Two-stage Stochastic Linear Programming Research Articles

STOCHOS: Stochastic opportunistic maintenance scheduling for offshore wind farms

Despite the promising outlook, the numerous economic and environmental benefits of offshore wind energy are still compromised by its high Operations and Maintenance (O&M) expenditures. On one hand, offshore-specific challenges such as site remoteness, harsh weather, transportation requirements, and production losses, significantly inflate the O&M costs relative to land-based wind farms. On the other hand, the uncertainties in weather conditions, asset degradation, and electricity prices largely constrain the farm operator’s ability to identify the time windows at which maintenance is possible, let alone optimal. In response, we propose STOCHOS, short for the stochastic holistic opportunistic scheduler—a maintenance scheduling approach tailored to address the unique challenges and uncertainties in offshore wind farms. Given probabilistic forecasts of key environmental and operational parameters, STOCHOS optimally schedules the offshore maintenance tasks by harnessing the opportunities that arise due to favorable weather conditions, on-site maintenance resources, and maximal operating revenues. STOCHOS is formulated as a two-stage stochastic mixed-integer linear program, which we solve using a scenario-based rolling horizon algorithm that aligns with the industrial practice. Tested on real-world data from the U.S. North Atlantic where several offshore wind farms are in-development, STOCHOS demonstrates considerable improvements relative to prevalent maintenance benchmarks, across various O&M metrics, including total cost, downtime, resource utilization, and maintenance interruptions.

Optimisation models for the day-ahead energy and reserve self-scheduling of a hybrid wind–battery virtual power plant

This work presents a suite of two optimisation models for the short-term self-scheduling and redispatch of a virtual power plant (VPP) composed of a wind farm and a Li-ion battery, that participates in the day-ahead energy and secondary regulation reserve markets of the Spanish power system. First, a two-stage stochastic mixed-integer linear programming model is used to obtain the VPP’s generation and reserve schedule and the opportunity cost of the energy stored in the battery. The model has an hourly resolution and a look-ahead period beyond the markets’ scheduling horizon and considers the hourly battery degradation costs as a function of both the depth of discharge and the discharge rate. Different strategies are evaluated to forecast the real-time use of the committed secondary regulation reserves. Second, a deterministic MILP model is used to determine the redispatch of the VPP using as input the generation and reserve schedule and the VPP’s storage opportunity cost provided by the former model and is executed on an hourly rolling basis. The results obtained show that the proposed models are effective for the short-term scheduling and redispatch of the VPP used with a low computational time, making them tractable and practical for their daily use.

Green-Resilient Supplier Selection and Order Allocation Under Disruption by Utilizing Conditional Value at Risk: Mixed Response Strategies

Two important decisions in supply chains and logistics systems design are the supplier selection and order allocation (SS&OA) problem and the vehicle routing problem (VRP). Supply disruption may reduce the capacity of suppliers, and the transportation network disruption may decrease the number of vehicles in the fleet and disrupt some routes. Also, increasing environmental regulations and environmental awareness makes companies pay more attention to green supply chain management (GSCM). In this paper, we integrate green and resilient supplier selection and order allocation decisions with vehicle routing decisions under disruption. We present a multiproduct two-stage risk-averse mixed-integer stochastic linear programming for the green and resilient supplier selection and order allocation integrated with vehicle routing (G&RSS&OA-V) problem. We consider resilient strategies before disruption, including multiple sourcing, supplier fortification, prepositioned inventory at the protected supplier, and contract with third-party logistics providers (3PLs). The objective function is to minimize the total mean-risk cost and the cost of greenhouse emissions. We use conditional value at risk (CVaR) as a risk measure to control the risk of worst-case cost. The most significant decisions of this model are the strategic decisions of determining the optimal suppliers and the operational decisions of vehicles routing under disruption simultaneously. Other decisions include determining which suppliers should be fortified, the amount to be transported to the hybrid manufacturing-distribution (HMD) center through the supplier or prepositioned emergency inventory, and the amount of lost sales. In order to validate the proposed model and its features, several numerical examples along with sensitivity analysis are performed by GAMS software, which shows the efficiency and application of the developed model, and some managerial insights are reported. The results of the sensitivity analysis show that as α increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 13.2%. As λ increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 35.6%. The increase of these two risk factors makes the proposed model more risk-averse. As the expected shortage cost increases by 150%, the mean-CVaR objective function cost increases to 36% while the amount of expected shortage decreases by 56%.

