Two-stage Stochastic Linear Programming Research Articles

Two-stage stochastic, large-scale optimization of a decentralized energy system: a case study focusing on solar PV, heat pumps and storage in a residential quarter

The expansion of fluctuating renewable energy sources leads to an increasing impact of weather-related uncertainties on future decentralized energy systems. Stochastic modeling techniques enable an adequate consideration of the uncertainties and provide support for both investment and operating decisions in such systems. In this paper, we consider a residential quarter using photovoltaic systems in combination with multistage air-water heat pumps and heat storage units for space heating and domestic hot water. We model the investment and operating problem of the quarter’s energy system as two-stage stochastic mixed-integer linear program and optimize the thermal storage units. In order to keep the resulting stochastic, large-scale program computationally feasible, the problem is decomposed in combination with a derivative-free optimization. The subproblems are solved in parallel on high-performance computing systems. Our approach is integrated in that it comprises three subsystems: generation of consistent ensembles of the required input data by a Markov process, transformation into sets of energy demand and supply profiles and the actual stochastic optimization. An analysis of the scalability and comparison with a state-of-the-art dual-decomposition method using Lagrange relaxation and a conic bundle algorithm shows a good performance of our approach for the considered problem type. A comparison of the effective gain of modeling the quarter as stochastic program with the resulting computational expenses justifies the approach. Moreover, our results show that heat storage units in such systems are generally larger when uncertainties are considered, i.e., stochastic optimization can help to avoid insufficient setup decisions. Furthermore, we find that the storage is more profitable for domestic hot water than for space heating.

A two-stage stochastic programming model for phlebotomist scheduling in hospital laboratories

ABSTRACTThis paper introduces a two-stage stochastic integer linear programming model to improve phlebotomist scheduling in laboratory facilities of healthcare delivery systems. The model developed enables laboratory management to determine optimal scheduling policies that minimize work overload. The stochastic programming model considers the uncertainty associated with the blood collection demand in laboratory environments when optimizing phlebotomist scheduling. The paper presents an application of the model to a hospital laboratory in urban North Carolina as a case study discussing the implications for hospital laboratory management.

Closed-Loop Supply Chain Network Design Under Demand and Return Uncertainty

This study considers an integrated closed-loop supply chain (CLSC) network design problem under uncertainty with regard to product demand and return quantities. To incorporate uncertainty in decision making, we formulate a two-stage stochastic mixed integer linear programming model to determine the optimal locations of (re)manufacturing and processing facilities along with their capacity levels and forward and reverse product flows in the CLSC network to minimize total design and expected operation costs. For the solution of the model and its analysis, we develop a Benders Decomposition approach enhanced for computational efficiency using induced constraints, strengthened Benders cuts, and multiple Benders cuts as well as mean-value scenario based lower-bounding inequalities obtained by dual subproblem disaggregation. Computational results illustrate that the enhancements provide substantial improvements in terms of solution times and quality. Using our model and the solution approach in a sample average approximation framework, we provide further analysis of network designs based on inspection location and recovery rates. Although product inspection at retailer or collection center locations generally reduce costs by avoiding unnecessary use of resources, our analysis also indicates that parameters such as product type and reason-for-return, expected recovery rates, inspection costs, and transportation costs can be instrumental in deciding where the return product inspection should take place and, in turn, dictating the overall cost as well as the structure of the CLSC network.

Detailed long‐term hydro‐thermal scheduling for expansion planning in the Nordic power system

The authors describe a method for long-term hydro-thermal scheduling allowing treatment of detailed large-scale hydro systems. Decisions for each week are determined by solving a two-stage stochastic linear programming problem considering uncertainty in weather and exogenous market prices. The overall scheduling problem is solved by embedding such two-stage problems in a rolling horizon simulator. The method is verified on data for the Nordic power system, studying the incremental changes in expected socio-economic surplus for expansions in both the transmission and generation systems. Comparisons are made with a widely used existing long-term hydro-thermal scheduling model. The results indicate that the model is well suited to valuate the flexibility of hydropower in systems with a high share of intermittent renewable generation.

