Uncertainty Set Research Articles

Relevance of Robustness and Uncertainties Analysis in the Optimal Design of Wave Energy Converters

The optimisation design of Wave Energy Converters (WEC) to reduce the cost of energy of the technology is a widely investigated topic. In literature classical optimisation strategies have been presented and applied to identify the optimal system parameters of WECs to optimise specific techno-economic metrics. The performance of the optimal identified devices relies on these nominal parameters and it can be strongly affected by construction and modelling uncertainties. In this context, the concept of robustness of the optimal solution plays a relevant role in the identification of a device whose performance is affected as little as possible by uncertainties of various kinds. In the first part of this paper different declinations of robustness concept are derived from other fields of application and described. The identified robustness indexes are then applied to optimal solutions obtained via classical optimisation to evaluate its importance in the design process of WECs.
 Strictly related to this kind of methodology is the Sensitivity Analysis (SA) technique, it aims to investigate how the input variation (due to uncertainties or external noise or additional environmental parameters) influences the output results of a defined numerical model and highlight the relative input parameters relevance. Sensitivity Analysis, therefore, can be a valuable tool applicable in the uncertainty set estimation to identify the variables most subject to such uncertainties and their prominence.
 The main objective of the work is to underline the importance of introduce the robustness evaluation of WECs during the optimisation process since classical optimisation techniques can lead to solutions that are affected by uncertainties.

A data-driven distributionally newsvendor problem for edge-cloud collaboration in intelligent manufacturing systems

In intelligent manufacturing systems, the industrial informatics has features of multi-source, multi-noise, and time series. It is difficult for small and medium enterprises (SMEs) to directly exploit the enormous amounts of data due to the limited budgets and computing capabilities. Edge intelligence is a key technique to power intelligent manufacturing systems and provide knowledge transferred from the cloud to SMEs at the edges. To address edge-cloud collaboration issue, we propose a refined data-driven distributionally robust newsvendor model based on φ-divergence measures and imprecise Dirichlet models (DRN-IDM). We construct new distributional uncertainty sets by effectively integrating local censored demand data and cloud knowledge, which helps SMEs to make intelligent production decisions and reduce significant decision deviations, even under a small censored data set. In particular, the novel demand uncertainty sets can depict the distance between distributions and probability intervals. Then, we transform the DRN-IDM model into a convex optimization model that is amenable to algorithmic implementation. Additionally, based on the coefficient of variation of limited historical data, we propose an adaptive demand information fusion procedure to achieve excellent synergy effect from cloud knowledge. We also validate the effectiveness of the DRN-IDM model and the practicability of adaptive procedure using extensive numerical studies with both simulated and real-life data. Furthermore, we measure the relative expected value of cloud knowledge and investigate the effect of censored demand samples. Our results verify the effectiveness condition of the DRN-IDM model and indicate that cloud knowledge can improve the precision and robustness of SMEs’ production decisions with small-scale censored data. Interestingly, the verified adaptive procedure can be applied in the learning criteria design of metaheuristics in intelligent manufacturing systems, and the reconstructed uncertainty set can narrow the search space to improve the convergence performance of algorithms.

Robust Generation Dispatch With Strategic Renewable Power Curtailment and Decision-Dependent Uncertainty

As renewable energy sources replace traditional power sources (such as thermal generators), uncertainty grows while there are fewer controllable units. To reduce operational risks and avoid frequent real-time emergency controls, a preparatory schedule of renewable generation curtailment is required. This paper proposes a novel two-stage robust generation dispatch (RGD) model, where the preparatory curtailment threshold is optimized in the pre-dispatch stage. The curtailment threshold will then influence the variation range of real-time renewable power outputs, resulting in a decision-dependent uncertainty (DDU) set. In the re-dispatch stage, the controllable units adjust their outputs within their reserve contributions to maintain power balancing. To overcome the difficulty in solving the RGD with DDU, an adaptive column-and-constraint generation (AC&CG) algorithm is developed. We prove that the proposed algorithm can generate the optimal solution in a finite number of iterations. Numerical studies show the advantages of the proposed model and algorithm and validate their practicability and scalability.

