Two-stage Robust Optimization Problem Research Articles

Machine Learning for K-Adaptability in Two-Stage Robust Optimization

Two-stage robust optimization problems constitute one of the hardest optimization problem classes. One of the solution approaches to this class of problems is K-adaptability. This approach simultaneously seeks the best partitioning of the uncertainty set of scenarios into K subsets and optimizes decisions corresponding to each of these subsets. In a general case, it is solved using the K-adaptability branch-and-bound algorithm, which requires exploration of exponentially growing solution trees. To accelerate finding high-quality solutions in such trees, we propose a machine learning-based node selection strategy. In particular, we construct a feature engineering scheme based on general two-stage robust optimization insights, which allows us to train our machine learning tool on a database of resolved branch-and-bound trees and to apply it as is to problems of different sizes and/or types. We experimentally show that using our learned node selection strategy outperforms a vanilla, random node selection strategy when tested on problems of the same type as the training problems as well as in cases when the K-value or the problem size differs from the training ones. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grants OCENW.GROOT.2019.015 and VI.Veni.191E.035]. 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.2022.0314 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0314 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Distributionally robust risk-resistant synergistic restoration strategy for integrated electricity and gas system with high-fidelity gas dynamic modeling

A synergistic restoration strategy offers a promising solution for expediting restoration of integrated electricity and gas system (IEGS) following blackouts, particularly under the increasing energy interdependence between these two subsystems. However, achieving synergistic restoration is challenging in 1) distinct operational characteristics between electric and gas flow rates, leading to difficulties in accurately modeling the restoration processes with significantly varying system states; 2) potential risks posed by uncertainties, such as fluctuating renewable energy and random contingencies, undermining the effectiveness of restoration strategy. To address these challenges, this paper proposes a novel distributionally robust risk-resistant synergistic restoration strategy for IEGS. Firstly, to accurately capture the dynamics of slow gas flow relative to instantaneous power flow, we propose a high-fidelity linear dynamic gas flow model with multiparametric disaggregation technique, demonstrating satisfactory accuracy in restoration scenarios. Then, the risks by uncertainties of renewable energy and contingencies are handled within distributionally robust framework. Moreover, the IEGS restoration problem is reformulated as a two-stage robust optimization problem by convex conservative approximation and strong duality theory, which is solved by a nested column-and-constraint generation algorithm. Finally, simulation results validate the effectiveness of the proposed restoration strategy, showing the improved restoration efficiency, security, and risk-resistant performance.

Probabilistic Forecasting-Based Reserve Determination Considering Multi-Temporal Uncertainty of Renewable Energy Generation

Operating reserve, among the most important ancillary services, is a powerful prescription for mitigating increasing uncertainty of renewable generation. Traditional reserve determination neglects uncertainty of renewable generation variations within the dispatch interval, which cannot gurantee sufficient reserve deliverability in real-time operation. This paper proposes a probabilistic forecasting-based reserve determination method to efficiently deal with multi-temporal uncertainty of renewable energy in power systems. A novel probabilistic forecasting approach is proposed by integrating Dirichlet process Gaussian mixture model into bootstrap-based extreme learning machine. A unified uncertainty model is constructed for depicting both interval-averaged uncertainty and intra-interval uncertainty by combing the probabilistic forecasts and linearized Itô-process model. In current business practices of many independent system operators, the coordinated determination of two well-designed reserve products, including regulating reserve and ramp capability reserve, is formulated in a two-stage robust optimization framework. A Bernstein polynomial-based model reformulation approach is then employed to handle the existing heterogeneity in the sub-models of each stage. Consequently, the integrated reserve determination is embedded in a generic two-stage robust optimization problem, which can be efficiently solved by the adopted modified column-and-constraint generation method. Finally, numerical simulations are implemented to validate the effectiveness and profitability of the proposed approach.

Two-stage robust optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage hybrid energy system