Coordination of heat and power scheduling in micro-grid considering inter-zonal power exchanges

This paper addresses optimal day-ahead scheduling of Combined Heat and Power (CHP) units of an Active Distribution Network (ADN) with Electric Storage Systems (ESS) and Thermal Storage Systems (TSS) considering Industrial Customers (ICs) inter-zonal power exchanges. The ADN operator may use CHP units to supply its ICs and based on smart grid conceptual model, it can transact electricity with upward wholesale electricity market and its downward ICs' systems; meanwhile its ICs can transact energy with each other through the ADN main grid. Basically, the optimal scheduling of CHP units problem is a Mixed Integer Non Linear Programing (MINLP) problem with many stochastic and deterministic variables. However, the electricity transactions between the ADN and its ICs in normal and contingency scenarios may highly complicate this problem. In this paper, linearization techniques are adopted to linearize equations and a two-stage Stochastic Mixed-Integer Linear Programming (SMILP) model is utilized to solve the problem. The first stage models behavior of operation parameters and minimizes the operation costs; check the feasibility of the ICs' requested firm and non-firm power exchanges, and the second stage considers the system's stochastic contingency scenarios. The competitiveness of ADN in the deregulated market can be improved by adjustment of the proposed decision variables in the two stage optimization procedure. The proposed method is applied to 18-bus, 33-bus IEEE test systems. The effectiveness of the proposed algorithm has been investigated.

An efficient linear model for optimal day ahead scheduling of CHP units in active distribution networks considering load commitment programs

The Optimal day-ahead Scheduling of Combined Heat and Power (OSCHP) units is a crucial problem in the energy management of Active Distribution Networks (ADNs), especially in the presence of Electrical and Thermal Energy Storages considering Load Commitment (LC) programs. The ADN operator may use Combined Heat and Power (CHP) units to supply its Industrial Customers (ICs) and can transact electricity with the upstream wholesale electricity market. The OSCHP problem is a Mixed Integer Non Linear Programming (MINLP) problem with many variables and constraints. However, the optimal operation of CHP units, Electrical and Thermal Energy Storages considering LC programs and contingency scenarios, may highly complicate this problem. In this paper, linearization techniques are adopted to linearize equations and a two-stage Stochastic Mixed-Integer Linear Programming (SMILP) model is utilized to solve the problem. The first stage models the behavior of operation parameters and minimizes the operation costs, verifies the feasibility of the ICs' requested power exchanges and the second stage considers LC programs and the system's stochastic contingency scenarios. The effectiveness of the proposed algorithm has been demonstrated considering 18-bus, 33-bus and 123-bus IEEE test systems.

New framework for optimal scheduling of combined heat and power with electric and thermal storage systems considering industrial customers inter-zonal power exchanges

Introducing Combined Heat and Power (CHP) units into Active Distribution Network (ADN) can significantly affect the problem of optimal generation scheduling. A new method for solving the problem of Optimal Scheduling of Combined Heat and Power (OSCHP) units of an ADN with Electric Storage Systems (ESSs) and Thermal Storage Systems (TSSs) considering Industrial Customers (ICs) Inter-Zonal Power Exchanges (IZPEs) is presented. The ADN operator may use CHP units to supply its ICs and based on smart grid conceptual model, it can transact electricity with upstream network. However, the electricity transactions between the ADN and its ICs in normal and contingency scenarios may highly complicate this problem. In this paper, linearization techniques are adopted to linearize equations and a two-stage stochastic mixed integer linear programming (SMILP) model is utilized to solve the problem to determine the optimal generation scheduling units. The first stage models the behaviour of operation parameters, minimizes the operation costs, and checks the feasibility of the ICs' requested firm and non-firm IZPEs, while the second stage considers system's stochastic contingency scenarios. The competitiveness of ADN in the deregulated market can be improved by adjusting the proposed decision variables in the two-stage optimization procedure. The proposed method is applied to 18- and 123-bus IEEE test systems to thoroughly demonstrate the benefits of implementing inter-zonal power exchanges.

Mathematical model to optimize design of integrated utility supply network and future global hydrogen supply network under demand uncertainty

Existing energy-efficient management and development of alternative energy infrastructure are the most important issues in modern industry. The former has been mainly studied in terms of optimal design of utility supply network. The latter has been consistently researched in the field of strategic planning of future global hydrogen supply network. In this work, we develop an integrated network model of huge chemical industrial complexes to combine various utility supply networks with a global hydrogen supply network. To construct an integrated network model, the steam methane reforming process is used as a linkage between the two networks. The developed model is a two-stage stochastic mixed integer linear programming that optimizes total cost and also considers demand uncertainty about water, electricity, steam, and hydrogen that are consumed in each network. Finally, a case study in South Korea is conducted to validate the proposed model; it suggests reasonable scenarios to decision makers.

