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Showing papers on "Stochastic programming published in 2013"


Journal ArticleDOI
TL;DR: In this paper, a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty is proposed, which only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data.
Abstract: Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources such as wind power and price responsive demand. To meet these challenges, we propose a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty. Compared to the conventional stochastic programming approach, the proposed model is more practical in that it only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data. The unit commitment solutions of the proposed model are robust against all possible realizations of the modeled uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and the outer approximation technique. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach.

1,454 citations


Journal ArticleDOI
TL;DR: Cachon et al. as mentioned in this paper studied robust linear optimization problems with uncertainty regions defined by φ-divergences and showed that the robust counterpart of a linear optimization problem with φ divergence uncertainty is tractable for most of the choices of φ typically considered in the literature.
Abstract: In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ-divergences for example, chi-squared, Hellinger, Kullback--Leibler. We show how uncertainty regions based on φ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with φ-divergence uncertainty is tractable for most of the choices of φ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach. This paper was accepted by Gerard P. Cachon, optimization.

617 citations


Journal ArticleDOI
TL;DR: The randomized stochastic gradient (RSG) algorithm as mentioned in this paper is a type of approximation algorithm for non-convex nonlinear programming problems, and it has a nearly optimal rate of convergence if the problem is convex.
Abstract: In this paper, we introduce a new stochastic approximation type algorithm, namely, the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a postoptimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and we show that such modification allows us to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.

599 citations


Journal ArticleDOI
TL;DR: The optimal bidding strategy of an electric vehicle (EV) aggregator participating in day-ahead energy and regulation markets using stochastic optimization is determined and a new battery model is proposed for better approximation of the battery charging characteristic.
Abstract: This paper determines the optimal bidding strategy of an electric vehicle (EV) aggregator participating in day-ahead energy and regulation markets using stochastic optimization. Key sources of uncertainty affecting the bidding strategy are identified and incorporated in the stochastic optimization model. The aggregator portfolio optimization model should include inevitable deviations between day-ahead cleared bids and actual real-time energy purchases as well as uncertainty for the energy content of regulation signals in order to ensure profit maximization and reliable reserve provision. Energy deviations are characterized as “uninstructed” or “instructed” depending on whether or not the responsibility resides with the aggregator. Price deviations and statistical characteristics of regulation signals are also investigated. Finally, a new battery model is proposed for better approximation of the battery charging characteristic. Test results with an EV aggregator representing one thousand EVs are presented and discussed.

399 citations


Journal ArticleDOI
TL;DR: In this article, a mixed-integer linear programming model is proposed that minimizes the total cost of a closed-loop supply chain (CLSC) network consisting of both forward and reverse supply chains.

391 citations


Journal ArticleDOI
TL;DR: The proposed multi-objective robust stochastic programming approach for disaster relief logistics under uncertainty can help in making decisions on both facility location and resource allocation in cases of disaster relief efforts.
Abstract: Humanitarian relief logistics is one of the most important elements of a relief operation in disaster management. The present work develops a multi-objective robust stochastic programming approach for disaster relief logistics under uncertainty. In our approach, not only demands but also supplies and the cost of procurement and transportation are considered as the uncertain parameters. Furthermore, the model considers uncertainty for the locations where those demands might arise and the possibility that some of the pre-positioned supplies in the relief distribution center or supplier might be partially destroyed by the disaster. Our multi-objective model attempts to minimize the sum of the expected value and the variance of the total cost of the relief chain while penalizing the solution's infeasibility due to parameter uncertainty; at the same time the model aims to maximize the affected areas' satisfaction levels through minimizing the sum of the maximum shortages in the affected areas. Considering the global evaluation of two objectives, a compromise programming model is formulated and solved to obtain a non-dominating compromise solution. We present a case study of our robust stochastic optimization approach for disaster planning for earthquake scenarios in a region of Iran. Our findings show that the proposed model can help in making decisions on both facility location and resource allocation in cases of disaster relief efforts.