Cyber-Physical Interdependent Restoration Scheduling for Active Distribution Network via Ad Hoc Wireless Communication

This paper proposes a post-disaster cyber-physical interdependent restoration scheduling (CPIRS) framework for active distribution networks (ADN) where the simultaneous damages on cyber and physical networks are considered. The ad hoc wireless device-to-device (D2D) communication is leveraged, for the first time, to establish cyber networks instantly after the disaster to support ADN restoration. The repair and operation crew dispatching, the remote-controlled network reconfiguration and the system operation with DERs can be effectively coordinated under the cyber-physical interactions. The uncertain outputs of renewable energy resources (RESs) are represented by budget-constrained polyhedral uncertainty sets. Through implementing linearization techniques on disjunctive expressions, a monolithic mixed-integer linear programming (MILP) based two-stage robust optimization model is formulated and subsequently solved by a customized column-and-constraint generation (C&CG) algorithm. Numerical results on the IEEE 123-node distribution system demonstrate the effectiveness and superiorities of the proposed CPIRS method for ADN.

Robust Scheduling of Thermostatically Controlled Loads With Statistically Feasible Guarantees

Flexible resources are increasingly significant for the reliable operation of power grids due to the high penetration of renewable energy. Thermostatically controlled loads (TCLs) are one of the common flexible resources, whose control has been extensively studied. Yet, much can be improved. We investigate the scheduling of TCLs facing uncertain temperatures and dynamic prices. Classical approaches often employ the chance-constrained program or robust optimization to handle such uncertainties. However, these approaches either require specific distribution knowledge or yield too conservative solutions. The distribution knowledge can be rather challenging to obtain especially when the uncertainties from different sources are correlated. To this end, we adopt the notion of statistical feasibility and propose a robust sample-based scheduling scheme for TCLs. Such a sample-based scheme relieves the reliance on the distribution knowledge and is able to characterize the coupling effects of the two uncertainty sources. Besides, through integrating the real-time domain knowledge into uncertainty set reconstruction, we relax the solutions’ conservation by exploring the consumers’ tolerance to the room temperature. Numerical studies highlight the remarkable performance of our proposed scheme. Specifically, our approach is able to simultaneously effectively reduce the electricity bills of consumers and satisfy the consumers’ tolerance to the room temperature.

Dryspells and Minimum Air Temperatures Influence Rice Yields and their Forecast Uncertainties in Rainfed Systems

Crop yield predictions on inter-seasonal to inter-annual horizons are subject to a diverse set of uncertainties associated with climate forecast scenarios. However, the uncertainties associated with climate-related parameters can be marginally controlled (reduced) over the growing season, i.e., from planting through harvest, leveraging a suite of climate forecast ensembles as forcings. In this study, we present a novel approach that combines a coupled hydrologic-crop modeling framework with probabilistic forecasts to characterize and reduce uncertainties in seasonal rice yields predictions. By weighting real-time (nowcast) and forecast climate forcings, the comprehensive framework accurately quantifies uncertainties associated with climate forecasts. At a provincial scale, the crop model extensively captured the uncertainties in yields whilst significantly complementing observations over the growing season. We observed that the spread of yield predictions gradually decreased over time (i.e., towards harvest) as subsequent timeframes incorporated a higher degree of present conditions/forcings and reduced reliance on forecasts. Furthermore, we investigated the information exchange between yields and hydrologic/drought variables over different time frames within the season. Notably, we found a higher synchronization of information transfer between yields, dryspells, and minimum air temperatures towards the end of season, indicating strong explicit links between these variables and crop yields. These outcomes have significant implications on crop yield forecasting and nowcasting, particularly in data-poor regions. By providing a better understanding of the uncertainties associated with seasonal climate forecasts and the interplay between hydrologic variables, drought conditions, and crop yields, this research can aid in improving decision-making processes related to agricultural planning, and risk management. Moreover, these insights can inform assessments of economic, social, and environmental impacts of drought in agricultural systems.

Robust multi‐echelon inventory management with multiple suppliers

AbstractWe study a periodic‐review multi‐supplier series inventory system in which the demand is restricted to partial sum uncertainty sets. We present and solve a robust rolling‐horizon model for the system. We propose an induction framework to characterize the closed‐form robust optimal solution of the problem. We show that the robust optimal policy combines the echelon base‐stock policy and a gap‐of‐echelon‐base‐stock policy for the uppermost stage and a modified echelon base‐stock policy for the other downstream stages. The policy structure is easy for the manager to understand and implement in practice. The policy parameters are directly determined by a sequence of nominal partial‐sum demands, and its computation is very effective. In addition, the policy does not rely on complete information about the demand distribution; its solution can be more robust than that of stochastic optimization methods, especially when demand is highly uncertain, and forecasting is difficult. Based on the structure of the robust optimal policy, we design two heuristic policies for the system and evaluate the policies' performance through an extensive numerical study using both synthetic and real data.