Two-Stage Stochastic Programming Model for Robust Personal Rapid Transit Network Design

This study investigated the personal rapid transit (PRT) guideway network design problem at a strategic level. PRT is known as an on-demand mobility method serving the first or last mile of travel demand, which can fluctuate significantly during a day. This on-demand property may make the system vulnerable to changes in the pattern of demand. The robustness of the PRT guideway network refers to the capability to fit different demand patterns. The motivation of this study was to take the uncertainty of demand into consideration in the process of network design to enhance the robustness of the PRT guideway network in operation. This problem was formulated as a two-stage stochastic linear programming model. The model was applied to a case study with real travel demand data from Singapore. The results of the comparison study show that the network designed via the proposed model is robust and cost-effective.

Parallel implementation of augmented Lagrangian method within L-shaped method for stochastic linear programs

In this paper, we study two-stage stochastic linear programming (SLP) problems with fixed recourse. The problem is often large scale as the objective function involves an expectation over a discrete set of scenarios. This paper presents a parallel implementation of the augmented Lagrangian method for solving SLPs. Our parallel method is based on a modified version of the L-shaped method and reducing linear master and recourse programs to unconstrained maximization of concave differentiable piecewise quadratic functions. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in MATLAB. Large scale SLP with several millions of variables and several hundreds of thousands of constraints are solved. The results of uniprocessor and multiprocessor computations are presented which show that the parallel algorithm is effective.

Modelling and solutions of coordinated economic dispatch with wind–hydro–thermal complex power source structure

This study presents an efficient optimisation strategy for solving coordinated economic dispatch problem with wind-hydro-thermal complex power source structure. The wind-hydro-thermal coordinated dispatch aims to minimise the total fuel costs of coal-fired thermal power units while satisfying all kinds of operating constraints. To better handle the random variables in the constraints introduced by wind power and load demand during analysis, a probabilistic analytical model is employed to describe the uncertainty of wind farm power output first; moreover, then an improved convolution method is applied to calculate the total stochastic power consisting of load demand and power output of wind farm. The two-stage stochastic linear programming method and stochastic chance constraints are employed to further form a new deterministic objective function with penalty items taken into account. An enhanced particle swarm optimisation method is applied in the solution of the proposed model. Finally, the simulations are performed on a 6-bus test system and a real-sized China power grid test system to investigate the effectiveness and validity of the proposed optimisation strategy.

Mitigating Uncertainty via Compromise Decisions in Two-Stage Stochastic Linear Programming: Variance Reduction

Stochastic Programming (SP) has long been considered a well-justified yet computationally challenging paradigm for practical applications. Computational studies in the literature often involve approximating a large number of scenarios by using a small number of scenarios to be processed via deterministic solvers, or running Sample Average Approximation on some genre of high performance machines so that statistically acceptable bounds can be obtained. In this paper we show that for a class of stochastic linear programming problems, an alternative approach known as Stochastic Decomposition (SD) can provide solutions of similar quality in far less computational time using ordinary desktop or laptop machines of today. In addition to these compelling computational results, we provide a stronger convergence result for SD, and introduce a new solution concept that we call the compromise decision. This new concept is attractive for algorithms that call for multiple replications in sampling-based convex optimization algorithms. For such replicated optimization, we show that the difference between an average solution and a compromise decision provides a natural stopping rule. We discuss three stopping criteria that enhance the reliability of the compromise decision, reducing bias and variance associated with the result. Finally our computational results cover a variety of instances from the literature, including a detailed study of SONET Switched Network (SSN), a network planning instance known to be more challenging than other test instances in the literature.

