Semidefinite Programming. methods and algorithms for energy management

La présente thèse a pour objet d explorer les potentialités d une méthode prometteuse de l optimisation conique, la programmation semi-définie positive (SDP), pour les problèmes de management d énergie, à savoir relatifs à la satisfaction des équilibres offre-demande électrique et gazier.Nos travaux se déclinent selon deux axes. Tout d abord nous nous intéressons à l utilisation de la SDP pour produire des relaxations de problèmes combinatoires et quadratiques. Si une relaxation SDP dite standard peut être élaborée très simplement, il est généralement souhaitable de la renforcer par des coupes, pouvant être déterminées par l'étude de la structure du problème ou à l'aide de méthodes plus systématiques. Nous mettons en œuvre ces deux approches sur différentes modélisations du problème de planification des arrêts nucléaires, réputé pour sa difficulté combinatoire. Nous terminons sur ce sujet par une expérimentation de la hiérarchie de Lasserre, donnant lieu à une suite de SDP dont la valeur optimale tend vers la solution du problème initial.Le second axe de la thèse porte sur l'application de la SDP à la prise en compte de l'incertitude. Nous mettons en œuvre une approche originale dénommée optimisation distributionnellement robuste , pouvant être vue comme un compromis entre optimisation stochastique et optimisation robuste et menant à des approximations sous forme de SDP. Nous nous appliquons à estimer l'apport de cette approche sur un problème d'équilibre offre-demande avec incertitude. Puis, nous présentons une relaxation SDP pour les problèmes MISOCP. Cette relaxation se révèle être de très bonne qualité, tout en ne nécessitant qu un temps de calcul raisonnable. La SDP se confirme donc être une méthode d optimisation prometteuse qui offre de nombreuses opportunités d'innovation en management d énergie.The present thesis aims at exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for addressing some difficult problems of energy management. We pursue two main objectives. The first one consists of using SDP to provide tight relaxations of combinatorial and quadratic problems. A first relaxation, called standard can be derived in a generic way but it is generally desirable to reinforce them, by means of tailor-made tools or in a systematic fashion. These two approaches are implemented on different models of the Nuclear Outages Scheduling Problem, a famous combinatorial problem. We conclude this topic by experimenting the Lasserre's hierarchy on this problem, leading to a sequence of semidefinite relaxations whose optimal values tends to the optimal value of the initial problem.The second objective deals with the use of SDP for the treatment of uncertainty. We investigate an original approach called distributionnally robust optimization , that can be seen as a compromise between stochastic and robust optimization and admits approximations under the form of a SDP. We compare the benefits of this method w.r.t classical approaches on a demand/supply equilibrium problem. Finally, we propose a scheme for deriving SDP relaxations of MISOCP and we report promising computational results indicating that the semidefinite relaxation improves significantly the continuous relaxation, while requiring a reasonable computational effort.SDP therefore proves to be a promising optimization method that offers great opportunities for innovation in energy management.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Computational Methods for Nonlinear Optimization Problems: Theory and Applications

