Distributionally Robust Optimization Research Articles

Sizing Energy Storage Systems to Dispatch Wind Power Plants

Integrating wind power plants into the electricity grid poses challenges due to the intermittent nature of wind energy generation. Energy storage systems (ESSs) have shown promise in mitigating the intermittent variability associated with wind power. This paper presents a distributionally robust optimization (DRO) model for sizing energy storage systems to dispatch wind power plants. The variable wind power is formulated as a moment-based ambiguity set. Dispatchability is described by the expected value of the insufficient power of wind power relative to the dispatch command, which is a sum of nonlinear functions and is taken as the optimal index. A deterministic semi-definite positive model is derived to solve the problem effectively. Numerical studies are conducted to demonstrate the effectiveness and advantages of the proposed method.

Low-carbon economic scheduling for AIES considering flexible load coordination and multiple uncertainties

The disordered energy use in traditional agricultural production has resulted in high energy purchase costs and carbon emissions. The coupling of integrated energy systems with agriculture offers the potential for flexible coordination of agricultural loads while agricultural loads are highly influenced by changes in the natural environment. In this context, this paper proposes a low-carbon and economic scheduling optimization method for agricultural integrated energy system (AIES), which comprehensively takes temperature, light intensity, and the coordination of agricultural loads into consideration. Firstly, solar and biomass resources are integrated through energy conversion equipment to support energy supply from both the electrical distribution network (EDN) and the gas network. At the same time, the modelling process of agricultural load is refined, which is divided into crop irrigation, greenhouse temperature control and residential consumption. The models of agricultural multi-energy complementary and energy storage equipment are established to achieve flexible conversion between different energy sources. In addition, the distributionally robust optimization (DRO) method based on multiple discrete scenarios is adopted to deal with the uncertainties arising from environmental condition fluctuations during the optimization process. Finally, the proposed model is tested on 4 various regions and 2 different seasons. Numerical results show that the proposed model can effectively enhance the economic and environmental performance of the AIES.

Data-driven distributionally robust day-ahead dispatch for active distribution networks based on improved conditional generative adversarial network

With the increasing penetration of renewable energy generation (REG), strong uncertainty poses a significant challenge to the rationality of day-ahead dispatch plans in distribution systems. We propose a data-driven, distributionally robust day-ahead dispatch method for active distribution networks (ADNs) based on an improved conditional generative adversarial network (CGAN). First, an improved CWGAN-GP model is constructed based on three-dimensional convolution (Conv3D) and the Wasserstein distance to obtain the day-ahead REG scenarios, considering the correlation features of photovoltaic (PV) and wind turbines (WTs), which reduces the conservatism of scenario generation. Accordingly, the Gaussian mixture model (GMM) clustering method based on moment information was used to construct an ambiguity set. Moreover, a two-stage distributionally robust day-ahead dispatch model for an ADN is established, which realizes the fusion of the data-driven method and distributionally robust optimization (DRO) mechanism model. Finally, the validity of the data-driven DRO (DDRO) method is verified through a case analysis.

Learning With Noisy Labels Over Imbalanced Subpopulations.

Learning with noisy labels (LNL) has attracted significant attention from the research community. Many recent LNL methods rely on the assumption that clean samples tend to have a "small loss." However, this assumption often fails to generalize to some real-world cases with imbalanced subpopulations, that is, training subpopulations that vary in sample size or recognition difficulty. Therefore, recent LNL methods face the risk of misclassifying those "informative" samples (e.g., hard samples or samples in the tail subpopulations) into noisy samples, leading to poor generalization performance. To address this issue, we propose a novel LNL method to deal with noisy labels and imbalanced subpopulations simultaneously. It first leverages sample correlation to estimate samples' clean probabilities for label correction and then utilizes corrected labels for distributionally robust optimization (DRO) to further improve the robustness. Specifically, in contrast to previous works using classification loss as the selection criterion, we introduce a feature-based metric that takes the sample correlation into account for estimating samples' clean probabilities. Then, we refurbish the noisy labels using the estimated clean probabilities and the pseudo-labels from the model's predictions. With refurbished labels, we use DRO to train the model to be robust to subpopulation imbalance. Extensive experiments on a wide range of benchmarks demonstrate that our technique can consistently improve state-of-the-art (SOTA) robust learning paradigms against noisy labels, especially when encountering imbalanced subpopulations. We provide our code in https://github.com/chenmc1996/LNL-IS.