The hybrid energy system of hydro-powers, pumped storages and renewable energies has become a new topic direction in modern power system developments. Consequently, it is essential to realize a rational and efficient allocation of different energy source capacities. Nevertheless, there is still a gap between the available studies and the requirement for further hybrid energy system development. This paper focuses on the optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage power system. In this direction, a bi-level programming model for the optimal capacity configuration of wind, photovoltaic, hydropower, pumped storage power system is derived. To model the operating mode of a pumped storage power station, two 0-1 variables are introduced. To handle the nonlinear and nonconvex lower level programing problem caused by the two 0-1 variables, it is proposed that the 0-1 variables are treated as some uncertain parameters. Also, by treating the 0-1 variables as some uncertain parameters, a two-stage robust optimization problem to decompose the original bi-level programing one into a master problem and a subproblem is finally introduced. The Karush-Kuhn-Tucker (KKT) conditions are then applied to simplify and linearize the min-max problem and nonlinear terms in the master problem. This results in both the master problem and the subproblem being formulated as mixed integer linear programming (MILP) problems. By utilizing the powerful Column-and-Constraint Generation (C&CG) algorithm, the two-stage robust optimization model is decomposed into an iterative procedure of solving the master problem and the subproblem sequentially. This approach eliminates the need for intricate optimization algorithms as commonly used in existing bi-level planning problems in hybrid energy systems. Finally, the effectiveness and advantages of the proposed model is verified by the numerical results on a case study.

Resilient Expansion Planning of Electricity Grid Under Prolonged Wildfire Risk

Public Safety Power Shut-off (PSPS) during potentially dangerous weather conditions is an emergency measure within areas prone to wildfires. While balancing the risk of wildfire ignition and the continuation of energy supply is a short-term operation challenge, this paper explores the optimal long-term resilient expansion planning strategies. With the quantified risk of wildfire ignition, the proposed expansion planning problem maximizes the supplied power to the end-users while limiting the risk of wildfire ignition. The presented scheme provides utility decision-makers with three groups of network expansion decisions, namely addition of new lines, modification of existing lines, and installation of Distributed Energy Resources (DERs). Given the uncertainty of DERs and wildfire risk, a two-stage robust optimization problem is proposed which ensures power system resilience against unfavorable events. The case studies illustrate how the presented model balances between shutting off customers, DER installation, and line addition/modification while inhibiting wildfire ignition risk. Besides, the results suggest that, with appropriate transmission switching, DER installation can be the optimal choice for meeting the growing demand while limiting wildfire ignition risk. The implications of different risk level choices are illustrated, and the impacts of uncertainties on the expansion decisions are explored. Finally, the results of applying the proposed model to the IEEE 118-bus test network are illustrated.

The robust cyclic job shop problem

This paper deals with the cyclic job shop problem where the task durations are uncertain and belong to a polyhedral uncertainty set. We formulate the cyclic job shop problem as a two-stage robust optimization model. The cycle time and the execution order of tasks executed on the same machines correspond to the here-and-now decisions and have to be decided before the realization of the uncertainty. The starting times of tasks corresponding to the wait-and-see decisions are delayed and can be adjusted after the uncertain parameters are known. In the last decades, different solution approaches have been developed for two-stage robust optimization problems. Among them, the use of affine policies, row and row-and-column generation algorithms are the most common. In this paper, we propose a branch-and-bound algorithm to tackle the robust cyclic job shop problem with cycle time minimization. The algorithm uses, at each node of the search tree, a robust version of the Howard’s algorithm to derive a lower bound on the optimal cycle time. Moreover, we design a row generation algorithm and a column-and-row generation algorithm and compare it to the branch-and-bound method. Finally, encouraging preliminary results on numerical experiments performed on randomly generated instances are presented.

Two-stage robust optimization strategy for spatially-temporally correlated data centers with data-driven uncertainty sets

Data center buildings (DCBs) in data center parks consume significant and ever-growing amounts of electric power for processing tremendous data workloads in a modern city. The workloads are divided into two types: interactive workloads (IWs) with spatially transferrable characteristics and batch workloads (BWs) with temporally shiftable characteristics. The unique spatially-temporally workloads show great demand response capability, enabling reducing the electricity cost of DCBs. However, the source-load uncertainties such as the number of arriving workloads, and photovoltaic output affect the cost-efficient operation of DCBs in the electricity market. Inadequate server numbers during worst-case scenarios may result in the inability to fulfill user contracts. This paper presents a model for DCBs that incorporates battery energy storage systems, photovoltaic generation, and electrical loads for processing spatially-temporally correlated workloads. Considering multiple uncertainties, a two-stage robust optimization strategy is proposed to coordinate workload and power for DCBs. To accurately determine the boundaries of the uncertainty sets, a data-driven method using the 1-Wasserstein metric is adopted. This method is reformulated into a distributionally robust chance-constrained programming model. The nested column-and-constraint generation (C&CG) method is used to resolve the established two-stage robust optimization problem with mixed-integer recourse. Finally, case studies verify the effectiveness of the proposed method.

