Two-stage Robust Optimization Approach Research Articles

A hybrid physical-data approach for solving dynamic optimal power flow considering uncertainties and different topology configurations

The security and economy of power system operation or planning are threatened by various uncertain factors, including the unpredictability of renewable energy, the variation of topology configurations, and significant load fluctuation. It is a challenge to solve dynamic optimal power flow efficiently for the operation and planning of power system considering uncertainties on both Renewable generation and Load, as well as different Topology configurations (RLT-DOPF). For fast solving the RLT-DOPF problem, we propose a hybrid physical-data approach using a physics-driven method — two-stage robust optimization approach based on an improved uncertainty set, and a data-driven method combined the Graph convolutional network with Long short term memory named GL-DOPF. The two-stage robust optimization approach can supply robust solutions for the RLT-DOPF problem under various topology configurations. Then the GL-DOPF model is trained using a training dataset that includes robust solutions. The solutions for RTL-DOPF are available with the trained GL-DOPF model and an AC power flow calculation fast. Simulations on IEEE 30/300-bus systems show that the speedup of the proposed hybrid physical-data approach is x25/29 as compared to a robust optimization method for RLT-DOPF under almost the same accuracy and the solutions for RTL-DOPF are feasible and robust. The training dataset for GL-DOPF generated from the proposed two-stage robust optimization model is less conservative than the original RO models and can cover the prediction errors of uncertainties. The effectiveness and robustness of the proposed hybrid physical-data approach for the RLT-DOPF problem clearly outperform.

Two-Stage Robust Liner Container Booking with Uncertain Customer Demand

Since third-party logistics (3PL) offers discounted prices for booking containers in advance, the container booking decisions are preferably made prior to the realization of customer demand by the companies with international trade business. When facing uncertain customer demand influenced by various factors, it is significant for companies to make proper container booking decisions and order fulfillment plans to minimize the total cost, including the container booking cost and the penalty cost of an unfulfilled order. In this paper, a two-stage robust liner container booking problem (2-RLCBP) was investigated, and an exact algorithm with high efficiency was designed to solve the problem based on the column-and-constraint generation (C&CG) framework. Compared to other methods with gaps of more than 70% existing after 3600 s, the proposed algorithm can get the optimal solution of the real case instance in 700 s. Furthermore, compared to the deterministic model, the two-stage robust optimization model can reduce more than 10% of the worst-case total cost when the budget level and demand deviation level are greater than 0.6, which verifies the effectiveness of the proposed robust approach. Furthermore, our work is the first to adopt a two-stage robust optimization approach for the problem of booking a liner container.

Day-ahead dispatch approach for cascaded hydropower-photovoltaic complementary system based on two-stage robust optimization

Cascade hydropower (CHP) is widely used to increase the integration of photovoltaic (PV) power in recent years. However, the randomness and variability of PV power output might cause power imbalances between generation plan and integrated power of complementary systems. It is crucial to provide robust day-ahead schemes to ensure sufficient adjustment capability of CHPs. In this paper, a day-ahead dispatch model is established for the CHP-PV complementary system to address the PV uncertainties based on the two-stage robust optimization approach. The uncertain PV power output is described by an ambiguity set without accurate probabilistic information. Two different stages in the dispatch model are integrated into a holistic optimization problem. In the first stage, the day-ahead startup/shutdown and operation zone schemes for CHP are determined. In the second stage, the re-dispatch problem is performed to compensate for PV power fluctuations and minimize water consumption. The nested column-and-constraint generation (NC&CG) algorithm is employed as the exact solution methodology. Simulation studies based on a system in southwest China are conducted to validate the effectiveness of the proposed dispatch model and the solution algorithm. The numerical results show that the robust solution can accommodate various scenarios of PV power output and the power imbalance can always be reduced.

