. This paper demonstrates that the complexity of the problems is linked to the number of hyperplanes selected to formulate the power production functions. To accelerate the resolution of the problem, a new heuristic based on the mean square algorithm is presented to reduce the number of hyperplanes required. This heuristic substantially reduces the number of constraints and the solving time is almost ten times faster. Numerical results show that the energy produced and the generated maintenance plannings are similar for both mathematical formulations, more precisely with one hyperplane for each generator combination versus a reduced number of hyperplanes

An optimization model to maximize energy generation in short-term hydropower unit commitment using efficiency points

This paper presents a linear mixed-integer formulation to solve the short-term hydropower unit commitment problem. It uses the pair of maximum efficiency points of water discharge and the power produced at the maximum storage for all possible combinations of turbines. The goal is to maximize total energy production for all periods. The objective function is calculated using the correction between the power produced at the current volume and the maximum storage and penalizes unit start-ups. Constraints on the maximum number of turbine changes are imposed to find a viable solution. Computational results are reported for 65 instances with two powerhouses of five turbines each located in the Saguenay-Lac-St-Jean region of the province of Quebec in Canada

Quantifying the impact of scenario tree generation and reduction methods on the solution of the short-term hydroscheduling problem

This paper studies the properties of a stochastic optimization model for the short-term hydropower generation and reduction problem with uncertain inflows. The price of energy is not considered. The uncertainty of the inflows is represented using scenario trees. Backward reduction and neural gas methods are used to generate and reduce a full scenario tree. The objective of this work is to evaluate the impact of scenario tree generation and reduction methods on the solution of the optimization. First, statistical tests are done where the expected volume, the variance and the standard deviation of each scenario tree are calculated and compared. Second, operational tests are realized, where the scenario trees are used as input to the stochastic programming model and the value of the objective function and solution are evaluated and compared. The model are tested on a 14 forecasted days and for a 10 days rolling-horizon for two powerhouses with five turbines each located in the Saguenay-Lac-St-Jean region of the province of Québec in Canada