Two-stage Stochastic Optimization Problem Research Articles

A two-stage stochastic collaborative planning approach for data centers and distribution network incorporating demand response and multivariate uncertainties

As a critical infrastructure to support the development of digital economy, the scale of data center (DC) power consumption has been growing continuously in recent years. The plug-in of a large number of DCs to the distribution network (DN) presents both opportunities and challenges for the secure, stable, and sustainable operation of the system. In order to address the planning issues arising from the integration of DCs into the DN, this paper proposes a two-stage (planning stage and operation stage) collaborative planning method for DCs and the DN while considering the source-demand uncertainty. Firstly, spatial-temporal demand response characteristic of DCs is explicitly elaborated, and based on the energy consumption model, a two-stage stochastic planning model considering conditional value at risk (CVaR) for DCs and the DN is constructed using probabilistic stochastic simulation, which obeys low-carbon and security, and satisfies the source-network-load constraints of the system. The siting and sizing for DCs and renewable generation, feeder expansion for the DN are determined to minimize the investment cost during the planning stage, while the power output for the devices considering uncertainties from load and renewable energy are optimized to minimize the operation cost during the operation stage. Secondly, the L-shaped decomposition algorithm is proposed to solve the two-stage stochastic optimization problem by reconstructing the problem into a master problem and subproblems, where the dual multiplies are calculated to generate feasibility cuts and optimality cuts to achieve iterative solving. Finally, numerical case studies on a 33-bus distribution system demonstrate that the participation of DCs in demand response can achieve a significant reduction in the investment and operation cost of the DN, and further promote the consumption of distributed renewable energy sources.

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Explainable subgradient tree boosting for prescriptive analytics in operations management

Motivated by the success of gradient boosting approaches in machine learning and driven by the need for explainable prescriptive analytics approaches in operations management (OM), we propose subgradient tree boosting (STB) as an explainable prescriptive analytics approach to solving convex stochastic optimization problems that frequently arise in OM. The STB approach combines the well-known method of subgradient descent in function space with sample average approximation, and prescribes decisions from a problem-specific loss function, historical demand observations, and prescriptive features. The approach provides a decision-maker with detailed explanations for the prescribed decisions, such as a breakdown of individual features’ impact. These explanations are particularly valuable in practice when the decision-maker has the discretion to adjust the recommendations made by a decision support system. We show how subgradients can be derived for common single-stage and two-stage stochastic optimization problems; demonstrate the STB approach’s applicability to two real-world, complex capacity-planning problems in the service industry; benchmark the STB approach’s performance against those of two prescriptive approaches—weighted sample average approximation (wSAA) and kernelized empirical risk minimization (kERM); and show how the STB approach’s prescriptions can be explained by estimating the impact of individual features. The results suggest that the quality of the STB approach’s prescriptions is comparable to that of wSAA’s and kERM’s prescriptions while also providing explanations.

Problem-driven scenario clustering in stochastic optimization

In stochastic optimisation, the large number of scenarios required to faithfully represent the underlying uncertainty is often a barrier to finding efficient numerical solutions. This motivates the scenario reduction problem: by finding a smaller subset of scenarios, reduce the numerical complexity while keeping the error at an acceptable level. In this paper we propose a novel and computationally efficient methodology to tackle the scenario reduction problem for two-stage problems when the error to be minimised is the implementation error, i.e. the error incurred by implementing the solution of the reduced problem in the original problem. Specifically, we develop a problem-driven scenario clustering method that produces a partition of the scenario set. Each cluster contains a representative scenario that best reflects the optimal value of the objective function in each cluster of the partition to be identified. We demonstrate the efficiency of our method by applying it to two challenging two-stage stochastic combinatorial optimization problems: the two-stage stochastic network design problem and the two-stage facility location problem. When compared to alternative clustering methods and Monte Carlo sampling, our method is shown to clearly outperform all other methods.

