Multi-stage Stochastic Programming Model Research Articles

On optimal participation in the electricity markets of wind power plants with battery energy storage systems

The recent cost reduction and technological advances in medium- to large-scale battery energy storage systems (BESS) makes these devices a true alternative for wind producers operating in electricity markets. Associating a wind power farm with a BESS (the so-called virtual power plant (VPP)) provides utilities with a tool that converts uncertain wind power production into a dispatchable technology that can operate not only in spot and adjustment markets (day-ahead and intraday markets) but also in ancillary services markets that, up to now, were forbidden to non-dispatchable technologies. What is more, recent studies have shown capital cost investment in BESS can be recovered only by means of such a VPP participating in the ancillary services markets. We present in this study a multi-stage stochastic programming model to find the optimal operation of a VPP in the day-ahead, intraday and secondary reserve markets while taking into account uncertainty in wind power generation and clearing prices (day-ahead, secondary reserve, intraday markets and system imbalances). A case study with real data from the Iberian electricity market is presented.

A multi-stage stochastic programming for lot-sizing and scheduling under demand uncertainty

A stochastic lot-sizing and scheduling problem with demand uncertainty is studied in this paper. Lot-sizing determines the batch size for each product and scheduling decides the sequence of production. A multi-stage stochastic programming model is developed to minimize overall system costs including production cost, setup cost, inventory cost and backlog cost. We aim to find the optimal production sequence and resource allocation decisions. Demand uncertainty is represented by scenario trees using moment matching technique. Scenario reduction is used to select scenarios with the best representation of original set. A case study based on a manufacturing company has been conducted to illustrate and verify the model. We compared the two-stage stochastic programming model to the multi-stage stochastic programming model. The major motivation to adopt multi-stage stochastic programming models is that it extends the two-stage stochastic programming models by allowing revised decision at each period based on the previous realizations of uncertainty as well as decisions. Stability test and weak out-of-sample test are applied to find an appropriate scenario sample size. By using the multi-stage stochastic programming model, we improved the quality of solution by 10–13%.

Assessing the benefits of labelling postponement in an export-focused winery

Supporting wine production operations in an increasingly global market has grown ever more challenging. Export-focused wineries supply many foreign clients, often requiring different labels for the same kind of wine. Order forecasts tend to be highly inaccurate, and wineries must be able to quickly react to changes, making lot-sizing an important consideration. One tool to reduce product misallocation is postponing product differentiation, where the natural decoupling point for premium wine is the labelling process. However, the double handling involved incurs additional costs and time penalties. We analyse the performance impact of postponing the labelling of bottled wines by developing a multi-stage mixed-integer stochastic programming model with full recourse for demand scenarios. The underlying data and policies are based on an unnamed Chilean export-focused winery. The model supports lot-sizing under several winery production policies. We experiment with different levels of capacity tightness, demand variability and demand correlation between wines, optimising for reducing order backlogs, inventory levels and line set-ups. While we find that some amount of postponement will always be recommended, the exact mix and degree depend on these external factors. Postponement has the most benefits when production capacity is moderately tight, demand variability is high and wines have negatively correlated demands.

ON INTEGRATED CHANCE CONSTRAINTS IN ALM FOR PENSION FUNDS

AbstractWe discuss the role ofintegrated chance constraints(ICC) as quantitative risk constraints in asset and liability management (ALM) for pension funds. We define two types of ICC: theone periodintegrated chance constraint (OICC) and themultiperiodintegrated chance constraint (MICC). As their names suggest, the OICC covers only one period, whereas several periods are taken into account with the MICC. A multistage stochastic linear programming model is therefore developed for this purpose and a special mention is paid to the modeling of the MICC. Based on a numerical example, we first analyze the effects of the OICC and the MICC on the optimal decisions (asset allocation and contribution rate) of a pension fund. By definition, the MICC is more restrictive and safer compared to the OICC. Second, we quantify this MICC safety increase. The results show that although the optimal decisions from the OICC and the MICC differ, the total costs are very close, showing that the MICC is definitely a better approach since it is more prudent.

