Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically

A Planner-Trader Decomposition for Multi-Market Hydro Scheduling

Peak/off-peak spreads on European electricity forward and spot markets are eroding due to the ongoing nuclear phaseout and the steady growth in photovoltaic capacity. The reduced profitability of peak/off-peak arbitrage forces hydropower producers to recover part of their original profitability on the reserve markets. We propose a bi-layer stochastic programming framework for the optimal operation of a fleet of interconnected hydropower plants that sells energy on both the spot and the reserve markets. The outer layer (the planner's problem) optimizes end-of-day reservoir filling levels over one year, whereas the inner layer (the trader's problem) selects optimal hourly market bids within each day. Using an information restriction whereby the planner prescribes the end-of-day reservoir targets one day in advance, we prove that the trader's problem simplifies from an infinite-dimensional stochastic program with 25 stages to a finite two-stage stochastic program with only two scenarios. Substituting this reformulation back into the outer layer and approximating the reservoir targets by affine decision rules allows us to simplify the planner's problem from an infinite-dimensional stochastic program with 365 stages to a two-stage stochastic program that can conveniently be solved via the sample average approximation. Numerical experiments based on a cascade in the Salzburg region of Austria demonstrate the effectiveness of the suggested framework

Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically

Modèles d’optimisation stochastique pour le problème de gestion de réservoirs

RÉSUMÉ : La gestion d’un système hydroélectrique représente un problème d’une grande complexité pour des compagnies comme Hydro-Québec ou Rio Tinto. Il faut effectivement faire un compromis entre plusieurs objectifs comme la sécurité des riverains, la production hydroélectrique, l’irrigation et les besoins de navigation et de villégiature. Les opérateurs doivent également prendre en compte la topologie du terrain, les délais d’écoulement, les interdépendances entre les réservoirs ainsi que plusieurs phénomènes non linéaires physiques. Même dans un cadre déterministe, ces nombreuses contraintes opérationnelles peuvent mener à des problèmes irréalisables sous certaines conditions hydrologiques. Par ailleurs, la considération de la production hydroélectrique complique considérablement la gestion du bassin versant. Une modélisation réaliste nécessite notamment de prendre en compte la hauteur de chute variable aux centrales, ce qui mène à un problème non convexe. En outre, de nombreuses sources d’incertitude entourent la réalisation d’un plan de production. Les prix de l’électricité sur les marchés internationaux, la disponibilité des turbines, la charge/demande du réseau ainsi que les apports en eau sont tous incertains au moment d’établir les soutirages et les déversés pour un horizon temporel donné. Négliger cette incertitude et supposer une connaissance parfaite du futur peut mener à des politiques de gestion beaucoup trop ambitieuses. Ces dernières ont tendance à engendrer des conséquences désastreuses comme le vidage ou le remplissage très rapide des réservoirs, ce qui conduit ensuite à des inondations ou des sécheresses importantes. Cette thèse considère le problème de gestion de réservoirs avec incertitude sur les apports. Elle tente spécifiquement de développer des modèles et des algorithmes permettant d’améliorer la gestion mensuelle de la rivière Gatineau, notamment en période de crue. Dans cette situation, il est primordial de considérer l’incertitude autour des apports, car ces derniers ont une influence marquée sur l’état hydrologique du système en plus d’être la cause d’évènements désastreux comme les inondations. La gestion des inondations est particulièrement importante pour la Gatineau, car la rivière coule près de la ville de Maniwaki qui a déjà vécu des inondations dans le passé et continue de présenter des risques importants. Cette rivière représente également une excellente étude de cas, car elle possède plusieurs barrages et réservoirs. La grande dimension du système rend difficile l’application de certains algorithmes populaires comme la programmation dynamique stochastique. Afin de minimiser le risque d’inondations, on propose initialement un modèle de programmation stochastique multi-étapes (multi-stage stochastic program) basé sur les règles de décision affine et les règles de décision affines liftées. On considère l’aversion au risque en évaluant la valeur à risque conditionnelle (conditional value-at-risk) aussi connue comme "CVaR". Ce travail considère une représen-tation polyhédrale de l’incertitude très simple basée sur la moyenne et la variance d’échantillon. Le deuxième article propose d’améliorer cette Représentation de l’incertitude en considérant explicitement la corrélation temporelle entre les apports. À cet effet, il introduit les modèles de séries chronologiques de type ARIMA et présente une manière de les incorporer efficacement dans un modèle multi-étapes avec règles de décision. On étend ensuite l’approche pour évaluer les processus GARCH, ce qui permet d’incorporer l’hétéroscédasticité. Le troisième travail raffine la représentation de l’incertitude utilisée dans le deuxième travail en s’appuyant sur un modèle ARMA calibré sur le logarithme des apports. Cette représentation non linéaire mène à un ensemble d’incertitude non convexe qu’on choisit d’approximer de façon conservatrice par un polyhèdre. Ce modèle offre néanmoins plusieurs avantages comme la possibilité de dériver une expression analytique pour l’espérance conditionnelle. Afin de considérer la hauteur de chute variable, on propose un algorithme de région de confiance très simple, mais efficace. Ces travaux montrent qu’il est possible d’obtenir de bons résultats pour le problème de gestion de réservoir en considérant les règles de décision linéaires en combinaison avec une représentation basée sur les processus ARIMA.----------ABSTRACT : The problem of designing an optimal release schedule for a hydroelectric system is extremely challenging for companies like Rio Tinto and Hydro-Québec. It is essential to strike an adequate compromise between various conflicting objectives such a riparian security, hydroelectric production as well as navigation and irrigation needs. Operators must also consider the topology of the terrain, water delays, dependence between reservoirs as well as non-linear physical phenomena. Even in a deterministic framework, it may be impossible to find a feasible solution under given hydrological conditions. Considering hydro-electricity generation further complicates the problem. Indeed, a realistic model must take into account variable water head, which leads to an intractable bilinear non-convex problem. In addition, there exists various sources of uncertainty surrounding the elaboration of the production plan. The price of electricity on foreign markets, availability of turbines, load of the network and water inflows all remain uncertain at the time of fixing water releases and spills over the given planning horizon. Neglecting this uncertainty and assuming perfect foresight will lead to overly ambitious policies. These decisions will in turn generate disastrous consequences such as very rapid emptying or filling of reservoirs, which in turn generate droughts or floods. This thesis considers the reservoir management problem with uncertain inflows. It aims at developing models and algorithms to improve the management of the Gatineau river, namely during the freshet. In this situation, it is essential to consider the randomness of inflows since these drive the dynamics of the systems and can lead to disastrous consequences like floods. Flood management is particularly important for the Gatineau, since the river runs near the town of Maniwaki, which has witnessed several floods in the past. This river also represents a good case study because it comprises various reservoirs and dams. This multi-dimensionality makes it difficult to apply popular algorithms such as stochastic dynamic programming. In order to minimize the risk of floods, we initially propose a multi-stage stochastic program based on affine and lifted decision rules. We capture risk aversion by optimizing the conditional value-at-risk also known as "CVaR". This work considers a simple polyhedral uncertainty representation based on the sample mean and variance. The second paper builds on this work by explicitly considering the serial correlation between inflows. In order to do so, it introduces ARIMA time series models and details their incorporation into multi-stage stochastic programs with decision rules. The approach is then extended to take into account heteroscedasticity with GARCH models The third work further refines the uncertainty representation by calibrating an ARMA model on the log of inflows. This leads to a non-convex uncertainty set, which is approximated with a simple polyhedron. This model offers various advantages such as increased forecasting skill and ability to derive an analytical expression for the conditional expectation. In order to consider the variable water head, we propose a successive linear programming (SLP) algorithm which quickly yields good solutions. These works illustrate the value of using affine decision rules in conjunction with ARIMA models to obtain good quality solutions to complex multi-stage stochastic problems