352 citations


Book
04 Dec 2013
TL;DR: This paper presents a meta-modelling framework called GAMS Codes, which automates the very labor-intensive and therefore time-heavy and expensive process of manually cataloging and automating the various stages of stochastic production.
Abstract: Introduction.- Renewable Energy Sources - Modeling and Forecasting.- Clearing the Day-Ahead Market with a High Penetration of Stochastic Production.- Balancing Markets.- Managing Uncertainty with Flexibility.- Impact of Stochastic Renewable Energy Generation on Market Quantities.- Trading Stochastic Production in Electricity Pools.- Virtual Power Plants.- Facilitating Renewable Integration by Demand Response.- Random Variables and Stochastic Processes.- Basics of Optimization.- Introduction to Stochastic Programming.- Introduction to Robust Optimization.- GAMS Codes.

323 citations


Journal ArticleDOI
TL;DR: In this article, a game theoretical model for the Stackelberg relationship between retailers (leaders) and consumers (followers) in a dynamic price environment is proposed, where both players in the game solve an economic optimisation problem subject to stochasticity in prices, weather-related variables and must-serve load.

316 citations


Journal ArticleDOI
TL;DR: A new demand side management technique, namely, a new energy efficient scheduling algorithm, is proposed to arrange the household appliances for operation such that the monetary expense of a customer is minimized based on the time-varying pricing model.
Abstract: High quality demand side management has become indispensable in the smart grid infrastructure for enhanced energy reduction and system control. In this paper, a new demand side management technique, namely, a new energy efficient scheduling algorithm, is proposed to arrange the household appliances for operation such that the monetary expense of a customer is minimized based on the time-varying pricing model. The proposed algorithm takes into account the uncertainties in household appliance operation time and intermittent renewable generation. Moreover, it considers the variable frequency drive and capacity-limited energy storage. Our technique first uses the linear programming to efficiently compute a deterministic scheduling solution without considering uncertainties. To handle the uncertainties in household appliance operation time and energy consumption, a stochastic scheduling technique, which involves an energy consumption adaptation variable , is used to model the stochastic energy consumption patterns for various household appliances. To handle the intermittent behavior of the energy generated from the renewable resources, the offline static operation schedule is adapted to the runtime dynamic scheduling considering variations in renewable energy. The simulation results demonstrate the effectiveness of our approach. Compared to a traditional scheduling scheme which models typical household appliance operations in the traditional home scenario, the proposed deterministic linear programming based scheduling scheme achieves up to 45% monetary expense reduction, and the proposed stochastic design scheme achieves up to 41% monetary expense reduction. Compared to a worst case design where an appliance is assumed to consume the maximum amount of energy, the proposed stochastic design which considers the stochastic energy consumption patterns achieves up to 24% monetary expense reduction without violating the target trip rate of 0.5%. Furthermore, the proposed energy consumption scheduling algorithm can always generate the scheduling solution within 10 seconds, which is fast enough for household appliance applications.

312 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a virtual power plant consisting of an intermittent source, a storage facility, and a dispatchable power plant, and casted the offering problem as a two-stage stochastic mixed-integer linear programming model.

310 citations


Journal ArticleDOI
TL;DR: This paper presents a unit commitment model for studying the impact of large-scale wind integration in power systems with transmission constraints and system component failures, and presents a scenario selection algorithm for selecting and weighing wind power production scenarios and composite element failures.
Abstract: In this paper we present a unit commitment model for studying the impact of large-scale wind integration in power systems with transmission constraints and system component failures. The model is formulated as a two-stage stochastic program with uncertain wind production in various locations of the network as well as generator and transmission line failures. We present a scenario selection algorithm for selecting and weighing wind power production scenarios and composite element failures, and we provide a parallel dual decomposition algorithm for solving the resulting mixed-integer program. We validate the proposed scenario selection algorithm by demonstrating that it outperforms alternative reserve commitment approaches in a 225 bus model of California with 130 generators and 375 transmission lines. We use our model to quantify day-ahead generator capacity commitment, operating cost impacts, and renewable energy utilization levels for various degrees of wind power integration. We then demonstrate that failing to account for transmission constraints and contingencies can result in significant errors in assessing the economic impacts of renewable energy integration. Subject classifications: unit commitment; stochastic programming; wind power; transmission constraints. Area of review: Environment, Energy, and Sustainability.

Posted Content
TL;DR: This paper discusses a variant of the algorithm which consists of applying a post-optimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and shows that such modification allows to improve significantly the large-deviation properties of the algorithms.
Abstract: In this paper, we introduce a new stochastic approximation (SA) type algorithm, namely the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming (SP) problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a post-optimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and show that such modification allows to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.

