Finite Set Of Scenarios Research Articles

Placement and Sizing of Inverter-Based Renewable Systems in Multi-Phase Distribution Networks

This paper develops a tractable formulation for optimal placement and sizing of inverter-based renewable systems in multi-phase distribution networks. The goal of the formulation is to minimize the cost of inverter installation, average power import, and average distributed generation curtailment. Three-phase and single-phase inverter models are presented that preserve the underlying mappings between renewable uncertainty to power injection. The uncertainty of distributed generators (DGs) and loads are characterized by a finite set of scenarios. Linear multi-phase power flow approximations are used in conjunction with scenario reduction techniques to arrive at a tractable two-stage stochastic formulation for optimal DG placement and sizing. First-stage decisions are locations for DG deployment and capacity sizes, and second-stage decisions include DG real power curtailment, reactive power support, as well as feeder voltage profile. The resulting formulation is a mixed-integer second-order cone program and can be solved efficiently either by existing optimization solvers or by relaxing the binary variables to the [0,1] interval. Simulation studies on standard multi-phase IEEE test feeders promise that optimal stochastic planning of DGs reduces costs during validation, compared to a scheme where uncertainty is only represented by its average value.

Stochastic Planning of Distributed PV Generation

Recent studies by electric utility companies indicate that maximum benefits of distributed solar photovoltaic (PV) units can be reaped when siting and sizing of PV systems is optimized. This paper develops a two-stage stochastic program that serves as a tool for optimally determining the placing and sizing of PV units in distribution systems. The PV model incorporates the mapping from solar irradiance to AC power injection. By modeling the uncertainty of solar irradiance and loads by a finite set of scenarios, the goal is to achieve minimum installation and network operation costs while satisfying necessary operational constraints. First-stage decisions are scenario-independent and include binary variables that represent the existence of PV units, the area of the PV panel, and the apparent power capability of the inverter. Second-stage decisions are scenario-dependent and entail reactive power support from PV inverters, real and reactive power flows, and nodal voltages. Optimization constraints account for inverter’s capacity, PV module area limits, the power flow equations, as well as voltage regulation. A comparison between two designs, one where the DC:AC ratio is pre-specified, and the other where the maximum DC:AC ratio is specified based on historical data, is carried out. It turns out that the latter design reduces costs and allows further reduction of the panel area. The applicability and efficiency of the proposed formulation are numerically demonstrated on the IEEE 34-node feeder, while the output power of PV systems is modeled using the publicly available PVWatts software developed by the National Renewable Energy Laboratory. The overall framework developed in this paper can guide electric utility companies in identifying optimal locations for PV placement and sizing, assist with targeting customers with appropriate incentives, and encourage solar adoption.

An effective two-stage stochastic multi-trip location-transportation model with social concerns in relief supply chains

The distribution of emergency aid from warehouses to relief centers to satisfy the needs of the victims in the aftermath of a disaster is a complex problem because it requires a rapid response to human suffering when resources are scarce amidst great uncertainty. In order to provide an effective response and use resources efficiently, this paper presents a novel model to optimize location, transportation, and fleet sizing decisions. In contrast with existing models, vehicles can be reused for multiple trips within micro-periods (blocks of hours) and/or over periods (days). Uncertainty regarding demand, incoming supply, and availability of routes is modeled via a finite set of scenarios, using two-stage stochastic programs. ‘Deprivation costs’ are used to represent social concerns and minimized via two objective functions. Mathematical programming based heuristics are devised to enable good-quality solutions within reasonable computing time. Experimental results based on data from the disastrous 2011 floods and landslides in the Serrana Region of Rio de Janeiro, Brazil, show that the model’s novel characteristics help get aid faster to victims and naturally enforce fairness in its distribution to disaster areas in a humanitarian spirit.