A Benders decomposition approach for solving a two-stage local energy market problem under uncertainty

New consumer-centric electricity market schemes have emerged as an alternative to manage the energy balancing locally in distribution networks with high distributed energy resources (DERs) penetration. Peer-to-peer (P2P) energy trading has become an attractive scheme for reducing user and distribution system operators (DSO) costs through the energy surplus exchange between the energy community users, where the uncertainty of DERs and the consumption are challenging issues that require suitable managing for a proper market operation. This article presents a novel two-stage stochastic mixed-integer linear programming model to address the scheduling day-ahead problem of an energy community operating under a P2P energy trading scheme. The formulation proposed minimizes the expected community cost considering the network limitations, ensuring fair trading in the local energy market (LEM), and allowing the prosumers to act as buyers or sellers depending on their load consumption and self-generation. In order to reduce the shared information by consumers/prosumers that trade energy, the proposed model is decomposed into a master problem (MP) and subproblems (SPs) to manage the network constraints and the market requirements, respectively. With this purpose, the Benders decomposition approach is implemented using the recently introduced Strengthened Benders cuts to address the binary variables related to the market and battery operation in the SPs. The model and the algorithm are tested in the 69-bus radial distribution system, considering from 3 to 39 agents trading energy to measure the model scalability. In this analysis, the algorithm convergence shows that the proposed methodology reduces the LEM’s shared information without increasing the energy community cost.

Decarbonizing the electricity sector in Qatar using PV combined with ice thermal and battery storage

High daytime electricity demand from space cooling synergetic with predictable and reliable solar insolation creates a unique opportunity to exploit solar PV-enabled decarbonization solutions in Qatar. This paper examines the economic viability of combining utility-scale PV with ice thermal and battery storage to decarbonize the electricity sector in Qatar, which exclusively runs on gas generation. The problem is formulated in a two-stage stochastic linear programming that minimizes annual system costs at a given gas price. Under the current gas price of $3.3/MMBtu (gas-generated electricity at $37/MWh), PV and ice storage deployed in Qatar could reduce gas generation use and peak demand by 43% and 18%, respectively, and cut the annual system costs by 20%. At a gas price of $6.5/MMBtu (equivalent to carbon pricing at $60/ton of CO2), gas generation can be reduced by 60% using PV and ice storage. Reducing gas generation further is challenging since both cooling and non-cooling demands peak in August, whereas PV generation peaks in June, producing less surplus generation at a time of need, and ice thermal storage cannot cost-effectively outcompete already existing gas generations for highly seasonal cooling needs. Battery storage becomes cost-effective above a gas price of $9.2/MMBtu (equivalent to carbon pricing at $110/ton of CO2); it is primarily used to manage the diurnal behavior of non-cooling loads and could decarbonize the electricity sector by around 90%.

Discrete Approximation and Convergence Analysis for a Class of Decision-Dependent Two-Stage Stochastic Linear Programs

Discrete Approximation and Convergence Analysis for a Class of Decision-Dependent Two-Stage Stochastic Linear Programs

Analytics with stochastic optimisation: experimental results of demand uncertainty in process industries

This study reports the test results of a two-stage stochastic linear programming (SLP) model with recourse using a user-friendly generic decision support system (DSS) in a North American steel company. This model has the flexibility to configure multiple material facilities, activities and storage areas in a multi-period and multi-scenario environment. The value of stochastic solution (VSS) with a real-world example has a potential benefit of US$ 24.61 million. Experiments were designed according to the potential joint probability distribution scenarios and the magnitude of demand variability. Overall, 144 SLP optimisation model instances were solved across four industries, namely, steel, aluminium, polymer and pharmaceuticals. The academic contribution of this research is two-fold: first, the potential contribution to profit in a steel company using an SLP model; and second, the optimisation empirical experiments confirm a pattern that the VSS and expected value of perfect information (EVPI) increase with the increase in demand variability. This study has implications for practicing managers seeking business solutions with prescriptive analytics using stochastic optimisation-based DSS. This study will attract more industry attention to business solutions, and the prescriptive analytics discipline will garner more scholarly and industry attention.