Incorporation of market signals for the optimal design of post combustion carbon capture systems

Recent studies have shown that fossil generators equipped with post-combustion carbon capture (PCC) systems are needed to reduce the cost of deep decarbonization. Such generators need to be flexible and responsive to grid conditions, particularly in a high variable renewable energy (VRE) environment. In this work, we evaluate the net present value (NPV) of retrofitting an existing natural gas combined cycle (NGCC) unit with a flexible PCC system while incorporating market signals from a high VRE grid. We use our industrial partner’s NGCC configuration as representative of existing NGCC units and Svante’s rapid-temperature swing adsorption (TSA) for PCC. Because of its ability to rapidly startup/shutdown and ramp-up/ramp-down, the chosen capture technology is very attractive for load-following operations. For a given set of market signals, we formulate a two-stage stochastic multi-period optimization problem, under the price-taker assumption, to simultaneously optimize the design of the capture system and operation of the entire plant. Rigorous models for the NGCC unit, PCC system, and compression system are developed using commercial process simulators and validated with either plant or vendor data. For computational tractability, we develop surrogate/reduced-order models for use in the optimization problem. The surrogate model for the NGCC plant is constructed by linearizing the rigorous dynamic model at 75% load, while data-driven nonlinear surrogate models for the capture and compression systems are constructed using simulation data from the rigorous models. The optimization problem, formulated as a mixed integer bilinear program, is implemented in the IDAES® integrated platform and solved to global optimality using Gurobi 9.5. Using this formulation, we determine the profitability of retrofitting an existing NGCC unit with the chosen capture system for multiple regions in the U.S. under two scenarios with different carbon prices. The results show that the optimal decision strongly depends on the region and on the carbon price, thereby demonstrating the importance of the inclusion of market signals in the design process.

A Distributed Mixed-Integer Framework to Stochastic Optimal Microgrid Control

This article deals with distributed control of microgrids composed of storages, generators, renewable energy sources, and critical and controllable loads. We consider a stochastic formulation of the optimal control problem associated with the microgrid that appropriately takes into account the unpredictable nature of the power generated by renewables. The resulting problem is a mixed-integer linear program and is NP-hard and nonconvex. Moreover, the peculiarity of the considered framework is that no central unit can be used to perform the optimization, but rather the units must cooperate with each other by means of neighboring communication. To solve the problem, we resort to a distributed methodology based on a primal decomposition approach. The resulting algorithm is able to compute high-quality feasible solutions to a two-stage stochastic optimization problem, for which we also provide a theoretical upper bound on the constraint violation. Finally, a Monte Carlo numerical computation on a scenario with a large number of devices shows the efficacy of the proposed distributed control approach. The numerical experiments are performed on realistic scenarios obtained from Generative Adversarial Networks (GANs) trained an open-source historical dataset of the EU.

Optimal sizing of an isolated microgrid with electric vehicles using stochastic programming

An isolated microgrid system with renewable sources and energy storage systems ensures sustainable access to electricity and is especially suitable to handle inadequate electrical infrastructure in rural areas. The capacity configuration of the system depends on the uncertainties of renewable energy sources and home user behaviour, and the stochastic behaviour of electric vehicle (EV) users. This study determines the optimal configuration of the isolated microgrid that comprises wind and photovoltaic generation systems as well as batteries and EVs by proposing a two-stage stochastic programming-based multi-objective optimization problem where the objective functions are minimization of the life cycle cost and reliability. The problem is solved using the deterministic equivalent of the stochastic programming model to find the exact solution. The scenarios in the model are generated for one year period to consider the effects of uncertainties throughout seasons. Cases for different roles and configurations of electric vehicles are simulated and analyzed to compare their impacts on the sizing of the microgrid. Also, the effects of charger size on the microgrid configuration are investigated through simulations. The simulation results show that a slight increase in LPSP values causes a drop in cost values of around 10%–20% in each EV scenario. The minimum cost is obtained when more efficient charge & discharge plugs for EVs are used. Furthermore, the robustness of the model results is observed through repeated experiments with different random parameters. • An isolated PV-wind-battery-EV-based microgrid system is designed by considering the uncertainties due to random demand, renewable generation and EV behaviour. • A two-stage stochastic multi-objective optimization problem is solved by minimizing the LCC and LPSP. • The impacts of EVs on the system design are investigated using different roles and configurations of EVs. • As bigger-sized chargers replace smaller ones, the cost of microgrid decreases.