Journal ArticleDOI
TL;DR: In this article, a robust optimization approach for transmission network expansion planning (TNEP) under uncertainties of renewable generation and load is presented. But this approach does not require knowledge of the probability distribution of the uncertain net injections; rather the uncertainties of the net injections are specified by a simple uncertainty set.
Abstract: This paper presents a robust optimization approach for transmission network expansion planning (TNEP) under uncertainties of renewable generation and load. Unlike conventional stochastic programming, the proposed approach does not require knowledge of the probability distribution of the uncertain net injections; rather the uncertainties of the net injections are specified by a simple uncertainty set. The solution algorithm is exact and produces expansion plans that are robust against all possible realizations of the net injections defined in the uncertainty set; it is based on a Benders decomposition scheme that iterates between a master problem that minimizes the cost of the expansion plan and a slave problem that minimizes the maximum curtailment of load and renewable generation. The paper demonstrates that when adopting the dc load flow model, both the master and the dual slave can be formulated as mixed-integer linear programs for which commercial solvers exist. Numerical results on several networks with uncertainties in their loads and renewable generation show that the proposed approach produces solutions that are superior to those from two recent techniques for robust TNEP design.

Journal ArticleDOI
TL;DR: In this article, the authors systematically cover the significant developments of the last decade, including surrogate modeling of electrical machines and direct and stochastic search algorithms for both single and multi-objective design optimization problems.
Abstract: This paper systematically covers the significant developments of the last decade, including surrogate modeling of electrical machines and direct and stochastic search algorithms for both single- and multi-objective design optimization problems. The specific challenges and the dedicated algorithms for electric machine design are discussed, followed by benchmark studies comparing response surface (RS) and differential evolution (DE) algorithms on a permanent-magnet-synchronous-motor design with five independent variables and a strong nonlinear multiobjective Pareto front and on a function with eleven independent variables. The results show that RS and DE are comparable when the optimization employs only a small number of candidate designs and DE performs better when more candidates are considered.

Journal ArticleDOI
TL;DR: In this article, a resource allocation algorithm design for energy-efficient communication in an orthogonal frequency division multiple access (OFDMA) downlink network with hybrid energy harvesting base station (BS) is studied.
Abstract: We study resource allocation algorithm design for energy-efficient communication in an orthogonal frequency division multiple access (OFDMA) downlink network with hybrid energy harvesting base station (BS). Specifically, an energy harvester and a constant energy source driven by a non-renewable resource are used for supplying the energy required for system operation. We first consider a deterministic offline system setting. In particular, assuming availability of non-causal knowledge about energy arrivals and channel gains, an offline resource allocation problem is formulated as a non-convex optimization problem over a finite horizon taking into account the circuit energy consumption, a finite energy storage capacity, and a minimum required data rate. We transform this non-convex optimization problem into a convex optimization problem by applying time-sharing and exploiting the properties of non-linear fractional programming which results in an efficient asymptotically optimal offline iterative resource allocation algorithm for a sufficiently large number of subcarriers. In each iteration, the transformed problem is solved by using Lagrange dual decomposition. The obtained resource allocation policy maximizes the weighted energy efficiency of data transmission (weighted bit/Joule delivered to the receiver). Subsequently, we focus on online algorithm design. A conventional stochastic dynamic programming approach is employed to obtain the optimal online resource allocation algorithm which entails a prohibitively high complexity. To strike a balance between system performance and computational complexity, we propose a low complexity suboptimal online iterative algorithm which is motivated by the offline algorithm. Simulation results illustrate that the proposed suboptimal online iterative resource allocation algorithm does not only converge in a small number of iterations, but also achieves a close-to-optimal system energy efficiency by utilizing only causal channel state and energy arrival information.

Journal ArticleDOI
TL;DR: A two-stage stochastic programming model used to solve the optimization problem with genetic algorithm performing the search on the first stage variables and a Monte Carlo method dealing with uncertainty in the second stage is proposed.

Proceedings ArticleDOI
26 Oct 2013
TL;DR: This work presents two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates that is optimal up to polylogarithmic factors.
Abstract: Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sub linear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sub linear in the supply.