Robust optimisation of electricity‐heating integrated energy system based on data‐driven PV interval estimation

AbstractShort‐term interval estimation can effectively and precisely quantify the uncertainties of renewable energy, accurately represent the range of fluctuations of uncertain variables in robust optimisation of electricity‐heating integrated energy system (EHIES) and it is getting crucial for reliable and flexible operation of renewable dominated new energy systems. The authors present a multivariate data‐driven short‐term PV power interval prediction model that consists of multiple layers, including one‐dimensional convolutional layer, ultra‐lightweight subspace attention mechanism (ULSAM), bidirectional long and short‐term memory (BiLSTM), quantile regression (QR) and kernel density estimation (KDE). The one‐dimensional convolutional layer and ULSAM can extract sequential features and highlight key information from the data; the BiLSTM processes time series data in both directions and conveys historical information; the QR and KDE models generate interval prediction with a given confidence level. Based on the proposed interval estimation, a refined PV uncertainty set can be established and adopted by robust optimal scheduling of EHIES utilising min‐max‐min algorithm. The simulation results have demonstrated the estimation accuracy and adaptability to various weather scenarios.

Adaptive robust optimization for lot-sizing under yield uncertainty

In manufacturing environments, uncertain production yield directly impacts the quality and feasibility of the production planning decisions. This paper investigates the use of adaptive robust optimization to hedge against uncertain yield when determining a production plan, and to react properly when updated information unfolds. We first derive a myopic adaptive robust policy for the inventory management problem, a special case of the lot-sizing problem where the setup and the production costs are omitted. We show that the policy is optimal under mild assumptions. Second, we address a multi-period single-item lot-sizing problem with a backorder and uncertain yield via adaptive robust optimization. We formulate an adaptive robust model based on the budgeted uncertainty set, where we exploit a linear approximation to transform the quadratic constraints into a mixed-integer linear program. We also propose a column and constraint generation algorithm to solve the adaptive model exactly. Finally, we demonstrate the performances of the proposed approaches and the value of the adaptive robust solutions through extensive numerical experiments.

A two‐stage adaptive‐robust optimization model for active distribution network with high penetration wind power generation

AbstractHigh penetration of uncertain wind power generation brings challenges to power system operational security and economy. Here, an adjustable box uncertainty set is first presented to characterize the spatiotemporal correlation of multiple wind power generations. Afterward, a two‐stage risk‐adjusted robust energy dispatching model is established as a min‐max‐min optimization problem for active distribution networks operating with high penetration wind power generation. Then a three‐layer Nest Column and Constraint Generation (NCCG) decomposition approach is designed to efficiently solve the proposed model. Finally, the proposed two‐stage adaptive robust optimization model and NCCG approach are validated by a distribution network with uncertain wind power generations, and the simulation results indicate the distribution network operation economy could be compromised with robustness by adjusting the conservative regulation parameter.

Assessment of residual geometrical errors of clinical target volumes and their impact on dose accumulation for head and neck radiotherapy

PurposeTo assess the residual geometrical errors (dr) and their impact on the clinical target volumes (CTV) dose coverage for head and neck cancer (HNC) proton therapy patients. MethodsWe analysed 28 HNC patients treated with 70 Gy (RBE) and 54.25 Gy (RBE) to the therapeutic CTV70 and prophylactic CTV54.25, respectively. Daily cone beam CTs were converted to high quality synthetic CTs (sCTs). The CTVs from the nominal CT were propagated to the corresponding sCTs using a hybrid deformable image registration (propagated CTVs) in RayStation 11B. For 11 patients, all propagated CTVs were reviewed by our HNC radiation oncologist (physician corrected CTVs).The residual geometrical error dr was quantified as a function of the daily CTVs volume overlap with the nominal plan CTV. The errors dr(propagated CTVs) and dr(physician corrected CTVs) and the difference in dice similarity coefficients (ΔDSC) were determined. Using clinical plans, dose coverage and the tumor control probability (TCP) for the nominal, accumulated and voxel-wise minimum scenarios were determined. ResultsThe difference in the residual geometrical error dr (propagated CTVs – physician corrected CTVs) and mean DSC (|ΔDSC|mean) were minor: Δdr(CTV70) = 0.16 mm, Δdr(CTV54.25) = 0.26 mm, |ΔDSC|mean < 0.9%. For all 28 patients, dr(CTV70) = 1.91 mm and dr(CTV54.25) = 1.90 mm. However, CTV54.25 above and below the cricoid cartilage differed substantially (1.00 mm c.f. 3.93 mm). The CTV54.25 coverage below the cricoid was then almost always lower, although the TCP of the accumulated dose was higher than the TCP of the voxel-wise minimum dose. ConclusionsSetup uncertainty setting of 2 mm is possible. The feasibility of using propagated CTVs for error determination is demonstrated.