En-Route Electric Vehicles Charging Navigation Considering the Traffic-Flow-Dependent Energy Consumption

En-route charging navigation system is significant for electric vehicle (EV) owners to relieve the range anxiety. Considering the limited battery capacity of EVs, accurate battery energy consumption prediction is crucial for charging navigation in that it determines the maximum traveling distance. This article develops a comprehensive EV energy consumption prediction model and the impacts of the traffic flow on the motor efficiency and driving resistances have been thoroughly considered. Moreover, a stochastic traffic flow model is proposed to improve the energy consumption prediction precision, in which the related uncertainties, such as travel time, vehicle speed, etc., are captured. Then, a novel cost-effective en-route EV charging navigation problem is formulated as a two-stage stochastic optimization problem. In addition, the EV charging behavior and its interaction with the routing selection are characterized in the navigation problem. Finally, an improved Benders Algorithm is modified to obtain the global optimal solution of the proposed navigation model. The simulation verification is carried out in the actual road network of a certain city and the IEEE 33-node coupling system. Results demonstrate the feasibility and effectiveness of the proposed model and solution algorithm.

Optimal scheduling of electrical and thermal resources and appliances in a smart home under uncertainty

The development of renewable energy sources, using the Combined Heat and Power (CHP), along with controllable appliances, has provided new opportunities for scheduling resources and consumption in smart homes. This paper proposes a framework for the optimal scheduling of electrical appliances and sources of electricity and heat in a smart home. The smart home is able to purchase electricity through bilateral contracts and spot market. Real-Time Pricing (RTP) and Inclining Block Rate (IBR) tariffs are considered in the spot market. Uncertainties related to production of renewable resources, spot market prices and usage time of inelastic non-shiftable appliances are modeled by scenarios. The presented framework is modeled as a two-stage stochastic optimization problem, with the aim of minimizing the expected cost of the smart home. Then, the problem is formulated in the form of MILP problem. The simulation results showed that the cogeneration of electricity and heat significantly reduced the expected cost of the smart home by 125% compared to the base case study. Meanwhile, the Peak to Average Ratio index (PAR) due to the distribution of consumption along the scheduling horizon has decreased by 35% compared to the base case study.

Robust DC optimal power flow with modeling of solar power supply uncertainty via R-vine copulas

We present a robust approximation of joint chance constrained DC optimal power flow in combination with a model-based prediction of uncertain power supply via R-vine copulas. It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in. This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (Eur J Oper Res (2022) https://doi.org/10.1016/j.ejor.2021.10.051). The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets. The resulting robust optimization problem has a known equivalent tractable reformulation. To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes. The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model. Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts. The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas. We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.

Two-Stage Stochastic Optimization for the Pre-Position and Reconfiguration of Microgrid Defense Resources against Natural Disasters.

With the aggravation and evolution of global warming, natural disasters such as hurricanes occur more frequently, posing a great challenge to large-scale power systems. Therefore, the pre-position and reconfiguration of the microgrid defense resources by means of Mobile Energy Storage Vehicles (MEVs) and tie lines in damaged scenarios have attracted more and more attention. This paper proposes a novel two-stage optimization model with the consideration of MEVs and tie lines to minimize the shed loads and the outage duration of loads according to their proportional priorities. In the first stage, tie lines addition and MEVs pre-position are decided prior to a natural disaster; in the second stage, the switches of tie lines and original lines are operated and MEVs are allocated from staging locations to allocation nodes according to the specific damaged scenarios after the natural disaster strikes. The proposed load restoration method exploits the benefits of MEVs and ties lines by microgrid formation to pick up more critical loads. The progressive hedging algorithm is employed to solve the proposed scenario-based two-stage stochastic optimization problem. Finally, the effectiveness and superiority of the proposed model and applied algorithm are validated on an IEEE 33-bus test case.