Game-theoretic genetic-priced optimization of multiple microgrids under uncertainties

To address an economic dispatch issue of multiple microgrids (MGs) in a game-theoretic framework under source-load uncertainties, a robust optimization scheme with genetic-priced mechanism is proposed. This scheme is a nested iterative algorithm with an outer loop calculating the pricing model in a game and an inner loop solving the robust optimization problem. Specifically, in the outer loop, all stakeholders in a grid-connected microgrid cluster (MGC), i.e. one energy trading center (ETC) and several MGs, are in a game where a genetic-priced model is developed. By utilizing this evolutionary pricing model, the ETC can fix electricity prices to maximize its profits and these prices are passed to the inner loop where a two-stage robust optimization approach is introduced to mitigate adverse effects of uncertainties in MGs induced by renewable energy resources (RERs) and loads. All optimization problems of MGs are solved and optimal values of electricity exchanged between ETC and MGs are passed to the outer loop. This optimization scheme can help address the robust economic dispatch problem in a game-theoretic framework. A grid-connected MGC is used as a case to illustrate the effectiveness of the proposed optimization scheme.

Effective Budget of Uncertainty for Classes of Robust Optimization

Robust optimization (RO) tackles data uncertainty by optimizing for the worst-case scenario of an uncertain parameter and, in its basic form, is sometimes criticized for producing overly conservative solutions. To reduce the level of conservatism in RO, one can use the well-known budget-of-uncertainty approach, which limits the amount of uncertainty to be considered in the model. In this paper, we study a class of problems with resource uncertainty and propose a robust optimization methodology that produces solutions that are even less conservative than the conventional budget-of-uncertainty approach. We propose a new tractable two-stage robust optimization approach that identifies the “ineffective” parts of the uncertainty set and optimizes for the “effective” worst-case scenario only. In the first stage, we identify the effective range of the uncertain parameter, and in the second stage, we provide a formulation that eliminates the unnecessary protection for the ineffective parts and, hence, produces less conservative solutions and provides intuitive insights on the trade-off between robustness and solution conservatism. We demonstrate the applicability of the proposed approach using a power dispatch optimization problem with wind uncertainty. We also provide examples of other application areas that would benefit from the proposed approach.

Two-stage robust optimization approach for flexible oxygen distribution under uncertainty in integrated iron and steel plants

Optimal oxygen distribution is one of the most important energy management problems in the modern iron and steel industry. Normally, the supply of the energy generation system is determined by the energy demand of manufacturing processes. However, the balance between supply and demand fluctuates frequently, owing to the uncertainty arising in manufacturing processes. In this study, we developed an optimal oxygen distribution strategy considering uncertain demands and proposed a two-stage robust optimization (TSRO) model with a budget-based uncertainty set that protects the initial distribution decisions with low conservativeness. The main goal of the TSRO model is to make “wait-and-see” decisions, maximizing energy profits, and make “here-and-now” decisions, minimizing operational stability and surplus/shortage penalty. To represent the uncertainty set of energy demands, we developed (1) a Gaussian process-based time-series model to forecast the demand intervals for continuous processes, and (2) a capacity-constrained scheduling model to generate multi-scenario demands for discrete processes. We performed extensive computational studies on TSRO and its components using well-synthesized instances from historical data. The results of model validation and analysis are promising and demonstrate that our approach is well adapted to solving industrial cases under uncertainty.

Managing reliable emergency logistics for hazardous materials: A two-stage robust optimization approach

This paper investigates the impact of possible system disruptions on the performance of an emergency logistics system for hazardous materials (hazmats). More specifically, we first present a time-varying risk assessment method that takes into consideration the population dynamism. Then the two-stage robust optimization approach is employed to formulate two mixed-integer programming models, namely the basic and expanded unit commitment models, for managing a reliable emergency response system. The locations of emergency facilities are determined in the first stage, and recourse decisions are made in the second stage after the uncertain disruptions are realized. A column-and-constraint-generation algorithm is used to solve the proposed models exactly and tested on various sized random instances. A real-world case study with a series of numerical analyses reveals managerial insights that can be applied to facilitate more effective and efficient emergency responses.

