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Showing papers on "Stochastic programming published in 2016"


Journal ArticleDOI
TL;DR: The AG method is generalized to solve nonconvex and possibly stochastic optimization problems and it is demonstrated that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general non Convex smooth optimization problems by using first-order information, similarly to the gradient descent method.
Abstract: In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.

578 citations


Journal ArticleDOI
TL;DR: This paper derives an equivalent reformulation for DCC and shows that it is equivalent to a classical chance constraint with a perturbed risk level, and analyzes the relationship between the conservatism of D CC and the size of historical data, which can help indicate the value of data.
Abstract: In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general $$\phi $$ź-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general $$\phi $$ź-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various $$\phi $$ź-divergence measures based on a capacitated lot-sizing problem with a quality-of-service requirement.

437 citations


Journal ArticleDOI
TL;DR: In this paper, a randomized stochastic projected gradient (RSPG) algorithm was proposed to solve the convex composite optimization problem, in which proper mini-batch of samples are taken at each iteration depending on the total budget of stochiastic samples allowed, and a post-optimization phase was also proposed to reduce the variance of the solutions returned by the algorithm.
Abstract: This paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a certain non-differentiable (but convex) component. In order to solve these problems, we propose a randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are taken at each iteration depending on the total budget of stochastic samples allowed. The RSPG algorithm also employs a general distance function to allow taking advantage of the geometry of the feasible region. Complexity of this algorithm is established in a unified setting, which shows nearly optimal complexity of the algorithm for convex stochastic programming. A post-optimization phase is also proposed to significantly reduce the variance of the solutions returned by the algorithm. In addition, based on the RSPG algorithm, a stochastic gradient free algorithm, which only uses the stochastic zeroth-order information, has been also discussed. Some preliminary numerical results are also provided.

375 citations


Journal ArticleDOI
TL;DR: In this paper, a mathematical formulation for optimal planning of a developed EH considering operation constraints is presented for deterministic and stochastic circumstances of wind power, electricity price, and the hub electricity demand.

258 citations


Journal ArticleDOI
TL;DR: This paper presents a two-stage stochastic programming approach to the optimal scheduling of a resilient MG, linearized which offers robustness, simplicity, and computational efficiency in optimizing the MG operation.
Abstract: In recent years, natural disasters around the world have underscored the need for operative solutions that can improve the power grid resilience in response to low-probability high-impact incidents. The advent of microgrids (MGs) in modern power systems has introduced promising measures that can fulfil the power network resiliency requirements. This paper presents a two-stage stochastic programing approach to the optimal scheduling of a resilient MG. The impact of natural disasters on the optimal operation of MGs is modeled using a stochastic programming process. Other prevailing uncertainties associated with wind energy, electric vehicles, and real-time market prices are also taken into account. The proposed hourly scheme attempts to mitigate damaging impacts of electricity interruptions by effectively exploiting the MG capabilities. Incorporating AC network constraints in the proposed model offers a better solution to the security-constrained operation of MGs. The proposed model is linearized which offers robustness, simplicity, and computational efficiency in optimizing the MG operation. The effectiveness of proposed approach is illustrated using a large-scale MG test bed with a realistic set of data.

240 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed a stochastic programming model for metro train rescheduling problem in order to jointly reduce the time delay of affected passengers, their total traveling time and operational costs of trains.
Abstract: In a heavily congested metro line, unexpected disturbances often occur to cause the delay of the traveling passengers, infeasibility of the current timetable and reduction of the operational efficiency. Due to the uncertain and dynamic characteristics of passenger demands, the commonly used method to recover from disturbances in practice is to change the timetable and rolling stock manually based on the experiences and professional judgements. In this paper, we develop a stochastic programming model for metro train rescheduling problem in order to jointly reduce the time delay of affected passengers, their total traveling time and operational costs of trains. To capture the complexity of passenger traveling characteristics, the arriving ratio of passengers at each station is modeled as a non-homogeneous poisson distribution, in which the intensity function is treated as time-varying origin-to-destination passenger demand matrices. By considering the number of on-board passengers, the total energy usage is modeled as the difference between the tractive energy consumption and the regenerative energy. Then, we design an approximate dynamic programming based algorithm to solve the proposed model, which can obtain a high-quality solution in a short time. Finally, numerical examples with real-world data sets are implemented to verify the effectiveness and robustness of the proposed approaches.

