Optimisation stochastique des systèmes multi-réservoirs par l'agrégation de scénarios et la programmation dynamique approximative

Les problèmes de gestion des réservoirs sont stochastiques principalement à cause de l’incertitude sur les apports naturels. Ceci entraine des modèles d’optimisation de grande taille pouvant être difficilement traitables numériquement. La première partie de cette thèse réexamine la méthode d’agrégation de scénarios proposée par Rockafellar et Wets (1991). L’objectif consiste à améliorer la vitesse de convergence de l’algorithme du progressive hedgging sur lequel repose la méthode. L’approche traditionnelle consiste à utiliser une valeur fixe pour ce paramètre ou à l’ajuster selon une trajectoire choisie a priori : croissante ou décroissante. Une approche dynamique est proposée pour mettre à jour le paramètre en fonction d’information sur la convergence globale fournie par les solutions à chaque itération. Il s’agit donc d’une approche a posteriori. La thèse aborde aussi la gestion des réservoirs par la programmation dynamique stochastique. Celle-ci se prête bien à ces problèmes de gestion à cause de la nature séquentielle de leurs décisions opérationnelles. Cependant, les applications sont limitées à un nombre restreint de réservoirs. La complexité du problème peut augmenter exponentiellement avec le nombre de variables d’état, particulièrement quand l’approche classique est utilisée, i.e. en discrétisant l’espace des états de « manière uniforme ». La thèse propose une approche d’approximation sur une grille irrégulière basée sur une décomposition simpliciale de l’espace des états. La fonction de valeur est évaluée aux sommets de ces simplexes et interpolée ailleurs. À l’aide de bornes sur la vraie fonction, la grille est raffinée tout en contrôlant l’erreur d’approximation commise. En outre, dans un contexte décision-information spécifique, une hypothèse « uni-bassin », souvent utilisée par les hydrologues, est exploitée pour développer des formes analytiques pour l’espérance de la fonction de valeur. Bien que la méthode proposée ne résolve pas le problème de complexité non polynomiale de la programmation dynamique, les résultats d’une étude de cas industrielle montrent qu’il n’est pas forcément nécessaire d’utiliser une grille très dense pour approximer la fonction de valeur avec une précision acceptable. Une bonne approximation pourrait être obtenue en évaluant cette fonction uniquement en quelques points de grille choisis adéquatement.Reservoir operation problems are in essence stochastic because of the uncertain nature of natural inflows. This leads to very large optimization models that may be difficult to handle numerically. The first part of this thesis revisits the scenario aggregation method proposed by Rochafellar and Wets (1991). Our objective is to improve the convergence of the progressive hedging algorithm on which the method is based. This algorithm is based on an augmented Lagrangian with a penalty parameter that plays an important role in its convergence. The classical approach consists in using a fixed value for the parameter or in adjusting it according a trajectory chosen a priori: decreasing or increasing. This thesis presents a dynamic approach to update the parameter based on information on the global convergence provided by the solutions at each iteration. Therefore, it is an a posteriori scheme. The thesis also addresses reservoir problems via stochastic dynamic programming. This scheme is widely used for such problems because of the sequential nature of the operational decisions of reservoir management. However, dynamic programing is limited to a small number of reservoirs. The complexity may increase exponentially with the dimension of the state variables, especially when the classical approach is used, i.e. by discretizing the state space into a "regular grid". This thesis proposes an approximation scheme over an irregular grid based on simplicial decomposition of the state space. The value function is evaluated over the vertices of these simplices and interpolated elsewhere. Using bounds on the true function, the grid is refined while controlling the approximation error. Furthermore, in a specific information-decision context, a "uni-bassin" assumption often used by hydrologists is exploited to develop analytical forms for the expectation of the value function. Though the proposed method does not eliminate the non-polynomial complexity of dynamic programming, the results of an industrial case study show that it is not absolutely necessary to use a very dense grid to appropriately approximate the value function. Good approximation may be obtained by evaluating this function at few appropriately selected grid points

Note on “Parameters estimators of irregular right-angled triangular distribution”

Simple estimators were given in [3] for the lower and upper limits of an irregular right-angled triangular distribution together with convenient formulas for removing their bias. We argue here that the smallest observation is not a maximum likelihood estimator (MLE) of the lower limit and we present a procedure for computing an MLE of this parameter. We show that the MLE is strictly smaller than the smallest observation and we give some bounds that are useful in a numerical solution procedure. We also present simulation results to assess the bias and variance of the ML

Controlled approximation of the value function in stochastic dynamic programming for multi-reservoir systems

We present a new approach for adaptive approximation of the value function in stochastic dynamic programming. Under convexity assumptions, our method is based on a simplicial partition of the state space. Bounds on the value function provide guidance as to where refinement should be done, if at all. Thus, the method allows for a trade-off between solution time and accuracy. The proposed scheme is experimented in the particular context of hydroelectric production across multiple reservoirs

Approximate stochastic dynamic programming for hydroelectric production planning

This paper presents a novel approach for approximate stochastic dynamic programming (ASDP) over a continuous state space when the optimization phase has a near-convex structure. The approach entails a simplicial partitioning of the state space. Bounds on the true value function are used to refine the partition. We also provide analytic formulae for the computation of the expectation of the value function in the “uni-basin” case where natural inflows are strongly correlated. The approach is experimented on several configurations of hydro-energy systems. It is also tested against actual industrial data

An Assessment of the Efficiency of Canadian Power Generation Companies with Bootstrap DEA

Power generation companies play an important role in the Canadian economy, as most of the economic activities in the manufacturing and service sectors are powered by electricity. The significance of the Canadian power generation industry shows that efficiency analysis is essential for efficiently managing power generation and distribution in Canada. However, there have been few attempts to study the relative efficiencies of the Canadian power generation companies. This study fills in this gap by assessing the overall technical, managerial, and scale efficiencies of a sample of Canadian power generation companies via the non-parametric bootstrap DEA methodology, with firm-level annual inputs and outputs data over an 18-year horizon. The results of our investigation indicate low levels of overall technical and managerial efficiencies but relatively high levels of scale efficiencies of the Canadian power generation companies over the entire study period. We also found that the 2007–2009 financial crisis impacted the relative performance of the Canadian power generation companies. Our results also allowed us to identify the benchmark power generation companies for each type of efficiency that the inefficient companies should target toward improving their efficiency

Risk-averse stochastic dual dynamic programming approach for the operation of a hydro-dominated power system in the presence of wind uncertainty

Operation planning models for hydro-dominated power systems usually use low temporal resolutions due to the excessive computational burden, thus ignoring short-term characteristics of such systems. As a result, in systems coupled with wind energy, such models may fail to accurately capture wind variability, and may not appropriately take into account potential consequences of uncertainty on the system operation. This paper addresses this drawback by (i) “controlling” the cost associated with the operation of a hydro-dominated power system equipped with wind power and batteries via a risk-measure and (ii) formulating the short-term load balance as probabilistic constraints in order to hedge against potential extreme wind power scenarios. The risk-averse scheme is embedded in the stochastic dual dynamic programming framework. Simulation results for a case study on a real industrial setting show that hedging the system against the short-term volatility of wind power contributes to mitigating the risk of excessive operations costs or load curtailments, and that the consideration of the decision maker risk profile contributes to decreasing the variability of the solutions. In addition, the results of the application also illustrate the potential of the scheme to assess the energy situation of a country or a region under the penetration of wind energy and batteries deployment. © 2019 Elsevier Lt