Virtual reference feedback tuning for linear discrete-time systems with robust stability guarantees based on set membership

In this paper we propose a novel methodology that allows us to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) virtual reference feedback tuning for enforcing suitable performance and (ii) the set membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through set membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as linear matrix inequality constraints within an optimization problem whose linear cost function relies on virtual reference feedback tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop.The proposed method is not direct due to the set membership identification. However, the uncertainty set is used with the objective of providing robust stability guarantees for the closed-loop system and it is not used for the definition of the cost function, which instead is based on data. The effectiveness of the method is shown with reference to three simulation examples. A comparison with other data-driven methods is carried out.

A Novel Weighted Data-Driven Robust Optimization Approach for Creating Adjustable Uncertainty Sets

Robust optimization provides a progressive and efficient approach to dealing with uncertainty which applies to various optimization problems. However, most existing robust optimization approaches generate a unique uncertainty set for each level of conservatism, disregarding supplementary information such as predicted values. This paper proposes a new weighted data-driven robust optimization approach for creating adjustable uncertainty sets. The proposed approach enables the adjustment of the boundary reduction rate of uncertainty sets, effectively pulling uncertainty sets towards areas of higher density or towards the predicted points. Additionally, a multi-stage clustering algorithm is proposed to fully cover the areas around the predicted point. A regularization parameter search algorithm is also developed to tune the conservatism degree. The numerical results indicate that by weighting historical data, adjustable uncertainty sets with the same fraction of data coverage can be created that ensures the feasibility of the model and reduces the extra conservatism.

Disease Bundling or Specimen Bundling? Cost- and Capacity-Efficient Strategies for Multidisease Testing with Genetic Assays

Problem definition: Infectious disease screening can be expensive and capacity constrained. We develop cost- and capacity-efficient testing designs for multidisease screening, considering (1) multiplexing (disease bundling), where one assay detects multiple diseases using the same specimen (e.g., nasal swabs, blood), and (2) pooling (specimen bundling), where one assay is used on specimens from multiple subjects bundled in a testing pool. A testing design specifies an assay portfolio (mix of single-disease/multiplex assays) and a testing method (pooling/individual testing per assay). Methodology/results: We develop novel models for the nonlinear, combinatorial multidisease testing design problem: a deterministic model and a distribution-free, robust variation, which both generate Pareto frontiers for cost- and capacity-efficient designs. We characterize structural properties of optimal designs, formulate the deterministic counterpart of the robust model, and conduct a case study of respiratory diseases (including coronavirus disease 2019) with overlapping clinical presentation. Managerial implications: Key drivers of optimal designs include the assay cost function, the tester’s preference toward cost versus capacity efficiency, prevalence/coinfection rates, and for the robust model, prevalence uncertainty. When an optimal design uses multiple assays, it does so in conjunction with pooling, and it uses individual testing for at most one assay. Although prevalence uncertainty can be a design hurdle, especially for emerging or seasonal diseases, the integration of multiplexing and pooling, and the ordered partition property of optimal designs (under certain coinfection structures) serve to make the design more structurally robust to uncertainty. The robust model further increases robustness, and it is also practical as it needs only an uncertainty set around each disease prevalence. Our Pareto designs demonstrate the cost versus capacity trade-off and show that multiplexing-only or pooling-only designs need not be on the Pareto frontier. Our case study illustrates the benefits of optimally integrated designs over current practices and indicates a low price of robustness. Funding: This work was supported by the National Science Foundation [Grant 1761842]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0296 .