221 citations


Journal ArticleDOI
TL;DR: A novel hybrid robust-stochastic programming (HRSP) approach to simultaneously model two different types of uncertainties by including stochastic scenarios for transportation costs and polyhedral uncertainty sets for demands and returns is developed.

212 citations


Journal ArticleDOI
TL;DR: In this article, an adjustable robust restoration optimization model with a two-stage objective is proposed, involving the uncertain DG outputs and load demands, where the first stage generates optimal strategies for recovery of outage power and the second stage seeks the worst-case fluctuation scenarios.
Abstract: Distributed generations (DGs) introduce significant uncertainties to restoration of active distribution networks, in addition to roughly estimated load demands. An adjustable robust restoration optimization model with a two-stage objective is proposed in this paper, involving the uncertain DG outputs and load demands. The first stage generates optimal strategies for recovery of outage power and the second stage seeks the worst-case fluctuation scenarios. The model is formulated as a mixed-integer linear programming problem and solved using the column-and-constraint generation method. The feasibility and reliability of the strategies obtained via this robust optimization model can be guaranteed for all cases in the predefined uncertainty sets with good performance. A technique known as the uncertainty budget is used to adjust the conservativeness of this model, providing a tradeoff between conservativeness and robustness. Numerical tests are carried out on the modified PG&E 69-bus system and a modified 246-bus system to compare the robust optimization model against a deterministic restoration model, which verifies the superiority of this proposed model.

196 citations


Journal ArticleDOI
TL;DR: In this paper, a mixed integer linear programming (MILP) formulation for stochastic unit commitment is proposed to optimize system operation by simultaneously scheduling energy production, standing/spinning reserves and inertia-dependent fast frequency response in light of uncertainties associated with wind production and generation outages.
Abstract: High penetration of wind generation will increase the requirement for fast frequency response services as currently wind plants do not provide inertial response. Although the importance of inertia reduction has been widely recognized, its impact on the system scheduling has not been fully investigated. In this context, this paper proposes a novel mixed integer linear programming (MILP) formulation for stochastic unit commitment that optimizes system operation by simultaneously scheduling energy production, standing/spinning reserves and inertia-dependent fast frequency response in light of uncertainties associated with wind production and generation outages. Post-fault dynamic frequency requirements, 1) rate of change of frequency, 2) frequency nadir and 3) quasi-steady-state frequency are formulated as MILP constraints by using the simplified model of system dynamics. Moreover the proposed methodology enables the impact of wind uncertainty on system inertia to be considered. Case studies are carried out on the 2030 Great Britain system to demonstrate the importance of incorporating inertia-dependent fast frequency response in the stochastic scheduling and to indicate the potential for the proposed model to inform reviews of grid codes associated with fast frequency response and future development of inertia-related market.

193 citations


Journal ArticleDOI
TL;DR: In this paper, a two-stage distributionally robust optimization model for the joint energy and reserve dispatch (D-RERD for short) of bulk power systems with significant renewable energy penetration is proposed.
Abstract: This paper proposes a two-stage distributionally robust optimization model for the joint energy and reserve dispatch (D-RERD for short) of bulk power systems with significant renewable energy penetration. Distinguished from the prevalent uncertainty set-based and worst-case scenario oriented robust optimization methodology, we assume that the output of volatile renewable generation follows some ambiguous distribution with known expectations and variances, the probability distribution function (pdf) is restricted in a functional uncertainty set. D-RERD aims at minimizing the total expected production cost in the worst renewable power distribution. In this way, D-RERD inherits the advantages from both stochastic optimization and robust optimization: statistical characteristic is taken into account in a data-driven manner without requiring the exact pdf of uncertain factors. We present a convex optimization-based algorithm to solve the D-RERD, which involves solving semidefinite programming (SDP), convex quadratic programming (CQP), and linear programming (LP). The performance of the proposed approach is compared with the emerging adaptive robust optimization (ARO)-based model on the IEEE 118-bus system. Their respective features are discussed in case studies.