Wasserstein Robust Classification with Fairness Constraints

Problem definition: Data analytics models and machine learning algorithms are increasingly deployed to support consequential decision-making processes, from deciding which applicants will receive job offers and loans to university enrollments and medical interventions. However, recent studies show these models may unintentionally amplify human bias and yield significant unfavorable decisions to specific groups. Methodology/results: We propose a distributionally robust classification model with a fairness constraint that encourages the classifier to be fair in the equality of opportunity criterion. We use a type-[Formula: see text] Wasserstein ambiguity set centered at the empirical distribution to represent distributional uncertainty and derive a conservative reformulation for the worst-case equal opportunity unfairness measure. We show that the model is equivalent to a mixed binary conic optimization problem, which standard off-the-shelf solvers can solve. We propose a convex, hinge-loss-based model for large problem instances whose reformulation does not incur binary variables to improve scalability. Moreover, we also consider the distributionally robust learning problem with a generic ground transportation cost to hedge against the label and sensitive attribute uncertainties. We numerically examine the performance of our proposed models on five real-world data sets related to individual analysis. Compared with the state-of-the-art methods, our proposed approaches significantly improve fairness with negligible loss of predictive accuracy in the testing data set. Managerial implications: Our paper raises awareness that bias may arise when predictive models are used in service and operations. It generally comes from human bias, for example, imbalanced data collection or low sample sizes, and is further amplified by algorithms. Incorporating fairness constraints and the distributionally robust optimization (DRO) scheme is a powerful way to alleviate algorithmic biases. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869] and the Chinese University of Hong Kong [Grant 4055191]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0230 .

Ship coping strategies for hurricane-induced port disruptions

PurposeDisruptions at ports may destroy the planned ship schedules profoundly, which is an imperative operation problem that shipping companies need to overcome. This paper attempts to help shipping companies cope with port disruptions through recovery scheduling.Design/methodology/approachThis paper studies the ship coping strategies for the port disruptions caused by severe weather. A novel mixed-integer nonlinear programming model is proposed to solve the ship schedule recovery problem (SSRP). A distributionally robust mean conditional value-at-risk (CVaR) optimization model was constructed to handle the SSRP with port disruption uncertainties, for which we derive tractable counterparts under the polyhedral ambiguity sets.FindingsThe results show that the size of ambiguity set, confidence level and risk-aversion parameter can significantly affect the optimal values, decision-makers should choose a reasonable parameter combination. Besides, sailing speed adjustment and handling rate adjustment are effective strategies in SSRP but may not be sufficient to recover the schedule; therefore, port skipping and swapping are necessary when multiple or longer disruptions occur at ports.Originality/valueSince the port disruption is difficult to forecast, we attempt to take the uncertainties into account to achieve more meaningful results. To the best of our knowledge, there is barely a research study focusing on the uncertain port disruptions in the SSRP. Moreover, this is the first paper that applies distributionally robust optimization (DRO) to deal with uncertain port disruptions through the equivalent counterpart of DRO with polyhedral ambiguity set, in which a robust mean-CVaR optimization formulation is adopted as the objective function for a trade-off between the expected total costs and the risk.