A two-stage robust optimization approach for the master surgical schedule problem under uncertainty considering downstream resources

This paper addresses a planning decision for operating rooms (ORs) that aim at supporting hospital management. Focusing on elective patients, we determined the master surgical schedule (MSS) on a one-week time horizon. We assigned the specialties to available sessions and allocated surgeries to them while taking into consideration the priorities of the outpatients in the ambulatory surgical discipline. Surgeries were selected from the waiting lists according to their priorities. The proposed approach considered operating theater (OT) restrictions, patients' priorities and accounted for the availability of both intensive care unit (ICU) beds and post-surgery beds. Since the management decisions of hospitals are usually made in an uncertain environment, our approach considered the uncertainty of surgery duration and availability of ICU bed. Two robust optimization approaches that kept the model computationally tractable are described and applied to deal with uncertainty. Computational results based on a medium-sized French hospital archives have been presented to compare the robust models to the deterministic counterpart and to demonstrate the price of robustness.

Coordinated operation and expansion planning for multiple microgrids and active distribution networks under uncertainties

Power systems have incorporated many new technologies in recent years and the introduction of energy storage systems and demand response provides additional challenges for their operation and expansion planning. Current distribution networks are complex active networks where many technologies are included, which must be considered and properly represented in expansion planning models. Thus, this paper proposes a robust model to solve the coordinated operation and expansion planning of active distribution networks with multiple microgrids, distributed energy resources (DERs), demand response, and N-1 generation contingency. The objective is to determine the best expansion planning proposals and the optimal daily operation (24-hour representation) for the DERs under the worst realization of uncertainties, using a contingency-constrained approach. Besides, the load is represented by non-controllable, controllable, and deferrable portions. The model is formulated as a tri-level problem that is solved using a two-stage robust optimization approach. The proposed method is illustrated using a modified version of the IEEE 123-bus test system. The results show that the coordinated operation and expansion planning decreases the total cost of the problem and provides more consistent results for demand response and DERs coordination. Simulations show a reduction of 13.8% in the total cost if the daily operation and demand response are represented, and the inclusion of contingencies provides a reduction of 70% in the EENS (expected energy not served) reliability index for the case study.

Approximate and robust bounded job start scheduling for Royal Mail delivery offices

Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling P||C_{max } problem which we call the bounded job start scheduling problem. Given a set of jobs, each specified by an integer processing time p_j, that have to be executed non-preemptively by a set of m parallel identical machines, the objective is to compute a minimum makespan schedule subject to an upper bound gle m on the number of jobs that may simultaneously begin per unit of time. With perfect input knowledge, we show that Longest Processing Time First (LPT) algorithm is tightly 2-approximate. After proving that the problem is strongly {mathcal {N}}{mathcal {P}}-hard even when g=1, we elaborate on improving the 2-approximation ratio for this case. We distinguish the classes of long and short instances satisfying p_jge m and p_j<m, respectively, for each job j. We show that LPT is 5/3-approximate for the former and optimal for the latter. Then, we explore the idea of scheduling long jobs in parallel with short jobs to obtain tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs, we derive a 1.985-approximation ratio. For general instances, we require machine augmentation to obtain better than 2-approximate schedules. In the presence of uncertain job processing times, we exploit machine augmentation and lexicographic optimization, which is useful for P||C_{max } under uncertainty, to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty aiming in a low number of used machines. Given a collection of schedules of makespan le D, this approach allows distinguishing which are the more robust. We substantiate both the heuristics and our recovery approach numerically using Royal Mail data. We show that for the Royal Mail application, machine augmentation, i.e., short-term van rental, is especially relevant.

A two-stage robust optimization approach for the berth allocation problem under uncertainty

This paper examines the berth allocation problem under uncertainty. We use uncertainty sets to describe the possible scenarios without depending on probabilistic information. Then we adopt a two-stage robust optimization approach where baseline schedule is made before and the recovery operation is made after the disruptions are known. Three two-stage robust models are constructed to generate less conservative solutions. Experimental results demonstrate that the schedule’s robustness can be improved with only a slight increase in the baseline cost. Finally, some management insights are presented based on the numerical results.