186 citations


Journal ArticleDOI
TL;DR: In this paper, a data-driven risk-averse stochastic unit commitment model is proposed, where risk aversion stems from the worst-case probability distribution of the renewable energy generation amount, and the corresponding solution methods to solve the problem are developed.
Abstract: Considering recent development of deregulated energy markets and the intermittent nature of renewable energy generation, it is important for power system operators to ensure cost effectiveness while maintaining the system reliability To achieve this goal, significant research progress has recently been made to develop stochastic optimization models and solution methods to improve reliability unit commitment run practice, which is used in the day-ahead market for ISOs/RTOs to ensure sufficient generation capacity available in real time to accommodate uncertainties Most stochastic optimization approaches assume the renewable energy generation amounts follow certain distributions However, in practice, the distributions are unknown and instead, a certain amount of historical data are available In this research, we propose a data-driven risk-averse stochastic unit commitment model, where risk aversion stems from the worst-case probability distribution of the renewable energy generation amount, and develop the corresponding solution methods to solve the problem Given a set of historical data, our proposed approach first constructs a confidence set for the distributions of the uncertain parameters using statistical inference and solves the corresponding risk-averse stochastic unit commitment problem Then, we show that the conservativeness of the proposed stochastic program vanishes as the number of historical data increases to infinity Finally, the computational results numerically show how the risk-averse stochastic unit commitment problem converges to the risk-neutral one, which indicates the value of data

Journal ArticleDOI
TL;DR: In this article, the economic value of heat pumps and electric boilers is assessed by simulating their day-to-day market performance using a novel operational strategy based on two-stage stochastic programming.

Journal ArticleDOI
TL;DR: In this paper, a stochastic multi-objective ORPD (SMO-ORPD) problem is studied in a wind integrated power system considering the loads and wind power generation uncertainties.

Journal ArticleDOI
TL;DR: This analysis explores the technical and practical implications of using EMODPS through a careful diagnostic assessment of the effectiveness and reliability of the overall EModPS solution design as well as of the resulting Pareto-approximate operating policies.
Abstract: Optimal management policies for water reservoir operation are generally designed via stochastic dynamic programming (SDP). Yet, the adoption of SDP in complex real-world problems is challenged by the three curses of dimensionality, modeling, and multiple objectives. These three curses considerably limit SDP’s practical application. Alternatively, this study focuses on the use of evolutionary multiobjective direct policy search (EMODPS), a simulation-based optimization approach that combines direct policy search, nonlinear approximating networks, and multiobjective evolutionary algorithms to design Pareto-approximate closed-loop operating policies for multipurpose water reservoirs. This analysis explores the technical and practical implications of using EMODPS through a careful diagnostic assessment of the effectiveness and reliability of the overall EMODPS solution design as well as of the resulting Pareto-approximate operating policies. The EMODPS approach is evaluated using the multipurpose Hoa ...

Book ChapterDOI
TL;DR: This chapter describes a stochastic ship routing problem with inventory management that involves finding a set of least cost routes for a fleet of ships transporting a single commodity when the demand for the commodity is uncertain.
Abstract: This chapter describes a stochastic ship routing problem with inventory management. The problem involves finding a set of least cost routes for a fleet of ships transporting a single commodity when the demand for the commodity is uncertain. Storage at supply and consumption ports is limited and inventory levels are monitored in the model. Consumer demands are at a constant rate within each time period, and in the stochastic problem, the demand rate for a period is not known until the beginning of that period. The demand situation over the time periods is described by a scenario tree with corresponding probabilities. A decomposition formulation is given and it is solved using a Branch and Price framework. A master problem (set partitioning with extra inventory constraints) is built, and the subproblems, one for each ship, are solved by stochastic dynamic programming and yield the columns for the master problem. Each column corresponds to one possible tree of actions for one ship giving its schedule loading/unloading quantities for all demand scenarios. Computational results are given showing that medium sized problems can be solved successfully.

Journal ArticleDOI
TL;DR: Recent advances that have addressed some of the barriers that have prevented optimization under uncertainty for mostly linear models are described.

Journal ArticleDOI
TL;DR: The paper addresses a variety of high-dimensional Markov chain Monte Carlo methods as well as deterministic surrogate methods, such as variational Bayes, the Bethe approach, belief and expectation propagation and approximate message passing algorithms.
Abstract: Modern signal processing (SP) methods rely very heavily on probability and statistics to solve challenging SP problems. SP methods are now expected to deal with ever more complex models, requiring ever more sophisticated computational inference techniques. This has driven the development of statistical SP methods based on stochastic simulation and optimization. Stochastic simulation and optimization algorithms are computationally intensive tools for performing statistical inference in models that are analytically intractable and beyond the scope of deterministic inference methods. They have been recently successfully applied to many difficult problems involving complex statistical models and sophisticated (often Bayesian) statistical inference techniques. This survey paper offers an introduction to stochastic simulation and optimization methods in signal and image processing. The paper addresses a variety of high-dimensional Markov chain Monte Carlo (MCMC) methods as well as deterministic surrogate methods, such as variational Bayes, the Bethe approach, belief and expectation propagation and approximate message passing algorithms. It also discusses a range of optimization methods that have been adopted to solve stochastic problems, as well as stochastic methods for deterministic optimization. Subsequently, areas of overlap between simulation and optimization, in particular optimization-within-MCMC and MCMC-driven optimization are discussed.