Journal ArticleDOI
TL;DR: A probabilistic framework to design an N-1 secure day-ahead dispatch and determine the minimum cost reserves for power systems with wind power generation is proposed and a reserve strategy according to which the reserves are deployed in real-time operation is identified.
Abstract: We propose a probabilistic framework to design an N-1 secure day-ahead dispatch and determine the minimum cost reserves for power systems with wind power generation. We also identify a reserve strategy according to which we deploy the reserves in real-time operation, which serves as a corrective control action. To achieve this, we formulate a stochastic optimization program with chance constraints, which encode the probability of satisfying the transmission capacity constraints of the lines and the generation limits. To incorporate a reserve decision scheme, we take into account the steady-state behavior of the secondary frequency controller and, hence, consider the deployed reserves to be a linear function of the total generation-load mismatch. The overall problem results in a chance constrained bilinear program. To achieve tractability, we propose a convex reformulation and a heuristic algorithm, whereas to deal with the chance constraint we use a scenario-based-approach and an approach that considers only the quantiles of the stationary distribution of the wind power error. To quantify the effectiveness of the proposed methodologies and compare them in terms of cost and performance, we use the IEEE 30-bus network and carry out Monte Carlo simulations, corresponding to different wind power realizations generated by a Markov chain-based model.

Journal ArticleDOI
TL;DR: Risk neutral and risk averse approaches to multistage (linear) stochastic programming problems based on the Stochastic Dual Dynamic Programming (SDDP) method are discussed.

Journal ArticleDOI
TL;DR: In this article, two alternative methodologies to efficiently generate electric load and wind-power production scenarios, which are used as input data for investment problems, are proposed. But, they do not consider the impact of wind power investment on the overall system.

Journal ArticleDOI
TL;DR: Numerical simulation results show that DP-derived heuristic rules are developed to coordinate shading blinds and natural ventilation, with simplified optimization strategies for HVAC and lighting systems, and can effectively reduce energy costs and improve human comfort.
Abstract: Buildings account for nearly 40% of global energy consumption. About 40% and 15% of that are consumed, respectively, by HVAC and lighting. These energy uses can be reduced by integrated control of active and passive sources of heating, cooling, lighting, shading and ventilation. However, rigorous studies of such control strategies are lacking since computationally tractable models are not available. In this paper, a novel formulation capturing key interactions of the above building functions is established to minimize the total daily energy cost. To obtain effective integrated strategies in a timely manner, a methodology that combines stochastic dynamic programming (DP) and the rollout technique is developed within the price-based coordination framework. For easy implementation, DP-derived heuristic rules are developed to coordinate shading blinds and natural ventilation, with simplified optimization strategies for HVAC and lighting systems. Numerical simulation results show that these strategies are scalable, and can effectively reduce energy costs and improve human comfort.

Journal ArticleDOI
Zhe Yu1, Liyan Jia1, Mary Murphy-Hoye2, Annabelle Pratt2, Lang Tong1 
TL;DR: The problem of modeling and stochastic optimization for home energy management is considered, and a model predictive control based heuristic is proposed for the scheduling of loads of different characteristics.
Abstract: The problem of modeling and stochastic optimization for home energy management is considered. Several different types of load classes are discussed, including heating, ventilation, and air conditioning unit, plug-in hybrid electric vehicle, and deferrable loads such as washer and dryer. A first-order thermal dynamic model is extracted and validated using real measurements collected over an eight months time span. A mixed integer multi-time scale stochastic optimization is formulated for the scheduling of loads of different characteristics. A model predictive control based heuristic is proposed. Numerical simulations coupled with real data measurements are used for performance evaluation and comparison studies.

Journal ArticleDOI
TL;DR: To solve the complicated nonlinear, non-smooth, and non-differentiable SDEED, an enhanced particle swarm optimization (PSO) algorithm is applied to obtain the best solution for the corresponding scenarios to improve the quality of the solutions attained by PSO.