Expansion planning of the transmission network with high penetration of renewable generation: A multi-year two-stage adaptive robust optimization approach

This paper addresses the multi-year two-stage expansion planning of the transmission network of a power system with high penetration of renewable generation modeling long-term uncertainty by using adaptive robust optimization. The multi-year nature of the problem is modeled by considering a comprehensive view of the planning horizon. A cardinality-constrained uncertainty set is used to model the future worst-case uncertainty realization of the peak power consumption of loads, along with the capacity and marginal production cost of generating units. Unlike previous works, we model certain features of the operation that are typically ignored in multi-year robust transmission network expansion planning problems, namely, the operational variability of renewable generating units, the operational flexibility of conventional generating units, and the non-convex operational feasibility sets of storage facilities. The solution procedure employed for this multi-year two-stage robust problem, which is formulated as a three-level problem, is based on the combination of the nested column-and-constraint generation algorithm with two exact acceleration techniques. We analyze the performance of the proposed model through the use of the IEEE 24-bus Reliability Test System and the IEEE 118-bus Test System. Numerical results show that the use of the multi-year approach leads to reductions in the total worst-case cost of up to 7% in comparison with the static and sequential static procedures. Moreover, an underestimation of the total worst-case cost of more than 8% is attained when ignoring certain operational constraints of conventional generating units and storage facilities. Lastly, a sensitivity analysis is presented in order to illustrate the impact of the maximum deviations of the uncertain parameters on the total worst-case cost.

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.

Worst-case Omega ratio under distribution uncertainty with its application in robust portfolio selection

AbstractOmega ratio, a risk-return performance measure, is defined as the ratio of the expected upside deviation of return to the expected downside deviation of return from a predetermined threshold described by an investor. Motivated by finding a solution protected against sampling errors, in this paper, we focus on the worst-case Omega ratio under distributional uncertainty and its application to robust portfolio selection. The main idea is to deal with optimization problems with all uncertain parameters within an uncertainty set. The uncertainty set of the distribution of returns given characteristic information, including the first two orders of moments and the Wasserstein distance, can handle data problems with uncertainty while making the calculation feasible.

Optimal scheduling of ethylene plants under uncertainty: An unsupervised learning-based data-driven strategy

Ethylene cracking furnaces are core units that determine the yield of key products, and its cycle scheduling optimization is of great significance to improve the efficiency of the whole plant. However, uncertainties in cracking batch durations may lead to sub-optimal or ineffective scheduling arrangements. This paper develops a novel data-driven stochastic robust optimization strategy to achieve the optimal scheduling of cracking furnaces under uncertainty. The proposed stochastic robust optimization model applies machine learning methods to extract data information and is represented as a bilevel problem: the outer layer is a two-stage stochastic programming, where typical scenarios are clustered by a three-level data mining approach; the inner layer is robust optimization, whose uncertainty set is constructed by an efficient kernel-based approach. An actual ethylene cracking system is investigated and the results show that the proposed method achieves the highest daily profit, performing well in data coverage (92.3%) and change rate (0.826%). Moreover, the sensitivity analysis for the number of clusters and regularization parameter values is performed.

Robust optimal consensus feedback mechanism with private interest in the context of uncertain cost

During the process of obtaining a consensus, the moderator may recommend modifications based on his personal interests to steer decision-makers (DMs) toward modifying their viewpoints so that the consensus evolves in the desired direction. In the process of revising opinions, the moderator must provide sufficient monetary remuneration to those who are willing to modify their opinions. Nevertheless, there is a lot of uncertainty and both internal and external factors affect the adjustment cost of decision-making. This paper proposes two robust optimal consensus feedback mechanisms with private interests in the context of uncertain cost (UCRFM), taking into account the moderator’s private interests and uncertain cost, minimizing the consensus cost under the established consensus threshold, and maximizing the fuzzy consensus measure under the given cost budget. Initially, two types of UCRFMs are presented by employing two uncertainty sets to characterize the uncertain cost, and then two UCRFMs are turned equivalently into a 0-1 programming model. Secondly, based on different uncertainty sets, the corresponding robust counterparts of UCRFMs are constructed to obtain the optimal total consensus cost and the maximum fuzzy consensus. Finally, comprehensive experimental research is conducted to confirm the effectiveness of the proposed approach, and a comparison between the models is given.