Journal ArticleDOI
01 Jan 2016-Energy
TL;DR: In this paper, the authors propose short-term decision-support models for aggregators that sell electricity to prosumers and buy back surplus electricity, where the aggregator can control flexible energy units at the prosumers.

Journal ArticleDOI
TL;DR: A multi-objective optimization model for the combined gas and electricity network planning is presented, wherein the Elitist Non-dominated Sorting Genetic Algorithm II is employed to capture the optimal Pareto front and the Primal–Dual Interior-Point (PDIP) method combined with the point-estimate method is adopted to evaluate the objective functions.

Journal ArticleDOI
TL;DR: The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis, including publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse.
Abstract: Currently, stochastic optimization on the one hand and multi-objective optimization on the other hand are rich and well-established special fields of Operations Research. Much less developed, however, is their intersection: the analysis of decision problems involving multiple objectives and stochastically represented uncertainty simultaneously. This is amazing, since in economic and managerial applications, the features of multiple decision criteria and uncertainty are very frequently co-occurring. Part of the existing quantitative approaches to deal with problems of this class apply scalarization techniques in order to reduce a given stochastic multi-objective problem to a stochastic single-objective one. The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis. We survey publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse. In the second case, modern risk measures play a prominent role, and generalizations of stochastic orders from the univariate to the multivariate case have recently turned out as a promising methodological tool. Modeling questions as well as issues of computational solution are discussed.

Journal ArticleDOI
TL;DR: In this article, a stochastic planning framework for the battery energy storage system (BESS) in distribution networks with high wind power penetrations was proposed, aiming to maximise wind power utilisation while minimise the investment and operation costs.
Abstract: In recent years, the battery energy storage system (BESS) has been considered as a promising solution for mitigating renewable power generation intermittencies. This study proposes a stochastic planning framework for the BESS in distribution networks with high wind power penetrations, aiming to maximise wind power utilisation while minimise the investment and operation costs. In the proposed framework, the uncertainties in wind power output and system load are modelled by the Monte-Carlo simulation, and a chance-constrained stochastic optimisation model is formulated to optimally determine the location and capacity of BESS while ensuring wind power utilisation level. Then, the Monte-Carlo simulation embedded differential evolution algorithm is used to solve the problem. Simulation studies performed on a 15-bus radial distribution system prove the efficiency of the proposed method.

Journal ArticleDOI
TL;DR: A two-stage stochastic UC model with high renewable penetration including reserve requirements for the efficient management of uncertainty is presented, and a minimum conditional value at risk term has been included in the model formulation.
Abstract: Isolated regions and islands are facing imported fossil-fuel dependency, higher electricity prices, and vulnerability to climate change. At the same time, they are increasing their renewable penetration and, therefore, risk for electric utilities. Integrating stochastic energy resources in noninterconnected systems may take advantage of an intelligent and optimized risk-averse unit commitment (UC) model. This paper presents a two-stage stochastic UC model with high renewable penetration including reserve requirements for the efficient management of uncertainty. In order to account for the uncertainty around the true outcomes of load, wind, and photovoltaic (PV) generation, a minimum conditional value at risk term has been included in the model formulation. A stochastic measure of the value of the stochastic solution is used to evaluate the benefits of using stochastic programming. The model considers the need for reserves dependent on the forecasting horizon and the amount of renewable generation. Active power demand, and wind and PV generations are considered as probability distribution functions. The model is applied to the Lanzarote–Fuerteventura system in the Canary Islands, Spain, and Crete, Greece.

Journal ArticleDOI
TL;DR: The numerous variants of the stochastic vehicle routing problem that have been studied in the literature are described and categorized.

Journal ArticleDOI
TL;DR: In this article, a stochastic programming-robust optimization model is presented to balance the profit estimates and the tractability to various circumstances in the supply chain of a power plant.