A two-stage stochastic programming model for seaport berth and channel planning with uncertainties in ship arrival and handling times

The limited availability of berths and channels is generally the bottleneck restricting the capacity of a seaport and thus resulting in traffic congestion. Optimizing the operations of the berths and channels has been recognized as a more economic avenue for mitigating seaport traffic congestion compared with channel dredging and berth expanding that needs significant capital and time costs. This paper presents a two-stage stochastic mixed integer linear programming model for the seaport berth and channel planning, aiming to minimize the expected total weighted completion times of ships under uncertain ship arrival times and ship handling durations. The first stage decides the berth allocation of ships under uncertainty. In the second stage, the channel planning, including the selection of lanes, assignment of tugboats, and sequencing of ships, is determined after the uncertainty has been realized. To effectively solve the model, we propose two tailored decomposition methods, that is, the stage decomposition method and the decomposition-based heuristic algorithm (DHA). Then, a lower bound of the problem is derived to evaluate the quality of the solution. Numerical experiments on Tianjin Port of China show the satisfactory performance of these two proposed methods. Especially, the DHA is able to obtain near-optimal solutions with the average optimality gap less than 3% within four-hour computational time for the instances up to 500 scenarios and 190 ship movements. Some managerial insights are obtained to guide the operations of the port.

Multi-Area Reserve Dimensioning Using Chance-Constrained Optimization

We propose a chance-constrained formulation for the problem of dimensioning frequency restoration reserves on a power transmission network. We cast our problem as a two-stage stochastic mixed integer linear program, and propose a heuristic algorithm for solving the problem. Our model accounts for the simultaneous sizing of both upward and downward reserves, and uncertainty driven by imbalances, contingencies and available transmission capacity. Our core methodology is further adapted in order to minimize inter-zonal flows and in order to split reserve requirements between automatic and manual frequency restoration reserves. We apply our methodology to the problem of sizing reserves in the four load frequency control areas of the Swedish power system. We demonstrate the benefits of our method in terms of decreasing reserve requirements in the absence of reserve sharing, we analyze the spatial allocation of reserves, and we perform various sensitivity analyses.

Stochastic Approximation Methods for the Two-Stage Stochastic Linear Complementarity Problem

The two-stage stochastic linear complementarity problem (TSLCP), which can be regarded as a special and important reformulation of two-stage stochastic linear programming, has arisen in various fields, such as stochastic programming, game theory, traffic equilibrium, and theoretical economics. Considerable effort has been devoted to designing numerical methods for solving TSLCPs. A popular approach is to integrate the progressive hedging algorithm (PHA) as a sub-algorithm into a discretization framework. In this paper, aiming to solve large-scale TSLCPs, we propose two kinds of stochastic methods: the stochastic approximation method based on projection (SAP) and the dynamic sampling SAP (DS-SAP), both of which offering more direct and improved control of the computational costs of the involved subproblems, especially compared with the PHA. In particular, the linear complementarity subproblems are solved inexactly during each iteration, and the convergence analysis of both SAP and DS-SAP with an inexactness criterion is presented. Moreover, numerical implementations and practical applications demonstrate the efficiency of our proposed methods.

Stochastic Decomposition Method for Two-Stage Distributionally Robust Linear Optimization

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen [Math. Oper. Res., 16 (1991), pp. 650--669] for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method).

Generalized adaptive partition-based method for two-stage stochastic linear programs: Geometric oracle and analysis

Adaptive Partition-based Methods (APM) are numerical methods that solve, in particular, two-stage stochastic linear problems (2SLP). We say that a partition of the uncertainty space is adapted to the current first stage control xˇ if we can aggregate scenarios while conserving the true value of the expected recourse cost at xˇ. The core idea of APM is to iteratively construct an adapted partition to all past tentative first stage controls. Relying on the normal fan of the dual admissible set, we give a necessary and sufficient condition for a partition to be adapted even for non-finite distribution, and provide a geometric method to obtain an adapted partition. Further, by showing the connection between APM and the L-shaped algorithm, we prove convergence and complexity bounds of the APM methods. The paper presents the fixed recourse case and ends with elements to forgo this assumption.