This dissertation is motivated by the lack of efficient global optimization techniques for polynomial optimization problems. The objective is twofold. First, a new mathematical foundation for obtaining a global or near-global solution will be developed. Second, several case studies will be conducted on a variety of real-world problems. Global optimization, convex relaxation and distributed computation are at the heart of this PhD dissertation. Some of the specific problems to be addressed in this thesis on both the theory and the application of nonlinear optimization are explained below: Graph theoretic algorithms for low-rank optimization problems: There is a rapidly growing interest in the recovery of an unknown low-rank matrix from limited information and measurements. This problem occurs in many areas of engineering and applied science such as machine learning, control, and computer vision. We develop a graph-theoretic technique in Part I that is able to generate a low-rank solution for a sparse Linear Matrix Inequality (LMI), which is directly applicable to a large set of problems such as low-rank matrix completion with many unknown entries. Our approach finds a solution with a guarantee on its rank, using the recent advances in graph theory. Resource allocation for energy systems: The flows in an electrical grid are described by nonlinear AC power flow equations. Due to the nonlinear interrelation among physical parameters of the network, the feasibility region represented by power flow equations may be nonconvex and disconnected. Since 1962, the nonlinearity of the network constraints has been studied, and various heuristic and local-search algorithms have been proposed in order to perform optimization over an electrical grid [Baldick, 2006; Pandya and Joshi, 2008]. Part II is concerned with finding convex formulations of the power flow equations using semidefinite programming (SDP). The potential of SDP relaxation for problems in power systems has been manifested in [Lavaei and Low, 2012], with further studies conducted in [Lavaei, 2011; Sojoudi and Lavaei, 2012]. A variety of graph-theoretic and algebraic methods are developed in Part II in order to facilitate performing fundamental, yet challenging tasks such as optimal power flow (OPF) problem, security-constrained OPF and the classical power flow problem. Synthesis of distributed control systems: Real-world systems mostly consist of many interconnected subsystems, and designing an optimal controller for them pose several challenges to the field of control theory. The area of distributed control is created to address the challenges arising in the control of these systems. The objective is to design a constrained controller whose structure is specified by a set of permissible interactions between the local controllers with the aim of reducing the computation or communication complexity of the overall controller. It has been long known that the design of an optimal distributed (decentralized) controller is a daunting task because it amounts to an NP-hard optimization problem in general [Witsenhausen, 1968; Tsitsiklis and Athans, 1984]. Part III is devoted to study the potential of the SDP relaxation for the optimal distributed control (ODC) problem Our approach rests on formulating each of different variations of the ODC problem as rank-constrained optimization problems from which SDP relaxations can be derived. As the first contribution, we show that the ODC problem admits a sparse SDP relaxation with solutions of rank at most 3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the low-rank SDP solution may be deployed to retrieve a near-global controller. Parallel computation for sparse semidefinite programs: While small- to medium-sized semidefinite programs are efficiently solvable by second-order-based interior point methods in polynomial time up to any arbitrary precision [Vandenberghe and Boyd, 1996a], these methods are impractical for solving large-scale SDPs due to computation time and memory issues. In Part IV of this dissertation, a parallel algorithm for solving an arbitrary SDP is introduced based on the alternating direction method of multipliers. The proposed algorithm has a guaranteed convergence under very mild assumptions. Each iteration of this algorithm has a simple closed-form solution, and consists of scalar multiplication and eigenvalue decomposition over matrices whose sizes are not greater than the treewdith of the sparsity graph of the SDP problem. The cheap iterations of the proposed algorithm enable solving real-world large-scale conic optimization problems

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Threat and decision models for informing power grid resilience under uncertainty

The electric power grid is lifeline infrastructure and a cornerstone of modern society. The ability of the grid to withstand severe contingencies is essential for the well-being of individuals, businesses, and communities as a whole, and recent extreme weather events have highlighted vulnerabilities that subject those parties to risk. As factors like climate change, renewable penetration, and mass electrification guide the transformation of power grids, improving or at least maintaining its resilience is crucial. In this work, we present end-to-end frameworks for simulating the uncertain effects of extreme weather on a power grid and then leveraging the simulations to guide resilience decision-making via optimization under uncertainty. The benefits of our frameworks are highlighted through a variety of case studies built on simulations of tropical storm and winter storm events and power flow optimization modeling.Mechanical Engineerin

Secure and cost-effective operation of low carbon power systems under multiple uncertainties

Power system decarbonisation is driving the rapid deployment of renewable energy sources (RES) like wind and solar at the transmission and distribution level. Their differences from the synchronous thermal plants they are displacing make secure and efficient grid operation challenging. Frequency stability is of particular concern due to the current lack of provision of frequency ancillary services like inertia or response from RES generators. Furthermore, the weather dependency of RES generation coupled with the proliferation of distributed energy resources (DER) like small-scale solar or electric vehicles permeates future low-carbon systems with uncertainty under which legacy scheduling methods are inadequate. Overly cautious approaches to this uncertainty can lead to inefficient and expensive systems, whilst naive methods jeopardise system security. This thesis significantly advances the frequency-constrained scheduling literature by developing frameworks that explicitly account for multiple new uncertainties. This is in addition to RES forecast uncertainty which is the exclusive focus of most previous works. The frameworks take the form of convex constraints that are useful in many market and scheduling problems. The constraints equip system operators with tools to explicitly guarantee their preferred level of system security whilst unlocking substantial value from emerging and abundant DERs. A major contribution is to address the exclusion of DERs from the provision of ancillary services due to their intrinsic uncertainty from aggregation. This is done by incorporating the uncertainty into the system frequency dynamics, from which deterministic convex constraints are derived. In addition to managing uncertainty to facilitate emerging DERs to provide legacy frequency services, a novel frequency containment service is designed. The framework allows a small amount of load shedding to assist with frequency containment during high RES low inertia periods. The expected cost of this service is probabilistic as it is proportional to the probability of a contingency occurring. The framework optimally balances the potentially higher expected costs of an outage against the operational cost benefits of lower ancillary service requirements day-to-day. The developed frameworks are applied extensively to several case studies. These validate their security and demonstrate their significant economic and emission-saving benefits.Open Acces