Column-generation and interior point methods applied to the long-term electric power-planning problem

Aquesta tesi s'adreça al problema de planificació de la generació elèctrica a llarg termini per a una companyia específica (SGC) que participa en un mercat liberalitzat organitzat en un pool. Els objectius de la tesi són: modelitzar aquest problema, i desenvolupar i implementar tècniques apropiades i eficients que el resolguin. Un planificació òptima a llarg termini és important, per exemple, per a la confecció de pressupostos, o per a la gestió de compres/consum de combustibles. Una altra aplicació és la de guiar la planificació a curt termini perquè aquesta tingui en compte decisions preses sota una òptica de llarg termini. La nostra proposta per a fer la planificació de la generació és optimitzar la generació esperada de cada unitat (o la unió de diverses unitats de característiques semblants) del pool per a cada interval en que dividim el llarg termini. El model bàsic per la planificació de la generació a llarg termini (LTGP) maximitza el benefici de totes les unitats del pool. La constricció més important és la satisfacció de la demanda, ja que el sistema està sempre balancejat. Utilitzem la formulació de Bloom i Gallant, la qual modela la càrrega a través d'una monòtona de càrrega per cada interval i requereix un número exponencial de constriccions lineals de desigualtat, anomenades LMCs. Altres constriccions (lineals) incloses en el model són: garantia de potència, límits en la disponibilitat de combustibles, emissions màximes de CO2 o una quota de mercat mínima per a la SGC. Una extensió d'aquest model és la planificació conjunta de l'assignació de manteniments de les unitats tèrmiques d'una SGC amb la planificació de la generació. El model conjunt és un problema quadràtic amb variables binàries i contínues. Per resoldre aquest model es proposa un parell d'heurístiques i s'ha implementat un prototipus de branch and bound en AMPL.Aquesta tesi també proposa una manera per coordinar el model LTGP proposat amb una planificació a curt termini. Es desenvolupa un model de curt que inclou els resultats de llarg termini. Donat que el model de planificació a llarg termini s'ha de resoldre sovint (principalment per passar informació acurada al model de curt), les tècniques emprades per a resoldre'l han de donar resultats fiables en un espai de temps curt. Les tècniques aplicades han estat:· Donat que les constriccions de recobriment i les fites de no negativitat defineixen un políedre convex els vèrtexs del qual són fàcils de trobar el model es transforma i les variables esdevenen els coeficients convexos que defineixen un punt. Aquest nou problema es resolt amb l'algoritme de Murtagh i Saunders, que és un procediment òptim. Aquest algoritme s'aplica sota un esquema de generació de columnes donat que el número de vèrtexs del políedre és comparable al número de constriccions. L'avantatge d'aquest mètode és que els vèrtexs es van generant a mesura que es necessiten.· L'aplicació de mètodes directes és computacionalment costós donat el número exponencial de LMCs. De totes maneres, a l'òptim només un conjunt reduït de constriccions de recobriment seran actives. Hem desenvolupat una heurística, anomenada heurística GP, la qual genera un subconjunt de constriccions, entre les quals hi ha les LMCs que són actives a l'òptim. L'heurística resol una seqüència de problemes quadràtics, els quals incrementen el número de LMCs considerades a cada iteració. Els problemes es resolen amb mètodes de punt interior que s'inicialitzen amb tècniques de warm start per tal d'accelerar la convergència cap a la nova solució. Aquest procediment resulta ser molt més eficient que el de generació de columnes. La modelització i els casos de prova estan basats en dades d'un sistema de pool pur i de mercat com ha estat a Espanya fins el juliol de 2006.This thesis presents an approach to the long-term planning of power generation for a company (SGC) participating in a liberalized market organized as a pool. The goal of this thesis is two-fold: to model the problem and to develop and implement appropriate and efficient techniques for solving it.The optimization of the long-term generation planning is important for budgeting and planning fuel acquisitions, and to give a frame where to fit short-term generation planning.Our proposal for planning long-term generation is to optimize the expected generation of each unit (or the merger of several units of the same type) in the power pool over each interval into which the long-term horizon is split.The basic model for long-term generation planning (LTGP) maximizes the profit for all the units participating in the pool. The most important constraint is matching demand, since the market always clears. The Bloom and Gallant formulation is used, which models the load with a load-duration curve for each interval and requires an exponential number of linear inequality constraints, called herein LMCs. Other (linear) constraints included in the model are: minimum generation time, limits on the availability of fuel, maximum CO2 emission limits or the market share of the SGC. This thesis also proposes the way in which coordination between the LTGP model developed and a short-term plan should be considered and provides a model for short-term electrical power planning adapted to the LTGP proposed and which includes the long-term results.Another decision that needs to be taken from a long-term point of view is the joint scheduling of thermal unit maintenances with the generation planning of a particular SGC. The results of a prototype of a Branch and Bound implemented in AMPL are included in this thesis.Long-term planning needs to be considered before short-term planning and whenever the real situation deviates from the forecasted parameters, so the techniques implemented must be efficient so as to provide reliable solutions in a short time. Two methods for handling the LMCs are proposed and compared:● A decomposition technique exploits the fact that the LMCs plus the non-negativity bounds define a convex polyhedron for each interval whose vertices are easy to find. Thus, the problem is transformed and the variables become the coefficients of a convex combination of the vertices. The transformed problem is quadratic with linear constraints, making it suitable to be solved with the Murtagh & Saunders algorithm, which gives an optimal solution. A column-generation approach is used because the number of vertices of the polyhedron is comparable to the number of LMCs. The advantage of this method is that it does not require previous computation of all of the vertices, but rather computes them as the algorithm iterates.● The application of direct methods is computationally difficult because of the exponential number of inequality LMCs. However, only a reduced subset of LMCs will be active at the optimizer. A heuristic, named GP heuristic, has been devised which is able to find a reduced set of LMCs including those that are active at the optimizer. It solves a sequence of quadratic problems in which the set of LMCs considered is enlarged at each iteration. The quadratic problems are solved with an interior point method, and warm starts are employed to accelerate the solution of the successively enlarged quadratic problems. This procedure is more efficient than the column generation one.The modeling and tests of this thesis are based on the pure pool system and market data from the Spanish system up to July 2006