Task assignment and sequencing decision model under uncertain available time of service providers

This paper studies an integrated decision problem on assigning and sequencing tasks to service providers. By considering the uncertain available time of the service providers, a stochastic programming model is proposed on the basis of a finite set of scenarios. Some realistic factors are also taken into account and formulated as some non-linear cost functions, which are linearised in this study. Moreover, a heuristic solution method is designed for solving some extremely large-scale problem cases within a reasonable period of time. Numerical experiments are performed to validate the effectiveness of the proposed model and the efficiency of the proposed solution method.

Scenario-based robust optimization of water flooding in oil reservoirs enjoys probabilistic guarantees

Model-based optimization of the water-flooding process in oil reservoirs suffers from high levels of uncertainty arising from strongly varying economic conditions and limited knowledge of the reservoir model parameters. To handle uncertainty, diverse robust optimization approaches that use an ensemble of uncertain parameter realizations (i.e., scenarios), have been adopted. However, in scenario-based approaches, the effect of considering a finite set of scenarios on the constraint violation and/or the performance degradation with respect to the unseen scenarios have not been studied. In this paper, we provide probabilistic guarantees on the worst-case performance degradation of a scenario-based solution. By using statistical learning, we analyze the impact of the number of scenarios on the probabilistic guarantees for the worst-case solution subject to both economic and geological uncertainties. For the economic uncertainty, we derive an explicit a-priori relationship between the probabilistic guarantee and the number of considered scenarios, while for the geological uncertainty, a-posteriori probabilistic upper bounds on the worst-case solution are given.

A multi-stage stochastic program for supply chain network redesign problem with price-dependent uncertain demands

In this paper, we address a multi-period supply chain network redesign problem in which customer zones have price-dependent stochastic demand for multiple products. A novel multi-stage stochastic program is proposed to simultaneously make tactical decisions including products’ prices and strategic redesign decisions. Existing uncertainty in potential demands of customer zones is modeled through a finite set of scenarios, described in the form of a scenario tree. The scenarios are generated using a Latin Hypercube Sampling method and then a forward scenario construction technique is employed to create a suitable scenario tree. The multi-stage stochastic problem is formulated as a mixed-integer linear programming model and then Benders decomposition algorithm is applied for solving it. Numerical results demonstrate the significance of the stochastic model as well as the good performance of Benders algorithm. The scenario tree construction method is also evaluated in terms of out-of-sample and in-sample stability. Finally, several key managerial and practical insights in terms of pricing issues are highlighted.

Fair emergency system design under uncertainty

In host situations, a public service system is designed so that the average disutility is minimized. In this paper, we deal with a special member of a category of public service systems known as the emergency service system. Designers of this sort of service system must take into consideration not only the disutility of the average user, but also the disutility of the worst situated user. Optimization of the average user disutility relates to the large weighted p-median problem. The necessity of solving large instances has led to the approximate approach based on a radial formulation, which enables the solving of larger instances in admissible time and making use of a universal IP-solver. The p-median problem objective denoted as a min-sum criterion often causes a situation where the average user’s disutility is minimal, but the disutility of the worst situated user can nonetheless be extremely high, which is considered unfair. The objective of a fair emergency service system design is to minimize perceived disutility of the worst situated system users under various scenarios, where the model uncertainty is connected with randomly occurring events. In this paper, we focus on two different modelling techniques, which enables designing a fair emergency service system under uncertainty with minimal disutility perceived by the worst situated users. We also deal with processing uncertainty, which can be modelled either by a finite set of scenarios or using fuzzy values.