Journal ArticleDOI
TL;DR: In this paper, an efficient solution approach based on Benders' decomposition is proposed to solve a network-constrained ac unit commitment problem under uncertainty, which is modeled through a suitable set of scenarios.
Abstract: This paper proposes an efficient solution approach based on Benders’ decomposition to solve a network-constrained ac unit commitment problem under uncertainty. The wind power production is the only source of uncertainty considered in this paper, which is modeled through a suitable set of scenarios. The proposed model is formulated as a two-stage stochastic programming problem, whose first-stage refers to the day-ahead market, and whose second-stage represents real-time operation. The proposed Benders’ approach allows decomposing the original problem, which is mixed-integer nonlinear and generally intractable, into a mixed-integer linear master problem and a set of nonlinear, but continuous subproblems, one per scenario. In addition, to temporally decompose the proposed ac unit commitment problem, a heuristic technique is used to relax the inter-temporal ramping constraints of the generating units. Numerical results from a case study based on the IEEE one-area reliability test system (RTS) demonstrate the usefulness of the proposed approach.

Journal ArticleDOI
TL;DR: The outcome is that a clever but simple implementation of the Benders approach can be very effective even without separability, as its performance is comparable and sometimes even better than that of the most effective and sophisticated algorithms proposed in the previous literature.

Journal ArticleDOI
TL;DR: A risk-constrained scenario-based stochastic programming framework is proposed using the conditional value at risk method to address various uncertainties in a microgrid that includes renewable energy, diesel generators, battery storage, and various loads.
Abstract: This paper presents a novel energy-management method for a microgrid that includes renewable energy, diesel generators, battery storage, and various loads. We assume that the microgrid takes part in a pool market and responds actively to the electricity price to maximize its profit by scheduling its controllable resources. To address various uncertainties, a risk-constrained scenario-based stochastic programming framework is proposed using the conditional value at risk method. The designed model is solved by two levels of stochastic optimization methods. One level of optimization is to submit optimal hourly bids to the day-ahead market under the forecast data. The other level of optimization is to determine the optimal scheduling using the scenario-based stochastic data of the uncertain resources. The proposed energy management system is not only beneficial for the microgrid and customers, but also applies the microgrid aggregator and virtual power plant. The results are shown to prove the validity of the proposed framework.

Journal ArticleDOI
TL;DR: A two-stage stochastic programming model is developed that incorporates the hybrid allocation policy and achieves high levels of accessibility and equity simultaneously and is illustrated on a case study based on real-world data from the 2011 Van earthquake in Turkey.
Abstract: In this study, we introduce a distribution network design problem that determines the locations and capacities of the relief distribution points in the last mile network, while considering demand-and network-related uncertainties in the post-disaster environment. The problem addresses the critical concerns of relief organizations in designing last mile networks, which are providing accessible and equitable service to beneficiaries. We focus on two types of supply allocation policies and propose a hybrid version considering their different implications on equity and accessibility. Then, we develop a two-stage stochastic programming model that incorporates the hybrid allocation policy and achieves high levels of accessibility and equity simultaneously. We devise a branch-and-cut algorithm based on Benders decomposition to solve large problem instances in reasonable times and conduct a numerical study to demonstrate the computational effectiveness of the solution method. We also illustrate the application of our model on a case study based on real-world data from the 2011 Van earthquake in Turkey.

Journal ArticleDOI
TL;DR: In this article, a two-stage stochastic programming approach is applied to efficiently optimize microgrid operations while satisfying a time-varying request and operation constraints, which aims at minimizing the expected cost of correction actions.

Book
21 Mar 2016
TL;DR: A tutorial on partially observable markov decision processes, a tutorial on controlled stochastic process encyclopedia of mathematics, and optimal control of partially observable piecewise.
Abstract: Covering formulation, algorithms, and structural results, and linking theory to real-world applications in controlled sensing (including social learning, adaptive radars and sequential detection), this book focuses on the conceptual foundations of partially observed Markov decision processes (POMDPs). It emphasizes structural results in stochastic dynamic programming, enabling graduate students and researchers in engineering, operations research, and economics to understand the underlying unifying themes without getting weighed down by mathematical technicalities. Bringing together research from across the literature, the book provides an introduction to nonlinear filtering followed by a systematic development of stochastic dynamic programming, lattice programming and reinforcement learning for POMDPs. Questions addressed in the book include: when does a POMDP have a threshold optimal policy? When are myopic policies optimal? How do local and global decision makers interact in adaptive decision making in multi-agent social learning where there is herding and data incest? And how can sophisticated radars and sensors adapt their sensing in real time?