Robust model predictive control for optimal energy management of island microgrids with uncertainties

As the increasing penetration of wind and PV generations in island microgrids, the intermittent nature of renewable energy resources and randomness of load demands are inevitable, therefore, maintaining system stability and reliability has become a challenging issue for microgrid operators. In addition, energy storage unit and demand side management technology are widely utilized in the island microgrids to alleviate the passive impacts introduced by renewable energy resources. Nevertheless, they produce uncertainties as well. To accommodate the combined uncertainties, a two-stage robust model predictive control based optimization approach is proposed in this paper. The mixed integer quadratic programming model is established in the first operation stage to minimize the operation cost under the joint worst case of uncertainty, then an economic dispatch model is used to minimize the adjustment cost after obtaining actual data in the second operation stage. Robust linearization methods with the consideration of three types of uncertainty scenarios and uncertainty budgets are utilized in the first operation stage. Finally, the case study indicates that the proposed approach is more robust and economical than the conventional two-stage robust optimization approach, then the sensitivity of typical parameters and important units are analyzed and discussed.

Robust Energy and Reserve Scheduling Considering Bulk Energy Storage Units and Wind Uncertainty

In the restructured power industry, bulk energy storage may play a crucial role to provide the flexibility required by system operators to cater for the unprecedented levels of uncertainty. Within the context of co-optimized electricity markets for energy and reserves under wind uncertainty, this paper addresses the incorporation of bulk energy storage units in day-ahead network-constrained energy and reserve scheduling. A novel two-stage robust optimization approach is presented whereby the nonconvex and time-coupled operation of storage devices is precisely modeled while accounting for the anticipativity of the two-stage setting. The resulting robust counterpart is cast as a mixed-integer trilevel program with lower-level binary variables. In order to address the nonconvexity of the recourse problem, this paper proposes the application of an exact nested column-and-constraint generation algorithm. Numerical results illustrate the effective performance of the proposed approach.

A two-stage robust approach for the reliable logistics network design problem

This paper examines a three-echelon logistics network in which all supply and transshipment nodes are subject to disruption. We use uncertainty sets to describe the possible scenarios without depending on probabilistic information. We adopt a two-stage robust optimization approach where location decisions are made before and recourse decisions are made after the disruptions are known. We construct three two-stage robust models, which are solved exactly by a column-and-constraint-generation algorithm. Numerical tests demonstrate that the proposed algorithm outperforms the Benders decomposition method in both solution quality and computational time, and that the system’s reliability can be improved with only a slight increase in the normal cost.

Lot sizing with storage losses under demand uncertainty

We address a variant of the single item lot sizing problem affected by proportional storage (or inventory) losses and uncertainty in the product demand. The problem has applications in, among others, the energy sector, where storage losses (or storage deteriorations) are often unavoidable and, due to the need for planning ahead, the demands can be largely uncertain. We first propose a two-stage robust optimization approach with second-stage storage variables, showing how the arising robust problem can be solved as an instance of the deterministic one. We then consider a two-stage approach where not only the storage but also the production variables are determined in the second stage. After showing that, in the general case, solutions to this problem can suffer from acausality (or anticipativity), we introduce a flexible affine rule approach which, albeit restricting the solution set, allows for causal production plans. A hybrid robust-stochastic approach where the objective function is optimized in expectation, as opposed to in the worst-case, while retaining robust optimization guarantees of feasibility in the worst-case, is also discussed. We conclude with an application to heat production, in the context of which we compare the different approaches via computational experiments on real-world data.