Dviejų etapų stochastinio tiesinio programavimo uždavinio agregavimo tyrimas

Darbe nagrinėjamas dviejų etapų stochastinio tiesinio programavimo uždavinys, kai antro etapo kintamieji priklauso nuo atsitiktinio parametro, pasiskirsčiusio pagal normalųjį (Gauso) dėsnį. Sprendžiant šį uždavinį, pritaikytas modifikuotas L pavidalo (L-shaped) algoritmas, realizuotas naudojant CPLEX optimizavimo paketą. Darbe nagrinėjamas scenarijų agregavimo metodas, siekiant sumažinti optimizavimo proceso iteracijų skaičių, naudojamų resursų kiekį ir optimalaus sprendinio gavimo skaičiavimo laiką. Agregavimo metodo efektyvumas ištirtas statistinio modeliavimo būdu.

Optimal day ahead scheduling of combined heat and power units with electrical and thermal storage considering security constraint of power system

The use of Combined Heat and Power (CHP) with an overall efficiency from 70 to 90% is one of the most effective solutions to optimize the energy consumption. Mainly due to interdependence of the power and heat in these systems, the optimal operation of CHP systems is a complex optimization problem that needs powerful solutions. This paper addresses optimal day-ahead scheduling of CHP units with Electric Storage Systems (ESSs) and Thermal Storage Systems (TSSs) considering security constraints. Basically, the optimal scheduling of CHP units problem is a Mixed Integer Non Linear (MINLP) problem with many stochastic and deterministic variables. In this paper, linearization techniques are adopted to linearize equations and a two-stage Stochastic Mixed-Integer Linear Programming (SMILP) model is utilized to solve the problem. The first stage models behavior of operation parameters and minimizes the operation costs meanwhile the second stage considers the system's stochastic contingency scenarios. The proposed method is applied to 18-bus, 24-bus IEEE test systems. The effectiveness of the proposed algorithm has been investigated.

Multi-objective stochastic multi-site supply chain planning under demand uncertainty considering downside risk

In this paper, a multi-period, multi-product, multi-site, multi-stage supply chain planning problem under demand uncertainty is considered. The problem is formulated as a two-stage stochastic linear programming model. In order to generate a robust supply chain planning solution, the downside risk is incorporated into the objective functions of the stochastic programming model as a risk measure. So, the proposed multi-objective stochastic model aims to simultaneously minimize the expected total cost, to minimize the lost customer demand level and to minimize the downside risk. The proposed solution approach yields to a front of Pareto optimal robust solutions. A fuzzy decision making approach is applied to select the most preferred solution among the Pareto optimal robust solutions. A numerical example from a real textile and apparel industry is addressed in order to illustrate the robustness of the supply chain network planning solutions and the effectiveness of the solution approach.

Assessing the economic value of co-optimized grid-scale energy storage investments in supporting high renewable portfolio standards

Worldwide, environmental regulations such as Renewable Portfolio Standards (RPSs) are being broadly adopted to promote renewable energy investments. With corresponding increases in renewable energy deployments, there is growing interest in grid-scale energy storage systems (ESS) to provide the flexibility needed to efficiently deliver renewable power to consumers. Our contribution in this paper is to introduce a unified generation, transmission, and bulk ESS expansion planning model subject to an RPS constraint, formulated as a two-stage stochastic mixed-integer linear program (MILP) optimization model, which we then use to study the impact of co-optimization and evaluate the economic interaction between investments in these three asset classes in achieving high renewable penetrations. We present numerical case studies using the 24-bus IEEE RTS-96 test system considering wind and solar as available renewable energy resources, and demonstrate that up to $180million/yr in total cost savings can result from the co-optimization of all three assets, relative to a situation in which no ESS investment options are available. Surprisingly, we find that co-optimized bulk ESS investments provide significant economic value through investment deferrals in transmission and generation capacity, but very little savings in operational cost. Finally, we observe that planning transmission and generation infrastructure first and later optimizing ESS investments—as is common in industry—captures at most 1.7% ($3million/yr) of the savings that result from co-optimizing all assets simultaneously.