Robust Task Scheduling for Delay-Aware IoT Applications in Civil Aircraft-Augmented SAGIN

Although 5G networks have enabled mobile users to get a better experience, task scheduling remains challenging for massive Internet of Things (IoT) devices in remote areas. This paper investigates the task scheduling problem for delay-aware IoT applications in civil aircraft-augmented space-air-ground integrated networks (CAA-SAGIN), where the normalized sky access platforms (SAPs) can collect and forward the terrestrial tasks. Specifically, we first propose an access control scheme for a non-preemptive priority queuing system and a transmission control scheme with cross-layer optimization. Secondly, considering the uncertain distribution of the transmission numbers and generated data, we formulate a robust two-stage stochastic optimization problem of delay minimization. With the proposed robust task scheduling with risk aversion (RTS-RA) algorithm, the original problem can be decomposed into two subproblems, which can be further transformed into tractable semi-definite program (SDP) problems respectively. Simulation results show that the cross-layer optimization scheme can achieve a good tradeoff between delay and throughput. Also, the RTS-RA algorithm outperforms the exiting offloading schemes in terms of end-to-end delay, transmitted data, and energy consumption with lower computational complexity.

Uncertainty-aware optimal dispatch of building thermal storage portfolios via smoothed variance-reduced accelerated gradient methods

The rapid penetration of uncertain renewable energy resources into the power grid has made generation planning and real-time power balancing a challenge, prompting the need for advanced control on the demand-side. Although a grid-integrated building portfolio has been studied by coordinating building-level flexible energy resources, to reap further benefits, the need to develop a scalable supervisory control framework with computational efficiency is paramount. In this work, we introduce an uncertainty-aware transactive control framework for a large-scale building portfolio with thermal energy storage (TES) using a decomposition-based approach. We propose a day-ahead decision making framework for the power procurement problem, which is cast as a two-stage stochastic optimization problem. To solve this problem, we propose a smoothed variance-reduced accelerated gradient method. Notably this framework allows for parallelization in computing the sampled gradient. Preliminary numerics demonstrate the scalability and computational efficiency of the proposed algorithm to apply to the control framework with respect to the existing algorithm. Substantial energy cost savings were observed for the stochastic control framework over all the validation scenarios. This study provides further insights into the supervisory controller development for building portfolio-based grid services that can help advance electrification goals through coordination of energy storage assets.

Optimization-Based Scenario Reduction for Data-Driven Two-Stage Stochastic Optimization

In the field of data-driven optimization under uncertainty, scenario reduction is a commonly used technique for computing a smaller number of scenarios to improve computational tractability and interpretability. However traditional approaches do not consider the decision quality when computing these scenarios. In “Optimization-Based Scenario Reduction for Data-Driven Two-Stage Stochastic Optimization,” Bertsimas and Mundru present a novel optimization-based method that explicitly considers the objective and problem structure for reducing the number of scenarios needed for solving two-stage stochastic optimization problems. This new proposed method is generally applicable and has significantly better performance when the number of reduced scenarios is 1%–2% of the full sample size compared with other state-of-the-art optimization and randomization methods, which suggests this improves both tractability and interpretability.