Unlock the Thermal Flexibility in Integrated Energy Systems: A Robust Nodal Pricing Approach for Thermal Loads

The thermal flexibility of buildings in integrated energy systems (IES) has great potential to improve the operational economy and wind power consumption. To incentivize the thermal flexibility of buildings under uncertainties, we propose a robust nodal pricing (RNP) model for thermal loads based on the Stackelberg game approach. The RNP model adopts a bilevel framework in which the IES operator plays the leader at the upper level while the thermal load aggregators (TLA) play the followers at the lower level. The IES operator problem optimizes the heat prices and dispatch plan, which is modeled as a two-stage robust optimization problem to address the uncertainties in the renewables and the loads. The TLA problem optimizes the thermal loads of buildings to minimize the energy cost and thermal comfort loss, which is modeled as a distributionally robust chance-constrained optimization problem to address the uncertainty in outdoor temperature. Then, we convert the TLA model into deterministic quadratic programming and prove that Slaters condition holds under a mild assumption. Using it, the TLA model is equivalently converted into Karush-Kuhn-Tucker conditions, leading to a reformulation of a classical two-stage robust optimization model for the RNP model. Case studies compare different pricing methods and verify the superiority of the proposed method.

Two-stage robust unit commitment with the cascade hydropower stations retrofitted with pump stations

Cascade hydropower stations are excellent flexible resources to regulate the drastic fluctuations of wind and photovoltaic power generation in the hybrid energy system. By constructing pump stations between two adjacent upstream and downstream reservoirs, the conventional cascade hydropower stations can be transformed into a cascade pumped hydro energy storage (CPHES) system, which further can promote the integration of clean energy resources. In this paper, a two-stage robust unit commitment model for the cascade hydropower stations retrofitted with pump stations is established to address the renewable energy uncertainties. A short-term scheduling framework is proposed for the hybrid energy system including CPHES, which coordinates the operating cost and clean energy curtailment. Besides, a modified nested column-and-constraint algorithm is employed to solve the two-stage robust optimization problem with integer resources. Numerical tests performed on a modified IEEE 24-bus system and a large-scale practical power system verify the effectiveness of the proposed scheduling model and algorithm.

Two-Stage Robust Optimization Model for Capacity Configuration of Biogas-Solar-Wind Integrated Energy System

Biogas-solar-wind integrated energy systems are effective for optimizing rural energy consumption and improving agricultural production. The performance of an integrated energy system generally depends on its capacity configuration. However, the uncertainties of renewable energy sources and loads deepen the coupling relationship between the capacity configuration and energy dispatching of integrated energy systems, which makes optimizing the capacity difficult. We propose a two-stage robust optimization model for the capacity configuration of a biogas-solar-wind integrated energy system that is applicable to rural areas. First, a framework of the biogas-solar-wind integrated energy system was designed and diverse evaluation indices were introduced. Then, the integrated energy system model was transformed into a two-stage robust optimization problem, where the column-and-constraint generation algorithm is used to solve the installed capacity problem of the system equipment in the first stage, and the nested column-and-constraint generation algorithm is used to optimize the energy dispatching schedule in the second stage. Finally, the proposed model was applied to a rural area in China to confirm the rationality and effectiveness of the optimization results.

An inexact column-and-constraint generation method to solve two-stage robust optimization problems

We propose a new inexact column-and-constraint generation (i-C&CG) method to solve two-stage robust optimization problems. The method allows solutions to the master problems to be inexact, which is desirable when solving large-scale and/or challenging problems. It is equipped with a backtracking routine that controls the trade-off between bound improvement and inexactness. Importantly, this routine allows us to derive theoretical finite convergence guarantees for our i-C&CG method. Numerical experiments demonstrate computational advantages of our i-C&CG method over state-of-the-art column-and-constraint generation methods.

Location-allocation-routing for emergency shelters based on geographical information system (ArcGIS) by NSGA-II (case study: Earthquake occurrence in Tehran (District-1))