Long-Term Electricity Procurement for Large Industrial Consumers under Uncertainty

Due to the need for adjusting plant operations to time-varying electricity prices and changing product demand, managing the procurement of electricity in power-intensive businesses has become a major challenge. Large industrial electricity consumers often enter into long-term contracts with favorable rates. However, such power contracts require the consumers to commit themselves to the amount that they are going to purchase months in advance when future demand is not yet known with certainty. In this work, we simultaneously optimize long-term electricity procurement and production planning while considering uncertainty in product demand. We propose a multiscale multistage stochastic programming model in which a 1-year planning horizon is divided into seasons, with each season represented by two characteristic weeks; also, each season corresponds to a stage at which the demand for that season is revealed. The progressive hedging algorithm is applied to solve industrially relevant large-scale instances of t...

A multistage stochastic programming model for a multi-period strategic expansion of biofuel supply chain under evolving uncertainties

We develop a multistage, stochastic mixed-integer model to support biofuel supply chain expansion under evolving uncertainties. By utilizing the block-separable recourse property, we reformulate the multistage program in an equivalent two-stage program and solve it using an enhanced nested decomposition method with maximal non-dominated cuts. We conduct extensive numerical experiments and demonstrate the application of the model and algorithm in a case study based on the South Carolina settings. The value of multistage stochastic programming method is also explored by comparing the model solution with the counterparts of an expected value based deterministic model and a two-stage stochastic model.

Optimal hedging strategies for salmon producers

We study the optimal hedging decisions for a risk-averse salmon producer. The hedging decisions are determined using a multistage stochastic programming model. The objective is to maximize the weighted sum of expected revenues from selling salmon either in the spot market or in futures contracts and Conditional Value-at-Risk (CVaR) of the revenues over the planning horizon. The scenario tree for the multistage stochastic programming model is generated based on a procedure that combines Principal Component Analysis and state space modelling. We present results for 3 different CVaR percentiles and different degrees of risk-aversion. The results indicate that salmon producers should use futures contracts to hedge price risk already at fairly low degrees of risk-aversion. The methods described in this paper will be useful as a decision support tool for determining fish companies' risk management and hedging strategies.

Multi-stage stochastic supply chain planning in textile and apparel industry under demand uncertainty with risk considerations

In this paper, a multi-product, multi-period, multi-site supply chain production and transportation planning problem under customer demand uncertainty is considered. The problem is modelled using a multi-stage stochastic programming model taking into account the production amount, the inventory and backorder levels as well as the quantity of products to be transported between the different plants and customers in each period. Risk management is also addressed by incorporating a risk metric into the stochastic programming model which leads to a multi-objective optimisation model. The objective aims simultaneously to maximise the expected profit as well as the worst case profit. A real example from textile and apparel industry is studied in order to illustrate the effectiveness of the proposed model and the robustness of the supply chain planning solutions.

Multi-stage stochastic supply chain planning in textile and apparel industry under demand uncertainty with risk considerations

In this paper, a multi-product, multi-period, multi-site supply chain production and transportation planning problem under customer demand uncertainty is considered. The problem is modelled using a multi-stage stochastic programming model taking into account the production amount, the inventory and backorder levels as well as the quantity of products to be transported between the different plants and customers in each period. Risk management is also addressed by incorporating a risk metric into the stochastic programming model which leads to a multi-objective optimisation model. The objective aims simultaneously to maximise the expected profit as well as the worst case profit. A real example from textile and apparel industry is studied in order to illustrate the effectiveness of the proposed model and the robustness of the supply chain planning solutions.

Multistage stochastic programming model and solution algorithm for the capacity expansion of railway network

Multistage stochastic programming model and solution algorithm for the capacity expansion of railway network

Multi-Stage Stochastic Programming With Nonanticipativity Constraints for Expansion of Combined Power and Natural Gas Systems

A novel multi-stage stochastic programming model is proposed for the expansion coplanning of gas and power networks considering the uncertainties in net load demand. Meanwhile, the nonanticipativity constraints are taken into account to guarantee the decisions should only depend on the information of realized uncertainties up to the present stage. Compared with the traditional two-stage stochastic programming model, the proposed multi-stage stochastic programming model yields sequential investment decisions with the uncertainties revealed gradually over time, such that the investment decisions are capable of keeping future options open and can shift from “never be changed” decisions to a flexible “wait and see” decisions. The test on three systems shows the effectiveness of the proposed multi-stage stochastic programming model.