Design of a reliable hierarchical location-allocation model under disruptions for health service networks: A two-stage robust approach

This study proposes a novel reliable hierarchical location-allocation model for health service network designs where facilities are subject to the risk of disruptions. To characterize the relationship between various levels of the concerned health network, a two-level multi-flow nested hierarchy with service referral is taken into account. The queuing system is considered in the model which handles uncertainty associated with demand as well as service. To ensure the service quality, a minimum limit reflected in the probability of patients’ waiting time is defined. The proposed model is formulated based on a two-stage robust optimization approach in which decisions are defined in two stages such that a first-stage solution should be robust against the possible realizations of the disruption that can only be revealed in a second stage. A solution procedure based on Benders decomposition type algorithm is implemented to solve the model. To illustrate the applicability of the proposed model, a practical case study is presented. The results show that the proposed model can be applied as a flexible decision support tool for decision makers to adopt appropriate strategies and policies in designing a reliable and robust health service network.

Strategic facility location, capacity acquisition, and technology choice decisions under demand uncertainty: Robust vs. non-robust optimization approaches

This paper presents a robust optimization (RO) modeling approach to develop insights into strategic capacity planning and resource acquisition decisions, including the facility location problem and the technology choice problem. The main purpose of the proposed model and numerical experiments is to examine the effects of economies of scale, economies of scope, and the combined effects of scale and scope under uncertain demand realizations using robust optimization. The type of capacities, or technology alternatives, that a firm can acquire can be classified on two basic dimensions. The first dimension relates to the effects of scale via distinction between labor-intensive technologies and capital-intensive technologies. The second dimension relates to the effects of scope via distinction between product-dedicated and flexible technologies. Moreover, each product-dedicated and flexible technology can have different levels of labor or capital-intensity, leading to the joint effects of economies of scale and economies of scope. Extensive computational results document how the optimal technology choice patterns depend on the cost structures and levels of model robustness specified to accommodate uncertain demand realizations. The results obtained by the two-stage robust optimization approach are compared to the results obtained by a non-robust approach.

Two-stage robust optimization approach to elective surgery and downstream capacity planning

Scheduling elective surgeries is a complicated task due to the coupled effects of multiple sources of uncertainty and the impact of the proposed schedule on the downstream units. In this paper, we propose a two-stage robust optimization model to address the existing uncertainty in surgery duration and length-of-stay in the surgical intensive care unit. The framework allows for the decision-maker to adjust the level of risk with which she/he is comfortable. We propose the adapted column-and-constraint generation method to obtain exact solutions for the proposed formulation. The performance and the quality of our proposed solution methodology are tested through computational experiments and simulation modeling.

A two-stage robust optimization approach for the mobile facility fleet sizing and routing problem under uncertainty

We propose a two-stage robust optimization model for the mobile facility fleet sizing and routing problem with demand uncertainty. A two-level cutting plane based method is developed, which includes an algorithm to generate problem-specific lower bound inequalities in the outer level, and a hybrid algorithm in the inner level that combines heuristic and exact methods to solve the recourse problem. Numerical tests show that the design and operation from the proposed method outperforms other solution approaches. The efficiency of the proposed solution algorithm in identifying the optimal solution is quantified and the robustness of the proposed model is demonstrated for varying degrees of uncertainty in demand.

Two-stage robust optimization for N-k contingency-constrained unit commitment

This paper proposes a two-stage robust optimization approach to solve the N- k contingency-constrained unit commitment (CCUC) problem. In our approach, both generator and transmission line contingencies are considered. Compared to the traditional approach using a given set of components as candidates for possible failures, our approach considers all possible component failure scenarios. We consider the objectives of minimizing the total generation cost under the worst-case contingency scenario and/or the total pre-contingency cost. We formulate CCUC as a two-stage robust optimization problem and develop a decomposition framework to enable tractable computation. In our framework, the master problem makes unit commitment decisions and the subproblem discovers the worst-case contingency scenarios. By using linearization techniques and duality theory, we transform the subproblem into a mixed-integer linear program (MILP). The most violated inequalities generated from the subproblem are fed back into the master problem during each iteration. Our approach guarantees a globally optimal solution in a finite number of iterations. In reported computational experiments, we test both primal and dual decomposition approaches. Our computational results verify the effectiveness of our proposed approach.