A two-stage stochastic programming model for periodic replenishment control system under demand uncertainty

The inventory control system consists of defining strategies to manage inventory replenishment such that costs involved in ordering, holding, and meeting the demands are optimized. Although several methods for inventory management are proposed in the literature, the assumptions made necessary can hinder their applicability in practice. Motivated by this fact, we propose a methodology for determining the optimal parameters of an inventory control system for single-item one-echelon supply chains using two-stage stochastic programming, considering periodic review and uncertain demand. The proposed approach is flexible enough to consider backlogs or lost sales cases without limitations on the number of outstanding orders and allows the consideration of uncertainties in a more comprehensive manner when compared to currently available methods. The optimal review periodicity and the inventory target level are determined using a nonlinear mixed-integer programming model, which takes into account the uncertain nature of the item demand levels through a finite set of scenarios. We present how this model can be reformulated as a deterministic equivalent mixed-integer linear programing model and how the Sample Average Approximation method can be used to incorporate the uncertainty into it. To validate the proposed approach, we perform a simulation-based sensitivity analysis in the neighborhood of the solutions obtained. We also compare them with those obtained applying the Hadley-Whitin method, considering the premises that are necessary for the later. Further, we present results from a case in which the stochastic demand is not stationary. The results obtained provide strong evidence of the potential of the proposed approach.

A stochastic multi-period capacitated multiple allocation hub location problem: Formulation and inequalities

This study focuses on the development of a modeling framework for multi-period stochastic capacitated multiple allocation hub location problems. We consider a planning horizon divided into several time periods. Uncertainty is assumed for the demands. The decisions to make concern the location of the hubs, their initial capacity, the capacity expansion of existing hubs and the transportation between origin–destination pairs. The goal is to minimize the total expected cost. For the situation in which uncertainty can be captured by a finite set of scenarios each occurring with some estimated probability we derive the extensive form of the deterministic equivalent. The resulting model is compact. However, it includes a set of binary variables that becomes too large for medium and large instances of the problem and thus hardly can it be tackled by a general optimization solver. For this reason, enhancements are proposed to the model making it possible to solve optimally instances that could not be solved using the initial model. This is confirmed by the computational tests performed using the well-known CAB data.

Log-robust portfolio management with parameter ambiguity

We present a robust optimization approach to portfolio management under uncertainty when randomness is modeled using uncertainty sets for the continuously compounded rates of return, which empirical research argues are the true drivers of uncertainty, but the parameters needed to define the uncertainty sets, such as the drift and standard deviation, are not known precisely. Instead, a finite set of scenarios is available for the input data, obtained either using different time horizons or assumptions in the estimation process. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values, using the worst-case nominal values among the possible scenarios) at the end of the time horizon in a one-period setting. Short sales are not allowed. We consider both the independent and correlated assets models. For the independent assets case, we derive a convex reformulation, albeit involving functions with singular Hessians. Because this slows computation times, we also provide lower and upper linear approximation problems and devise an algorithm that gives the decision maker a solution within a desired tolerance from optimality. For the correlated assets case, we suggest a tractable heuristic that uses insights derived in the independent assets case.

A multi-stage interval-stochastic programming model for planning end-of-life vehicles allocation

The amount of material entering end-of-life vehicles (ELVs) management systems all over the EU has been reduced due to increased export of used vehicles to non-EU countries. According to the latest data, only 6.23 million ELVs were processed in 2012 (e.g., 8.37 million ELVs were processed in 2009). Considering that currently 342 active vehicle recycling factories exist in the EU, as well as that in the last ten years the total processing capacity has been increased several times, it is clear that there is a serious conflict. This paper proposes a multi-stage interval-stochastic programming model for planning end-of-life vehicles allocation. The developed model is able to reflect dynamics in terms of decisions for ELV allocation from a multi-region waste management system to multiple vehicle recycling factories within a multi-period context. It is capable of incorporating multiple policies within the optimization framework. Uncertainties expressed as probability distributions and discrete intervals are effectively handled, based on a multi-layered scenario tree with a finite set of scenarios. A semi-hypothetical case study is conducted in order to demonstrate the potentials and applicability of the proposed model. Influences of parameter uncertainty on model solutions are thoroughly examined. Comprehensive analyses of various policy situations, associated with different levels of economic penalties and system failure risks, are presented. The proposed model is effective for solving difficult ELV allocation problems with uncertainties included as part of the formulation. It can help quantify the relationships between ELV management system profit and disruption risk of the vehicle recycling factories, and thus provide optimal ELV allocation schemes.