Managing Energy Storage in Microgrids: A Multistage Stochastic Programming Approach

A microgrid is a small-scale version of a centralized power grid that generates, distributes and regulates electricity flow to local entities using distributed generation and the main grid. Distributed energy storage systems can be used to mitigate adverse effects of intermittent renewable sources in a microgrid in which operators dynamically adjust electricity procurement and storage decisions in response to randomly-evolving demand, renewable supply and pricing information. We formulate a multistage stochastic programming (SP) model whose objective is to minimize the expected total energy costs incurred within a microgrid over a finite planning horizon. The model prescribes the amount of energy to procure, store and discharge in each decision stage of the horizon. However, for even a moderate number of stages, the model is computationally intractable; therefore, we customize the stochastic dual dynamic programming (SDDP) algorithm to obtain high-quality approximate solutions. Computation times and optimization gaps are significantly reduced by implementing a dynamic cut selection procedure and a lower bound improvement scheme within the SDDP framework. An extensive computational study reveals significant cost savings as compared to myopic and non-storage policies, as well as policies obtained using a two-stage SP model. The study also demonstrates the scalability of our solution procedure.

A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach

We introduce a stochastic multi-stage fixed charge transportation problem, in which a producer has to satisfy an uncertain demand within a deadline. At each time period, a fixed transportation cost can be paid to buy a transportation capacity. If the transportation capacity is used, the supplier also pays an uncertain unit transportation cost. A unit inventory cost is charged for the unsatisfied demand. The aim is to determine the transportation capacities to buy and the quantity to send at each time period in order to minimize the expected total cost. We prove that this problem is NP-hard, we propose a multi-stage stochastic optimization model formulation, and we determine optimal policies for particular cases, with deterministic unit transportation costs or demand and zero fixed costs. Furthermore, we provide the worst–case analysis of the rolling horizon approach, a classical heuristic approach for solving multi-stage stochastic programming models, applied to this NP-hard problem and to polynomially solvable particular cases. Worst–case results show that the rolling horizon approach can be very suboptimal. We also provide experimental results.

Application of Scenario Tree Reduction Via Quadratic Process to Medium-Term Hydrothermal Scheduling Problem

In systems with hydraulic predominance, the medium-term hydrothermal scheduling problem (MTHS) is usually modeled as a stochastic programming model where the objective is to obtain an optimal operation policy over a planning horizon. Inflows follow a stochastic process that is approximated by a multistage scenario tree and a multistage stochastic programming model is employed to compute the generation policy. The difficulty in solving the MTHS depends on the size of the scenario tree. To obtain a small but representative scenario tree, we employ the scenario reduction algorithm of Kovacevic and Pichler with quadratic process distances defined by appropriate metrics exploiting relevant features of the problem. Numerical assessments of the MTHS problem, using a reduced hydrothermal configuration extracted from the Brazilian system, show that reduced trees obtained by eliminating 80% scenarios provide approximate solutions to the problem with less than 1% accuracy errors and CPU time reduction of around 90%.

Asset liability management for open pension schemes using multistage stochastic programming under Solvency-II-based regulatory constraints

Open private pension schemes are subject to risk-based regulation. In this context, asset and liability management (ALM) frameworks for pension plan operators are increasingly based on multistage stochastic programming (MSP). The significant advances in MSP modeling notwithstanding, previous works ignore risk-based regulatory constraints such as those in the Solvency II Directive. In this work, we propose an ALM model for open pension schemes based on an MSP model with a thorough representation of a risk-based regulation. Our proposal aims to define a dynamic optimal asset allocation including a detailed depiction of bond coupon payments, based on insolvency risk measures over a planning horizon. We present a realistic case study based on the Brazilian market, where the regulator imposes Solvency-II-compatible constraints on credit, underwriting, and operational risks. We develop a computationally tractable MSP model with explicit regulatory constraints, which induce risk aversion for even risk-neutral open pension plan operators.