A Novel Pre-Storm Island Formation Framework to Improve Distribution System Resilience Considering Tree-Caused Failures

This paper presents a new framework for island formation prior to windstorms, which considers tree-caused failures of distribution networks. In the proposed framework, both direct and indirect effects of windstorms on distribution lines are quantified. Thus, a novel discrete Markov chain model is proposed for representing the failure modes of trees in each time interval of windstorm duration. This model categorizes “healthy”, “uprooted”, “stem breakage”, and “branch breakage” states of a tree. In addition, a new line-tree interaction model is presented for calculating tree-caused failure probability of overhead lines. The results of the proposed Markov model are taken as inputs by the developed line-tree interaction model. In these models, the different characteristics of windstorms are taken into account. Tree vulnerability to windstorms is characterized by different factors such as their species, height, and critical wind speeds. Windstorm duration is sectionalized into multiple time intervals, and the proposed models are applied to trees and distribution system components in each interval. Moreover, the interdependency between the intervals is captured by the Markov model. The results of the models are used by an optimization model, thereby dividing a distribution system into multiple islands before storm onset. Subsequently, the framework is extended as a two-stage stochastic optimization problem to address the uncertainties of loads. In addition, this framework considers the allocation of mobile emergency resources. The proposed models are implemented on the IEEE 33- and 123-bus test systems, as well as a practical distribution feeder, and their effectiveness is demonstrated through several case studies.

State-Variable Modeling for a Class of Two-Stage Stochastic Optimization Problems

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

Technical Note—Two-Stage Sample Robust Optimization

In “Two-Stage Sample Robust Optimization,” Bertsimas, Shtern, and Sturt investigate a simple approximation scheme, based on overlapping linear decision rules, for solving data-driven two-stage distributionally robust optimization problems with the type-infinity Wasserstein ambiguity set. Their main result establishes that this approximation scheme is asymptotically optimal for two-stage stochastic linear optimization problems; that is, under mild assumptions, the optimal cost and optimal first-stage decisions obtained by approximating the robust optimization problem converge to those of the underlying stochastic problem as the number of data points grows to infinity. These guarantees notably apply to two-stage stochastic problems that do not have relatively complete recourse, which arise frequently in applications. In this context, the authors show through numerical experiments that the approximation scheme is practically tractable and produces decisions that significantly outperform those obtained from state-of-the-art data-driven alternatives.

Production network, technology choice, capacity investment and inventory sourcing decisions: operational hedging under demand uncertainty

Our research is motivated by capacity investment and inventory sourcing decisions that firms make while investing in multi-plant production facilities to offer products and services to the markets. We study the strategic choice between investing alternative technologies in a production network and examine for conditions under which flexible technology to be chosen over dedicated technology and vice-versa. We model a firm’s technology choice, capacity investment, and inventory sourcing decisions in face of demand uncertainty as a two-stage stochastic optimization problem for a production network. We conduct numerical studies to examine a situation under which flexible technology to be chosen over dedicated technology in a production network. We observe that optimal amount of capacity to be acquired in a production network with flexible technology is less compared to a network with dedicated technologies. However, optimal levels of inventories to be sourced in a production network with flexible technology is more compared to a network with dedicated technologies. We show that a production network with flexible technology can hedge against demand uncertainty using operational measures such as extra inventory and flexible capacity; in case products demand is either negatively or positively correlated then the value of investing in flexible technology in a production network diminishes.

Submodular reassignment problem for reallocating agents to tasks with synergy effects

We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given k submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s representation theorem for distributive lattices.

A multicountry, multicommodity stochastic game theory network model of competition for medical supplies inspired by the Covid-19 pandemic

In this paper, we construct the first stochastic Generalized Nash Equilibrium model for the study of competition among countries for limited supplies of medical items (PPEs, ventilators, etc.) in the disaster preparedness and response phases in the Covid-19 pandemic. The government of each country is faced with a two-stage stochastic optimization problem in which the first stage is prior to the pandemic declaration and the second stage is post the pandemic declaration. We provide the theoretical constructs, a qualitative analysis, and an algorithm, accompanied by convergence results. Both illustrative examples are presented as well as algorithmically solved numerical examples, inspired by the need for N95 masks and ventilators. The results reveal that, in addition to the preparedness of countries before the pandemic declaration, their ability to adapt to the conditions in different scenarios has a significant impact on their overall success in the management of the pandemic crisis. The framework can capture competition for other medical supplies, including Covid-19 vaccines and possible treatments, with modifications to handle perishability.