Temporally and Distributionally Robust Optimization for Cold-Start Recommendation

Collaborative Filtering (CF) recommender models highly depend on user-item interactions to learn CF representations, thus falling short of recommending cold-start items. To address this issue, prior studies mainly introduce item features (e.g., thumbnails) for cold-start item recommendation. They learn a feature extractor on warm-start items to align feature representations with interactions, and then leverage the feature extractor to extract the feature representations of cold-start items for interaction prediction. Unfortunately, the features of cold-start items, especially the popular ones, tend to diverge from those of warm-start ones due to temporal feature shifts, preventing the feature extractor from accurately learning feature representations of cold-start items. To alleviate the impact of temporal feature shifts, we consider using Distributionally Robust Optimization (DRO) to enhance the generation ability of the feature extractor. Nonetheless, existing DRO methods face an inconsistency issue: the worse-case warm-start items emphasized during DRO training might not align well with the cold-start item distribution. To capture the temporal feature shifts and combat this inconsistency issue, we propose a novel temporal DRO with new optimization objectives, namely, 1) to integrate a worst-case factor to improve the worst-case performance, and 2) to devise a shifting factor to capture the shifting trend of item features and enhance the optimization of the potentially popular groups in cold-start items. Substantial experiments on three real-world datasets validate the superiority of our temporal DRO in enhancing the generalization ability of cold-start recommender models.

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes chi^2-divergences as a special case. We prove that our algorithm finds an epsilon-stationary point with an improved computational complexity than existing methods. Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

Trust Region Methods for Nonconvex Stochastic Optimization beyond Lipschitz Smoothness

In many important machine learning applications, the standard assumption of having a globally Lipschitz continuous gradient may fail to hold. This paper delves into a more general (L0, L1)-smoothness setting, which gains particular significance within the realms of deep neural networks and distributionally robust optimization (DRO). We demonstrate the significant advantage of trust region methods for stochastic nonconvex optimization under such generalized smoothness assumption. We show that first-order trust region methods can recover the normalized and clipped stochastic gradient as special cases and then provide a unified analysis to show their convergence to first-order stationary conditions. Motivated by the important application of DRO, we propose a generalized high-order smoothness condition, under which second-order trust region methods can achieve a complexity of O(epsilon(-3.5)) for convergence to second-order stationary points. By incorporating variance reduction, the second-order trust region method obtains an even better complexity of O(epsilon(-3)), matching the optimal bound for standard smooth optimization. To our best knowledge, this is the first work to show convergence beyond the first-order stationary condition for generalized smooth optimization. Preliminary experiments show that our proposed algorithms perform favorably compared with existing methods.

Distributionally robust PV planning and curtailment considering cyber attacks on electric vehicle charging under PV/load uncertainties

A manipulated charging behavior of electric vehicles (EVs) due to an adversary along with the uncertain photovoltaic (PV) generation outputs and loads may lead to unstable power distribution system operations with voltage and current violations. To resolve this issue, this paper proposes an optimization framework where the following two types of uncertainties are addressed: (i) the natural uncertainties of PV generation outputs/loads and (ii) artificial uncertainties of load altering attacks (LAAs) on EV charging stations (EVCSs) via the manipulation of EV charging control signals. The proposed framework is formulated as a Wasserstein metric-integrated distributionally robust optimization (DRO)-based Volt/VAR optimization (VVO) problem. The proposed DRO-based VVO framework combined with PV planning and curtailment aims to minimize substation energy and voltage imbalance along with the complete removal of the constraint violations while handling uncertain PV generation outputs/loads and LAAs. To use off-the-shelf optimization solvers, tractable reformulation of the chance constraints of the voltage, current, and curtailed PV real power of the original DRO problem is provided. Numerical examples tested over IEEE 13-bus and 37-bus systems with PV systems and EVCSs show the efficiency of the proposed DRO framework in terms of substation energy, voltage imbalance, and PV planning/curtailment cost under stochastic LAAs.

Optimizing bike rebalancing strategies in free-floating bike-sharing systems: An enhanced distributionally robust approach