Optimal Monitoring and Mitigation of Systemic Risk in Financial Networks

This paper studies the problem of optimally allocating a cash injection into a financial system in distress. Given a one-period borrower-lender network in which all debts are due at the same time and have the same seniority, we address the problem of allocating a fixed amount of cash among the nodes to minimize the weighted sum of unpaid liabilities. Assuming all the loan amounts and asset values are fixed and that there are no bankruptcy costs, we show that this problem is equivalent to a linear program. We develop a duality-based distributed algorithm to solve it which is useful for applications where it is desirable to avoid centralized data gathering and computation. Since some applications require forecasting and planning for a wide variety of different contingencies, we also consider the problem of minimizing the expectation of the weighted sum of unpaid liabilities under the assumption that the net external asset holdings of all institutions are stochastic. We show that this problem is a two-stage stochastic linear program. To solve it, we develop two algorithms based on Monte Carlo sampling: Benders decomposition algorithm and projected stochastic gradient descent. We show that if the defaulting nodes never pay anything, the deterministic optimal cash injection allocation problem is an NP-hard mixed-integer linear program. However, modern optimization software enables the computation of very accurate solutions to this problem on a personal computer in a few seconds for network sizes comparable with the size of the US banking system. In addition, we address the problem of allocating the cash injection amount so as to minimize the number of nodes in default. For this problem, we develop two heuristic algorithms: a reweighed l1 minimization algorithm and a greedy algorithm. We illustrate these two algorithms using three synthetic network structures for which the optimal solution can be calculated exactly. We also compare these two algorithms on three types random networks which are more complex.

Two-stage stochastic mixed-integer linear programming: The conditional scenario approach

In this paper we consider the two-stage stochastic mixed-integer linear programming problem with recourse, which we call the RP problem. A common way to approximate the RP problem, which is usually formulated in terms of scenarios, is to formulate the so-called Expected Value (EV) problem, which only considers the expectation of the random parameters of the RP problem. In this paper we introduce the Conditional Scenario (CS) problem which represents a midpoint between the RP and the EV problems regarding computational tractability and ability to deal with uncertainty. In the theoretical section we have analyzed some useful bounds related to the RP, EV and CS problems. In the numerical example here presented, the CS problem has outperformed both the EV problem in terms of solution quality, and the RP problem with the same number of scenarios as in the CS problem, in terms of solution time.

A new cross decomposition method for stochastic mixed-integer linear programming

Two-stage stochastic mixed-integer linear programming (MILP) problems can arise naturally from a variety of process design and operation problems. These problems, with a scenario based formulation, lead to large-scale MILPs that are well structured. When first-stage variables are mixed-integer and second-stage variables are continuous, these MILPs can be solved efficiently by classical decomposition methods, such as Dantzig/Wolfe decomposition (DWD), Lagrangian decomposition, and Benders decomposition (BD), or a cross decomposition strategy that combines some of the classical decomposition methods. This paper proposes a new cross decomposition method, where BD and DWD are combined in a unified framework to improve the solution of scenario based two-stage stochastic MILPs. This method alternates between DWD iterations and BD iterations, where DWD restricted master problems and BD primal problems yield a sequence of upper bounds, and BD relaxed master problems yield a sequence of lower bounds. The method terminates finitely to an optimal solution or an indication of the infeasibility of the original problem. Case study of two different supply chain systems, a bioproduct supply chain and an industrial chemical supply chain, show that the proposed cross decomposition method has significant computational advantage over BD and the monolith approach, when the number of scenarios is large.

Stochastic fuzzy multi-objective backbone selection and capacity allocation problem under tax-band pricing policy with different fuzzy operators

In this paper, we investigate a multi-objective optimization problem that a telecom bandwidth broker (BB) faces when acquiring and selling bandwidth in an uncertain market environment in which there exists several backbone providers (BPs) and end users. The proposed model incorporates two important goals: maximizing expected profit and minimizing expected loss capacity within realistic constraints such as BPs' capacity, meeting the end users' bandwidth requests and satisfying the Quality of Service requests of end users', considering stochastic capacity loss rates of BPs. The fuzzy set theory and stochastic programming techniques are employed to handle the non-deterministic nature of telecommunication network setting due to the presence of vagueness and randomness of information. The model is formulated in such a way that it simultaneously considers the randomness in demand and determines the allocation of end users' bandwidth requests into purchased capacity based on tax-band pricing scheme. As solution strategies, two different fuzzy operators, namely max---min and weighted additive max---min, are integrated into a resulting two-stage multi-objective stochastic linear programming model. Then, algorithms are provided to solve and to compare methodologies with deterministic approaches. Finally, the proposed algorithms are tested on several randomly generated test scenarios to provide managerial insight to decision makers of BB companies.