Journal ArticleDOI
Yuhan Liu1
TL;DR: An operational law of uncertain random variables is presented, and an expected value formula is shown by using probability and uncertainty distributions.
Abstract: Uncertain random variable is a tool to deal with a mixture of uncertainty and randomness. This paper presents an operational law of uncertain random variables, and shows an expected value formula by using probability and uncertainty distributions. This paper also provides a framework of uncertain random programming that is a type of mathematical programming involving uncertain random variables. Finally, some applications of uncertain random programming are discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors propose an offering strategy for a wind power producer with market power that participates in the day-ahead market as a price-maker, and in the balancing market as an deviator.
Abstract: As a result of subsidies and technological maturity, renewable electricity producers have grown in some jurisdictions to clearly dominant positions in the market. Under this context, we propose an offering strategy for a wind power producer with market power that participates in the day-ahead market as a price-maker, and in the balancing market as a deviator. Uncertainty pertaining to wind power production and balancing market price is represented through a set of correlated scenarios. The proposed model is a stochastic mathematical program with equilibrium constraints (MPEC) that can be recast as a tractable mixed-integer linear programming (MILP) problem, which is solvable using available optimization software. Results from an illustrative example and two case studies show the effectiveness of the proposed model.

Journal ArticleDOI
TL;DR: A stochastic programming model is proposed to determine how supplies should be positioned and distributed among a network of cooperative warehouses to more effectively preposition supplies in preparation for their distribution at an operational level.

Journal ArticleDOI
TL;DR: This work introduces two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and provides convergence analysis for each of them.
Abstract: We consider a Cartesian stochastic variational inequality problem with a monotone map. Monotone stochastic variational inequalities arise naturally, for instance, as the equilibrium conditions of monotone stochastic Nash games over continuous strategy sets or multiuser stochastic optimization problems. We introduce two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and provide convergence analysis for each of them. Of these, the first is a stochastic iterative Tikhonov regularization method which necessitates the update of the regularization parameter after every iteration. The second method is a stochastic iterative proximal-point method, where the centering term is updated after every iteration. The Cartesian structure lends itself to constructing distributed multi-agent extensions and conditions are provided for recovering global convergence in limited coordination variants where agents are allowed to choose their steplength sequences, regularization and centering parameters independently, while meeting a suitable coordination requirement. We apply the proposed class of techniques and their limited coordination versions to a stochastic networked rate allocation problem.

Journal ArticleDOI
TL;DR: A stochastic programming approach to solve a multi-period multi-product multi-site aggregate production planning problem in a green supply chain for a medium-term planning horizon under the assumption of demand uncertainty.

Proceedings ArticleDOI
07 Nov 2013
TL;DR: A new and rigorous SC correlation (SCC) measure is introduced, and it is shown that, contrary to intuition, correlation can be exploited as a resource in SC design and can be significantly more efficient and more accurate than traditional SC circuits.
Abstract: Stochastic computing (SC) is a re-emerging computing paradigm which enables ultra-low power and massive parallelism in important applications like real-time image processing. It is characterized by its use of pseudo-random numbers implemented by 0-1 sequences called stochastic numbers (SNs) and interpreted as probabilities. Accuracy is usually assumed to depend on the interacting SNs being highly independent or uncorrelated in a loosely specified way. This paper introduces a new and rigorous SC correlation (SCC) measure for SNs, and shows that, contrary to intuition, correlation can be exploited as a resource in SC design. We propose a general framework for analyzing and designing combinational circuits with correlated inputs, and demonstrate that such circuits can be significantly more efficient and more accurate than traditional SC circuits. We also provide a method of analyzing stochastic sequential circuits, which tend to have inherently correlated state variables and have proven very hard to analyze.

Journal ArticleDOI
TL;DR: This work presents a modeling framework, generalized disjunctive programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions.
Abstract: Discrete-continuous optimization problems are commonly modeled in algebraic form as mixed-integer linear or nonlinear programming models. Since these models can be formulated in different ways, leading either to solvable or nonsolvable problems, there is a need for a systematic modeling framework that provides a fundamental understanding on the nature of these models. This work presents a modeling framework, generalized disjunctive programming (GDP), which represents problems in terms of Boolean and continuous variables, allowing the representation of constraints as algebraic equations, disjunctions and logic propositions. An overview is provided of major research results that have emerged in this area. Basic concepts are emphasized as well as the major classes of formulations that can be derived. These are illustrated with a number of examples in the area of process systems engineering. As will be shown, GDP provides a structured way for systematically deriving mixed-integer optimization models that exhibit strong continuous relaxations, which often translates into shorter computational times. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3276–3295, 2013