Scenario-based robust optimization of a water supply system under risk of facility failure

In this paper, we propose a scenario-based robust optimization model for the design of a water supply system considering the risk of facility failure, which is represented as an uncertainty set generated by a finite set of scenarios. New facilities are planned to be built to hedge against the possible failure of existing system facilities that would potentially damage the capacity of the system to meet given user demands. The goal is to build facilities that are both cost-effective and make the system robust. The system robustness is defined as the ability to satisfy user demands for every data realization in the uncertainty set. The proposed model is shown to be equivalent to a large-scale mixed-integer linear program that is solved by a Benders decomposition algorithm. Computational results demonstrate the efficiency of the proposed algorithm, and show that substantial improvement in system robustness can be achieved with minimal increase in system cost.

A two-stage stochastic transportation problem with fixed handling costs and a priori selection of the distribution channels

In this paper, a transportation problem comprising stochastic demands, fixed handling costs at the origins, and fixed costs associated with the links is addressed. It is assumed that uncertainty is adequately captured via a finite set of scenarios. The problem is formulated as a two-stage stochastic program. The goal is to minimize the total cost associated with the selected links plus the expected transportation and fixed handling costs. A prototype problem is initially presented which is then progressively extended to accommodate capacities at the origins and multiple commodities. The results of an extensive set of computational tests are reported and discussed.

Sell or Hold: A simple two-stage stochastic combinatorial optimization problem

The sell or hold problem (SHP) is to sell k out of n indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. We show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number of scenarios in the second stage is constant. A max{1/2,k/n}-approximation algorithm is presented for the scenario-based SHP.

Risk conscious solution of planning problems under uncertainty by hybrid multi-objective evolutionary algorithms

Abstract We consider the risk conscious solution of planning problems with uncertainties in the problem data. The problems are formulated as two-stage stochastic mixed-integer models in which some of the decisions (first-stage) have to be made under uncertainty and the remaining decisions (second-stage) can be made after the realization of the uncertain parameters. The uncertain model parameters are represented by a finite set of scenarios. The risk conscious optimization problem under uncertainty is solved by a stage decomposition approach using a multi-objective evolutionary algorithm which optimizes the expected scenario costs and the risk criterion with respect to the first-stage decisions. The second-stage scenario decisions are handled by mathematical programming. Results from numerical experiments for two real-world problems are shown.

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.

Portfolio optimization for wealth-dependent risk preferences

Empirical and theoretical studies of preference structures of investors have long shown that personal and corporate utility is typically multimodal, implying that the same investor can be risk-averse at certain levels of wealth while risk-seeking at others. In this paper, we consider the problem of optimizing the portfolio of an investor with an indefinite quadratic utility function. The convex and concave segments of this utility reflect the investor’s attitude towards risk, which changes based on deviations from a fixed goal. Uncertainty is modeled via a finite set of scenarios for the returns of securities. A global optimization approach is developed to solve the proposed nonconvex optimization problem. We present computational results which investigate the effect of short sales and demonstrate that the proposed approach systematically produces portfolios with higher values of skewness than the classical expectation-variance approach.

A two-stage stochastic programming model for scheduling replacements in sow farms

This paper presents a formulation and resolution of a two-stage stochastic linear programming model with recourse for sow farms producing piglets. The proposed model considers a medium-term planning horizon and specifically allows optimal replacement and schedule of purchases to be obtained for the first stage. This model takes into account sow herd dynamics, housing facilities, reproduction management, herd size with initial and final inventory of sows and uncertain parameters such as litter size, mortality and fertility rates. These last parameters are explicitly incorporated via a finite set of scenarios. The proposed model is solved by using the algebraic modelling software OPL Studio from ILOG, in combination with the solver CPLEX to solve the linear models resulting from different instances considered. The article also presents results obtained with previous deterministic models assessing the suitability of the stochastic approach. Finally, the conclusions drawn from the study including an outlook are presented.