Bike-sharing systems are important components of urban transportation systems that facilitate short-distance travel. In recent years, free-floating bike-sharing systems have become popular. However, a major problem in such systems is the increasing number of idle bikes due to their spatial dispersion, which leads to a loss of demand and difficulty in bike rebalancing. To overcome these drawbacks, operators reserve parking areas, and laborers are incorporated into the bike rebalancing operation. We propose a novel dynamic bike rebalancing strategy based on trucks and laborers for free-floating bike-sharing systems. This strategy combines bike collection by laborers and bike transportation by trucks for rebalancing operations. In addition, we consider broken bike detection by laborers and user dynamics in two extension models. Because of the demand uncertainty in bike-sharing systems, we employ an enhanced distributionally robust optimization (EDRO) approach to protect against uncertainty. A target-level constraint is added to the general distributionally robust optimization (DRO) model to further improve the out-of-sample performance. An adaptive robust strategy is also employed to avoid overly conservative solutions. Furthermore, a linear decision rule is adopted to reformulate the EDRO model into a mixed-integer programming model, which guarantees computational tractability. The results of numerical experiments show that the EDRO model outperforms other benchmark models on a series of measures. Moreover, we verify the value of laborers in dynamic rebalancing operations and explore the effect of different bike supply levels and several other parameters on the value of laborers. We also test how the target-level parameter affects the model performance.

Distributionally Robust Optimization methods on robust medical diagnosis systems

In the medical field, modern recommendation systems face significant challenges due to distributional shifts in data. We propose utilizing Distributionally Robust Optimization (DRO) and Distributionally and Outlier Robust Optimization (DORO) methods to address this issue. This paper aims to develop suitable DRO and DORO frameworks for the medical domain and validate their effectiveness through extensive experiments. We employ the DDXPlus dataset for our investigations and cluster patients based on age, sex, and initial evidence to partition the data into distinct distributions. Using a simple three-layer neural network, we incorporate CVaR and CHISQ as DRO methods and their respective DORO forms. The experimental results show that the overall DRO approach demonstrates more significant enhancements while all four methods exhibit improvements over the original distributional scenarios. Our research contributes to optimizing deep learning models in the medical domain and enhancing their robustness. Furthermore, we intend to use these methods to estimate and provide best-fit patient therapies, addressing real-world medical challenges. The application of these approaches has the potential to enhance the performance and practicality of medical recommendation systems, offering improved medical services to patients.

Consignment contract in a supply chain with price, demand and yield randomness: A distributionally robust approach

We investigate a supply chain consisting of a supplier and an e-commence platform who sign a consignment contract with uncertainty in price, demand, and yield. Due to the challenges of actually determining the joint distribution of random variables, we assume that only the partial distributional information is available for these three uncertainties. To address the issue of distributional ambiguity and to more accurately account for arbitrary correlations among these uncertainties, we introduce a moment-based distributionally robust optimization (DRO) approach that assumes only the first and second moments of these three uncertainties are known. Given an exogenous consignment rate, we derive the closed-form expression for the supplier’s optimal production decision and her worst-case expected profit. We study the impact of random price and random yield within the ambiguity set on the supplier’s production decision and the worst-case expected profit. From the e-commerce platform’s perspective, we further examine the contract design problem when the consignment rate is endogenous and analyze equilibrium solutions for both contracting parties. Our results show that the additional consideration of price or yield uncertainty within the ambiguity set does not always mitigate or enhance the degree of conservatism of the contracting parties, which is closely related to the correlation between any two types of uncertainty, as well as the degree of uncertainty itself.

Distributionally robust optimization scheduling of port energy system considering hydrogen production and ammonia synthesis

In order to effectively address the uncertainty risks of port energy system caused by intermittence and fluctuation of renewable energy, this paper proposes a scheduling method for port energy system based on distributionally robust optimization (DRO) considering ammonia synthesis after hydrogen production by water electrolysis (P2H2A), and uses real data from Tianjin Port for example analysis. The calculation results show that 1 h selected for the scheduling interval of P2H2A is reasonable, it can ensure that the ammonia synthesis reaction transitions smoothly to the new steady state, and the temperature and pressure of the ammonia converter meet safety constraints. The two-stage scheduling of port energy system based on DRO can be divided into pre-scheduling in the day-ahead stage and rescheduling in the intraday stage, which can improve the capacity of anti-risk for stochastic optimization and overcome the conservatism of robust optimization, and consider economy and robustness. Moreover, the rescheduling decision can be transformed to a prediction error function, the result of two-stage scheduling based on DRO is the pre-scheduling result, which is between the cost of stochastic optimization and robust optimization. As the Wasserstein distance-based sphere radius increases, the pre-scheduling cost of DRO gradually deviates from risk neutral stochastic optimization and leans towards risk averse robust optimization. When the Wasserstein distance-based sphere radius remains constant, the variance gradually decreases as the number of scenarios increases, which can promote the Wasserstein distance-based fuzzy set to converge to the true distribution. When the number of scenarios is greater than 15, the pre-scheduling cost will no longer fluctuate significantly, and the calculation time is in the range of 1200 s–6600 s. It can meet the demands of day-ahead scheduling calculation time. Therefore, the scheduling model has outstanding advantages in the computing time to improve the flexibility and economy of Tianjin Port's energy system scheduling, considering ammonia synthesis after hydrogen production using renewable energy.