The purpose of this study is to locate temporary shelters and assign injured people to the shelters. Routing people from destroyed areas to newly constructed shelters (in order to conduct rescue operations in the event of an earthquake) based on a case study of one of Tehran's densest areas (District 1) and the extent to which it is vulnerable to the occurrence of a possible earthquake is also considered. Some criteria influenced by the natural and physical characteristics of the desired area have been developed to accomplish this. The characteristics of this area are considered using Geographic Information Systems (GIS) software for determining the affected areas (as the demand points), and the ideal locations for shelters construction. Based on mathematical modeling, the optimal locations for shelters have been determined to allocate injured individuals to those shelters. Due to the existence of uncertainty regarding activated fault, their magnitude, severity, and the amount of demand for rescue services, multiple scenarios for earthquake occurrence have been considered, and the model is presented as a two-stage robust scenario-based optimization problem. The Non-Dominated Sorting Genetic Algorithm (NSGA-II) is proposed to solve the problem. The computational results demonstrate the effectiveness of the proposed methodology. Suitable locations for construction of shelters, allocating shelters to affected areas, the optimal route for collecting people, the arrival time of each vehicle in each affected area, the total rescue time, the required budget, and the number of injured people collected from each area are the outputs of the model. This study aims to develop an adaptable and rational system for crisis relief in cities with varying characteristics.

Disjoint Bilinear Optimization: A Two-Stage Robust Optimization Perspective

In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this end, a solution scheme based on a blending of three popular robust optimization techniques is proposed. For disjoint bilinear optimization problems with a polyhedral feasible region and a general convex feasible region, we show that, under mild regularity conditions, the convex relaxations of the original bilinear formulation and its two-stage robust reformulation obtained from a reformulation-linearization-based technique and linear decision rules, respectively, are equivalent. For generic bilinear optimization problems, the convex relaxations from the reformulation-linearization-based technique are generally tighter than the one from linear decision rules. Numerical experiments on bimatrix games, synthetic disjoint bilinear problem instances, and convex maximization problems demonstrate the efficiency and effectiveness of the proposed solution scheme. Summary of Contribution: Computing solutions for disjoint bilinear optimization problems are of much interest in real-life applications, yet they are, in general, computationally intractable. This paper proposes a computationally tractable approximation as well as a convergent algorithm to the optimal values of such problems. Extensive computational experiments on (i) (constrained) bimatrix games, (ii) synthetic disjoint bilinear problems, and (iii) convex maximization problems are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

A Lagrangian dual method for two-stage robust optimization with binary uncertainties

This paper presents a new exact method to calculate worst-case parameter realizations in two-stage robust optimization problems with categorical or binary-valued uncertain data. Traditional exact algorithms for these problems, notably Benders decomposition and column-and-constraint generation, compute worst-case parameter realizations by solving mixed-integer bilinear optimization subproblems. However, their numerical solution can be computationally expensive not only due to their resulting large size after reformulating the bilinear terms, but also because decision-independent bounds on their variables are typically unknown. We propose an alternative Lagrangian dual method that circumvents these difficulties and is readily integrated in either algorithm. We specialize the method to problems where the binary parameters switch on or off constraints as these are commonly encountered in applications, and discuss extensions to problems that lack relatively complete recourse and to those with integer recourse. Numerical experiments provide evidence of significant computational improvements over existing methods.

Two-stage robust energy storage planning with probabilistic guarantees: A data-driven approach

Shorter-term (e.g., hourly) uncertainties, which are not explicitly accounted for in conventional power system planning practice, become imperative in the longer-term planning with deepening penetration of renewable energy resources. This paper addresses this central issue in power system planning: the challenges induced by the increasing short-term and long-term uncertainties and the pivotal opportunities from the rapidly growing flexible resources (e.g., storage devices). By leveraging the abundant operation data, we propose a data-driven power system planning framework based on robust optimization and the scenario approach. The proposed framework considers a broad range of operation conditions and provides rigorous theoretical guarantees on the future risk of planning decisions. By connecting two-stage robust optimization with the scenario approach theory, we show that the operation risk level of the robust solution can be adaptable to the risk preference set by planners. The theoretical guarantees hold true for any distribution, and the proposed approach is scalable towards real-world power systems. Furthermore, we show that the column-and-constraint generation algorithm, which is a popular algorithm to solve two-stage robust optimization problems, is capable of tightening theoretical guarantees. We substantiate this framework through a planning problem of energy storage in a power grid with significant renewable penetration. Case studies are performed on large-scale test systems (modified IEEE 118-bus system) to illustrate the theoretical bounds as well as the scalability of the proposed algorithm.

Two-Stage robust optimization problems with two-stage uncertainty

We consider two-stage robust optimization problems, which can be seen as games between a decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution.We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing two-stage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. For the former, we show that a wide range of robust combinatorial optimization problems can be decomposed into polynomially many subproblems, which can be solved in polynomial time for example in the case of (representative) selection. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where first- and second-stage adversarial costs are equal can remain solvable in polynomial time.