Robust Optimisation for Hydroelectric System Operation under Uncertainty

In this paper, we propose an optimal dispatch scheme for a cascade hydroelectric power system that maximises the head levels of each dam, and minimises the spillage effects taking into account uncertainty in the net load variations, i.e., the difference between the load and the renewable resources, and inflows to the cascade. By maximising the head levels of each dam the volume of water stored, which is a metric of system resiliency, is maximised. In this regard, the operation of the cascade hydroelectric power system is robust to the variability and intermittency of renewable resources and increases system resilience to variations in climatic conditions. Thus, we demon- strate the benefits of coupling hydroelectric and photovoltaic resources. To this end, we introduce an approximate model for a cascade hydroelectric power system. We then develop correlated probabilistic forecasts for the uncertain output of renewable resources, e.g., solar generation, using historical data based on clustering and Markov chain techniques. We incorporate the gen- erated forecast scenarios in the optimal dispatch of the cascade hydroelectric power system, and define a robust variant of the developed system. However, the robust variant is intractable due to the infinite number of constraints. Using tools from robust optimisation, we reformulate the resulting problem in a tractable form that is amenable to existing numerical tools and show that the computed dispatch is immunised against uncertainty. The efficacy of the proposed approach is demonstrated by means of an actual case study involving the Seven Forks system located in Kenya, which consists of five cascaded hydroelectric power systems. With the case study we demonstrate that the Seven Forks system may be coupled with solar generation since the “price of robustness” is small; thus demonstrating the benefits of coupling hydroelectric systems with solar generatio

Dynamic probabilistic constraints under continuous random distributions

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented

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