Advanced Optimization and Statistical Methods in Portfolio Optimization and Supply Chain Management

This dissertation is on advanced mathematical programming with applications in portfolio optimization and supply chain management. Specifically, this research started with modeling and solving large and complex optimization problems with cone constraints and discrete variables, and then expanded to include problems with multiple decision perspectives and nonlinear behavior. The original work and its extensions are motivated by real world business problems.The first contribution of this dissertation, is to algorithmic work for mixed-integer second-order cone programming problems (MISOCPs), which is of new interest to the research community. This dissertation is among the first ones in the field and seeks to develop a robust and effective approach to solving these problems. There is a variety of important application areas of this class of problems ranging from network reliability to data mining, and from finance to operations management.This dissertation also contributes to three applications that require the solution of complex optimization problems. The first two applications arise in portfolio optimization, and the third application is from supply chain management. In our first study, we consider both single- and multi-period portfolio optimization problems based on the Markowitz (1952) mean/variance framework. We have also included transaction costs, conditional value-at-risk (CVaR) constraints, and diversification constraints to approach more realistic scenarios that an investor should take into account when he is constructing his portfolio. Our second work proposes the empirical validation of posing the portfolio selection problem as a Bayesian decision problem dependent on mean, variance and skewness of future returns by comparing it with traditional mean/variance efficient portfolios. The last work seeks supply chain coordination under multi-product batch production and truck shipment scheduling under different shipping policies. These works present a thorough study of the following research foci: modeling and solution of large and complex optimization problems, and their applications in supply chain management and portfolio optimization.Ph.D., Business Administration -- Drexel University, 201

Active-set prediction for interior point methods

This research studies how to efficiently predict optimal active constraints of an inequality constrained optimization problem, in the context of Interior Point Methods (IPMs). We propose a framework based on shifting/perturbing the inequality constraints of the problem. Despite being a class of powerful tools for solving Linear Programming (LP) problems, IPMs are well-known to encounter difficulties with active-set prediction due essentially to their construction. When applied to an inequality constrained optimization problem, IPMs generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination, as well as increasing ill-conditioning of the solution process. We show that, existing techniques for active-set prediction, however, suffer from difficulties in making an accurate prediction at the early stage of the iterative process of IPMs; when these techniques are ready to yield an accurate prediction towards the end of a run, as the iterates approach the solution set, the IPMs have to solve increasingly ill-conditioned and hence difficult, subproblems. To address this challenging question, we propose the use of controlled perturbations. Namely, in the context of LP problems, we consider perturbing the inequality constraints (by a small amount) so as to enlarge the feasible set. We show that if the perturbations are chosen judiciously, the solution of the original problem lies on or close to the central path of the perturbed problem. We solve the resulting perturbed problem(s) using a path-following IPM while predicting on the way the active set of the original LP problem; we find that our approach is able to accurately predict the optimal active set of the original problem before the duality gap for the perturbed problem gets too small. Furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active set for the perturbed problem or for the original one if no perturbations are used. Proof-of-concept algorithms are presented and encouraging preliminary numerical experience is also reported when comparing activity prediction for the perturbed and unperturbed problem formulations. We also extend the idea of using controlled perturbations to enhance the capabilities of optimal active-set prediction for IPMs for convex Quadratic Programming (QP) problems. QP problems share many properties of LP, and based on these properties, some results require more care; furthermore, encouraging preliminary numerical experience is also presented for the QP case

Exploiting Structures in Mixed-Integer Second-Order Cone Optimization Problems for Branch-and-Conic-Cut Algorithms

This thesis studies computational approaches for mixed-integer second-order cone optimization (MISOCO) problems. MISOCO models appear in many real-world applications, so MISOCO has gained significant interest in recent years. However, despite recent advancements, there is a gap between the theoretical developments and computational practice. Three chapters of this thesis address three areas of computational methodology for an efficient branch-and-conic-cut (BCC) algorithm to solve MISOCO problems faster in practice. These chapters include a detailed discussion on practical work on adding cuts in a BCC algorithm, novel methodologies for warm-starting second-order cone optimization (SOCO) subproblems, and heuristics for MISOCO problems.The first part of this thesis concerns the development of a novel warm-starting method of interior-point methods (IPM) for SOCO problems. The method exploits the Jordan frames of an original instance and solves two auxiliary linear optimization problems. The solutions obtained from these problems are used to identify an ideal initial point of the IPM. Numerical results on public test sets indicate that the warm-start method works well in practice and reduces the number of iterations required to solve related SOCO problems by around 30-40%.The second part of this thesis presents novel heuristics for MISOCO problems. These heuristics use the Jordan frames from both continuous relaxations and penalty problems and present a way of finding feasible solutions for MISOCO problems. Numerical results on conic and quadratic test sets show significant performance in terms of finding a solution that has a small gap to optimality.The last part of this thesis presents application of disjunctive conic cuts (DCC) and disjunctive cylindrical cuts (DCyC) to asset allocation problems (AAP). To maximize the benefit from these powerful cuts, several decisions regarding the addition of these cuts are inspected in a practical setting. The analysis in this chapter gives insight about how these cuts can be added in case-specific settings