AWS IoT analytics platform for microgrid operation management

Microgrid (MG) represents a promising opportunity for integrating renewable energy systems with the electric power grid. However, numerous complexities need to be addressed in the process. The electrical grid is complex, vulnerable, and centralized. Thus, the integration is challenging owing to the stochastic nature of renewable energy generation, which affects the possibility of reliable forecasting. The wastage due to poor estimation of clean energy generation discourages new investments in this area. However, recent advancements in big data technologies enable processing a large amount of data captured from multiple sources in real-time. It opens the possibility of improving the operational optimization of MGs and the performance of forecasting models. The overall MG problem is formulated using a two-stage stochastic mixed-integer linear programming problem with recourse. Amazon Web Services (AWS) IoT analytics platform inputs data in real-time and runs a sophisticated wind generation forecast analysis. The stochastic model is solved using the Sample Average Approximation (SAA) algorithm. The innovative methodology leads to significant improvements in the total average operating cost by integrating AWS IoT Analytics compared to traditional methods that use historical data. Computations are performed for different power-grid settings, including a power-outage, and different power generators units capacities with total operation average cost savings of 7.6% and 5.9%, respectively. Sensitivity analysis showed that the SSA algorithm could solve all the instances by providing high-quality solutions. The AWS IoT strategy outperformed at 7.7 % and 3.6% for optimality gap and CPU time, respectively. We examined an actual case in Peru for an agricultural application to assess the performance of a stochastic optimization model with a real-time IoT wind generation forecast strategy. The results revealed the following capabilities of our novel framework: (1) it can realize higher cost savings from the MG operating systems; (2) improve real-time renewable energy forecasting; (3) facilitate robust decision-making under conditions of uncertainty.

Optimal location and operation of waste-to-energy plants when future waste composition is uncertain

In many countries, waste management is increasingly geared towards a circular economy, aiming for a sustainable society with less waste generation, fewer landfills, and a higher rate of recycling. Waste-to-Energy (WtE) plants, which convert waste into heat and energy, can contribute to the circular economy by utilizing types of waste that cannot be recycled. Due to the varying quality of sorting and socio-economic conditions in individual regions, the waste composition differs between regions and has an uncertain future development. Waste composition significantly affects the operation of WtE plants due to differences in energy potential. This paper supports strategic capacity planning for waste energy recovery by introducing a two-stage stochastic mixed-integer linear programming model that captures waste composition uncertainty through scenarios of possible future development. The results of the model provide insights into the economics of operation and identify important factors in the sustainability of the waste handling system. The model is demonstrated on an instance with six scenarios for waste management in the Czech Republic for the year 2030. The solution of the proposed model is to build 14 new WtE plants with a total capacity of 1970 kt in addition to the four existing plants with a capacity of 831 kt. The annual energy recovery capacity is expected to increase almost four times to satisfy EU directives that restrict waste landfilling.

A distributionally ambiguous two-stage stochastic approach for investment in renewable generation

The optimal expansion of a power system with reduced carbon footprint entails dealing with uncertainty about the distribution of the random variables involved in the decision process. Optimisation under ambiguity sets provides a mechanism to suitably deal with such a setting. For two-stage stochastic linear programs, we propose a new model that is between the optimistic and pessimistic paradigms in distributionally robust stochastic optimisation. When using Wasserstein balls as ambiguity sets, the resulting optimisation problem has nonsmooth convex constraints depending on the number of scenarios and a bilinear objective function. We propose a decomposition method along scenarios that converges to a solution, provided a global optimisation solver for bilinear programs with polyhedral feasible sets is available. The solution procedure is applied to a case study on expansion of energy generation that takes into account sustainability goals for 2050 in Europe, under uncertain future market conditions.

Stochastic program for disassembly lot-sizing under uncertain component refurbishing lead times

Planning disassembly operations for a given demand in components is challenging in practice because the quality of recovered components is very uncertain, and thus the duration of refurbishing operations is unpredictable. In this paper, we address the capacitated disassembly lot-sizing problems under uncertain refurbishing durations. More precisely, we consider a two-level disassembly system with a single type of end-of-life product, a dynamic demand, and stochastic refurbishing lead times for all components. To deal with the static decision frameworks, this problem is modeled as a two-stage stochastic Mixed-Integer Linear Program (MILP), where the objective is to minimize the expected total cost. To alleviate the scalability issues, we propose a reformulation of the inventory constraint that significantly reduces the number of scenarios. In addition, to solve large scale problems, we couple this reformulation with Monte–Carlo sampling. We provide a rolling horizon approach to deal with the static decision framework, where disassembly decisions are updated when new information unfolds. Experimental results show the effectiveness of the proposed models and the convergence of the resulting Sample Average Approximation (SAA) estimator.