Decomposition-Based Approaches for a Class of Two-Stage Robust Binary Optimization Problems

In this paper, we study a class of two-stage robust binary optimization problems with objective uncertainty, where recourse decisions are restricted to be mixed-binary. For these problems, we present a deterministic equivalent formulation through the convexification of the recourse-feasible region. We then explore this formulation under the lens of a relaxation, showing that the specific relaxation we propose can be solved by using the branch-and-price algorithm. We present conditions under which this relaxation is exact and describe alternative exact solution methods when this is not the case. Despite the two-stage nature of the problem, we provide NP-completeness results based on our reformulations. Finally, we present various applications in which the methodology we propose can be applied. We compare our exact methodology to those approximate methods recently proposed in the literature under the name [Formula: see text]adaptability. Our computational results show that our methodology is able to produce better solutions in less computational time compared with the [Formula: see text]adaptability approach, as well as to solve bigger instances than those previously managed in the literature. Summary of Contribution: Our manuscript describes an exact solution approach for a class of robust binary optimization problems with mixed-binary recourse and objective uncertainty. Its development reposes first on a reformulation of the problem, then a carefully constructed relaxation of this reformulation. Our solution approach is designed to exploit the two-stage and binary structure of the problem for effective resolution. In its execution, it relies on the branch-and-price algorithm and its efficient implementation. With our computational experiments, we show that our proposed exact solution method outperforms the existing approximate methodologies and, therefore, pushes the computational envelope for the class of problems considered.

A Wind Power Accommodation Capability Assessment Method for Multi-Energy Microgrids

With the increase in wind generation installed capacity, the uncertainty of wind power has brought great challenges to wind power accommodation. The accurate assessment result of wind power accommodation capability (WPAC) has important guiding significance to the formulation of the day-ahead wind power trading plan for the multi-energy microgrid (MEMG). This paper proposes a WPAC assessment method for the MEMG. A liner energy flow model is first derived based on the equivalent infinitesimal replacement and the mild assumption that supply temperatures of different heat load nodes are nearly the same. Then, the operational risk model easily embedded into the assessment model is established based on the probability distributions of prediction error corresponding to multiple wind power output ranges. Finally, the WPAC assessment model is constructed as a two-stage robust optimization problem and solved by the column-and-constraint generation (C&CG) algorithm. Simulations on a MEMG composed of the IEEE 33-bus power system and the 23-node district heating network (DHN) verify the effectiveness of the proposed method.

A robust dispatch model for integrated electricity and heat networks considering price-based integrated demand response

Wind power integration may be restricted due to the asynchrony of electricity and heat loads and the strong coupling between the power generation and heat production of combined heat and power (CHP) units. The price-based integrated demand response (PBIDR) can adjust electricity and heat loads to accommodate wind power. But, it is tricky to formulate appropriate electricity and heat prices to make the system having enough flexibility and achieves the best economic benefits. This paper proposes a robust dispatch model for integrated electricity and heat networks (IEHNs) based on PBIDR. The linear model of the district heating network (DHN) is first derived based on the equivalent infinitesimal replacement theorem and the assumption that the supply temperatures at load nodes are approximately equal. Then, a computationally tractable PBIDR model is formulated in terms of the theory of price elasticity of demand and the concept of thermal sensation vote (TSV). Finally, the dispatch model is constructed as a two-stage robust optimization problem to hedge the uncertainty of the wind power output, the ambient temperature, and the PBIDR. The effectiveness of the proposed model is verified by two different scale test systems.

Optimal Design of Distributed Energy Resource Systems under Uncertainties Based on Two-Stage Robust Optimization

Distributed energy resource (DER) systems are widely used owing to their excellent economic and environmental performance. However, uncertainties in the system generate difficulties in the optimal design of DER systems. In practice, the distribution of uncertain parameters is generally unknown. In this work, a two-stage robust optimization (RO) model was proposed for the optimal design of DER systems considering uncertainties in renewable energy intensity, energy prices, and load demands. Three uncertainty sets (i.e., the box, ellipsoid, and convex-hull uncertainty sets) were adopted to describe the distribution of uncertain parameters, and the proposed two-stage RO problem was solved using affine decision rules. A typical hospital in Lianyungang, Jiangsu Province, China, was selected as the case study object, and the effectiveness of the model was verified. The case study results showed that uncertainties in energy prices and load demands have a significant impact on system configuration and economic performance, and mainly affect the installed capacities of gas boilers, absorption chillers, and storages. Uncertainty set will affect the optimization results and an appropriate uncertainty set should be adopted to describe uncertainties precisely and increase accuracy of results.