Multistage‐multiresponse models for dynamic quality chain design problems

AbstractQuality of final products depends on several decision variables and design factors from several stages of operations. Multiresponse optimization is a well‐grounded method for offline quality design that can consider several inputs and outputs. This study introduces a new methodology for quality optimization in multistage systems with multiple response variables. Several stochastic parameters, including response surfaces and covariates, are to be considered in this methodology. For this purpose, scenario‐based multistage stochastic programming models are developed in three cases addressing (1) decision on control factors, (2) decision on control factors as well as targets of intermediate stages, and (3) scenario‐based decision on control factors. An example is presented at the end to illustrate the mechanism of the methodology and applicability of the three cases.

An inexact cost-risk balanced model for regional energy structure adjustment management and resources environmental effect analysis-a case study of Shandong province, China

Energy structure adjustment is a significant way to resolve constrained resources and environment, improve the security of energy supply, and achieve sustainable development, which is a complicated itself but further challenged by existence of uncertainties. In this study, an inexact multistage stochastic cost-risk balanced programming model is developed for energy system management under considering the policies of coal-consumption control and pollutants emission reduction. The method is an integration of inexact multistage stochastic programming and a risk management tools from modern financial theory, and could tackle uncertainties described as interval values and probability distributions. The method could reflect energy system dynamic characteristic in a multi-period problem, and provide an effective linkage between conflicting economic cost and the system stability. The proposed method is applied to a real case of planning electric power structure adjustment in Shandong province, China, where the energy system is faced with lots of difficulties complexities in electric power structure adjustment. The impact of coal-consumption control level and pollutants emission reduction scenarios on energy system structure adjustment, and resources and environmental effect were compared and analyzed. The results are valuable for supporting the adjustment or justification of the existing power generation schemes of the complicated regional energy system.

Optimal operation of a CHP plant participating in the German electricity balancing and day-ahead spot market

During the last years, operators of combined heat and power (CHP) plants are challenged by changing circumstances in the electricity markets. While ensuring a stable heat and power supply for customers, CHP plant operators seek to maximise the profitability of the CHP plant operation. We propose a comprehensive concept and illustrate its potential for increasing the profitability when operating a CHP plant with heat storage by participating in multiple electricity markets. For this purpose, the structure of the decision-making process consisting of bid submission, market clearing and CHP plant operation planning is represented. A multistage stochastic mixed-integer linear programming (MILP) model is developed that simultaneously optimises the operation of the CHP plant with heat storage and bidding in sequential electricity markets. Price uncertainty is captured by means of stochastic processes. The proposed concept is applied to the real case of a municipal energy supply company that participates in the German day-ahead spot and balancing market. Results of the case study illustrate how trading in sequential electricity markets can result in an increased profitability of the CHP plant operation, where additional revenues from trading offset higher generation costs. Furthermore, we exemplify how the optimal operation of the CHP plant and heat storage device are adjusted according to revenue potential in multiple electricity markets.

A multistage stochastic programming framework for cardinality constrained portfolio optimization

This paper presents a multistage stochastic programming model to deal with multi-period, cardinality constrained portfolio optimization. The presented model aims to minimize investor's expected regret, while ensuring achievement of a minimum expected return. To generate scenarios of market index returns, a random walk model based on the empirical distribution of market-representative index returns is proposed. Then, a single index model is used to estimate stock returns based on market index returns. Afterward, historical returns of a number of stocks, selected from Frankfurt Stock Exchange (FSE), are used to implement the presented scenario generation method, and solve the stochastic programming model. In addition, the impact of cardinality constraints, transaction costs, minimum expected return and predetermined investor's target wealth are investigated. Results show that the inclusion of cardinality constraints and transaction costs significantly influences the investors risk-return tradeoffs. This is also the case for investors target wealth.

Bulk ship fleet renewal and deployment under uncertainty: A multi-stage stochastic programming approach

Faced with simultaneous demand and charter cost uncertainty, an industrial shipping company must determine a suitable fleet size, mix, and deployment strategy to satisfy demand. It acquires vessels by time chartering and voyage chartering. Time chartered vessels are acquired for different durations, a decision made before stochastic parameters are known. Voyage charters are procured for a single voyage after uncertain parameters are realized. We introduce the first multi-stage stochastic programming model for the bulk ship fleet renewal problem and solve it in a rolling horizon fashion. Computational results indicate that our approach outperforms traditional methods relying on expected value forecasts.