Resilience-oriented operation of power systems: Hierarchical partitioning-based approach

As an achievement of innovations resulting from partitioning mechanisms, these mechanisms can contribute to the more flexible operation of power systems in local communities. The ever-increasing frequency and severity of unexpected real-time failures have created challenges for partitioned-based power system operators, affecting each partition’s resiliency. With this in mind, this paper presents an adaptive local operation strategy (ALOS) for resilient scheduling of the renewable-dominated partitioned-based power systems under normal and islanding modes in a decentralized manner. The main objective of the developed ALOS lies in reaching an affordable preparedness level in each partition to deal with unscheduled islanding mode, which can occur subsequent to real-time failures at common lines between adjacent partitions on transmission level. To this end, a set of resilience-target constraints is presented to prepare sufficient spinning reserve capacity in each partition to ensure continuity of supply during islanding mode. The proposed strategy is formulated as a two-stage stochastic mixed-integer linear program (MILP), and the nested formation algorithm is employed to execute it in a hierarchical fashion based on the privacy-preserving protocols. Besides, the tri-state compressed air energy storage (CAES) system is also included in the proposed strategy to mitigate the negative consequences caused by real-time failures and uncertain sources. Numerical results conducted on the IEEE 30-bus test system reveal that the proposed ALOS can enhance the resilience of each partition in responding to unscheduled islanding mode by efficiently utilizing all available capacities on the generation side. Furthermore, the DIgSILENT PowerFactory is used to identify the worst possible series of events and to evaluate the effectiveness of the proposed resilience-promoting proactive strategy in dealing with these events.

Two-stage stochastic energy procurement model for a large consumer in hydrothermal systems

The Energy Procurement (EP) problem faced by a large consumer is concerned with planning the energy procurement in the various energy markets available, such that its short- and medium-term demands are met, and the risks involved in such trading are mitigated. Although a number of EP models have been proposed for purely thermal systems, no specific model has been addressed for solving this problem for a large consumer located in a hydro-dominated system. In this paper we discuss the main specific features and issues involving EP problems for hydro-dominated markets. A central issue is the estimation of future energy prices in the pool market. In hydro-dominated systems, uncertainties in incremental water inflows into reservoirs affect directly such prices, as well as the estimated demands, and these are difficult correlations to be captured by a price estimation model. In this paper, we propose a Price Scenario Generation (PSG) model to estimate future pool prices, which is able to capture spatial and temporal correlation among uncertain prices, water inflows and demands. The estimated prices obtained by the PSG are introduced in the proposed Energy Procurement Model for Hydrothermal Systems (EPMHS), which calculates the optimal procurement decisions, involving the portions of energy traded in the pool, bilateral and futures markets, as well as the self-produced energy. The proposed EPMHS is formulated as a sequence of mixed-integer two-stage stochastic linear programming problems, where pool prices are handled as uncertain parameters. The EPMHS represents trading risks using the Conditional Value at Risk (CVaR) metric. We also propose a strategy for including yearly estimation of water inflows into the EPMHS, since prices in hydro-dominated markets are generally driven by water inflows forecasts. The model proposed is applied to the generation system of the northeast region of Brazil, and the results reveal coherent correlations between hydro and economic variables.

A robust multiobjective integrated master surgery schedule and surgical case assignment model at a publicly funded hospital

This paper considers an integrated operating room (OR) planning and advanced scheduling problem in a surgery department comprising several specialties that share a fixed number of ORs and post-surgery beds. It jointly considers the allocation of surgical specialties to OR blocks together with the assignment of the subsets of patients from each specialty’s waiting list to the OR blocks over a one-week planning horizon. By integrating the master surgical schedule problem (MSSP) and surgical case assignment problem (SCAP), we extend the stability of the scheduling process. We consider both uncertain surgery durations and emergency arrivals. The aim of this study is to maximize both patient service levels and hospital efficiency. A multi-objective integer linear programming (MOILP) model is developed to minimize patient waiting times as well as to maximize the OR utilization rates. An Lp-metric methodology is applied to solve the final integrated objective function. The MOILP is transformed into a robust optimization model using a transformation framework. In addition, we develop a two-stage stochastic integer linear programming model. The effectiveness of three models is compared and demonstrated through an extensive numerical experiment carried out on a large set of instances based on real data.

Design of Sustainable Renewable-Based Utility Plants in the Face of Uncertainty

This work presents the formulation of a two-stage stochastic mixed-integer linear programming (MILP) model to include uncertainty in the design of renewable-based utility plants. The model is based on a superstructure that integrates technologies to process biomass, waste, solar radiation, and wind and considers uncertainty in availability of the renewable resources and on the utility demands. The uncertain parameter space is calculated based on a monthly probability density function for each uncertain parameter and discretized into different levels. It is shown that as uncertainty is considered in the model formulation, design flexibility improves with respect to the deterministic-based designs, although the flexibility is achieved at the expense of higher underused facilities and therefore unused investment cost.