Optimization of energy storage capacity using distributionally robust optimization for converter-based grids

As the penetration of converters into the grid continues to increase, converter-based power sources are replacing synchronous machine-based power sources, leading to the establishment of a converter-based grid (CBG) for regulating grid frequency and voltage. At this stage, energy storage becomes a necessity to support the operation of the CBG. The grid-interfacing inverters are transitioning from the conventional grid-following (GFL) control to the grid-forming (GFM) control. within the context of this research paper, considering the premise that energy storage capacity must meet CBG stability constraints and accommodate the integration of renewable energy sources, a capacity optimization model is constructed based on distributional robust optimization. The primary objective of this model is to minimize the overall cost with the allocation of energy storage capacity as the central focus. The study includes case analyses conducted within an IEEE-14 node CBG. The results of the case analyses confirm that the determined energy storage capacity can ensure the stable operation of the CBG and meet the demands for integrating renewable energy sources.

Value of Resilience for Self-Sustained Highway Transportation Systems

Nowadays, self-sustained highway transportation systems (HTSs) are calling for the inter-operation of road networks, distribution networks (DNs), and electric vehicles (EVs) towards the unexpected impacts inducted by extreme weather events (EWEs). A novel joint resilience assessment scheme is proposed to quantify the resilience of self-sustained HTSs under tropical cyclone (TC) events. The impacts of TCs on transmission lines and highways are formulated using a two-step ambiguity set. In the first step, a multitask learning (MTL) based model with a novel training strategy is proposed to simulate the future TC tracks; the impacts of TCs are then calculated in the second step based on the simulation results. Using the ambiguity set, the self-sustained HTSs resilience assessment problem is formulated based on the dynamic optimal traffic assignment problem, where EVs are integrated as third-party emergency resources before the advent of TCs. This scheme is then formulated as a two-stage distributionally robust optimization (DRO) problem and reformulated as a mixed-integer linear programming (MILP) problem. Case studies have been conducted on a self-sustained HTS consisting of a modified IEEE-14 bus test system with 2 wind farms (WFs) and a 14-node transportation network with 4 charging stations (CSs). Numerical results indicate that the proposed scheme can quantify the resilience of self-sustained highway transportation systems under the joint operation of EVs regarding the unmet travel demand and load-shedding costs. In comparison to the statistical TC forecasting models, the proposed MTL-based model is more accurate and can reduce the system cost in most cases.

Cross-database and cross-channel electrocardiogram arrhythmia heartbeat classification based on unsupervised domain adaptation

The classification of electrocardiogram (ECG) plays a crucial role in the development of an automatic cardiovascular diagnostic system. However, the considerable variances in ECG signals between individuals pose a significant challenge. Changes in data distribution limit cross-domain utilization of a model. In this study, we propose a solution to classify ECG in an unlabeled dataset by leveraging knowledge obtained from labeled source domain. We present a domain-adaptive deep network based on cross-domain feature discrepancy optimization. Our method comprises three stages: pre-training, cluster-centroid computing, and adaptation. In pre-training, we employ a Distributionally Robust Optimization (DRO) technique to deal with the vanishing worst-case training loss. To enhance the richness of the features, we concatenate three temporal features with the deep learning features. The cluster computing stage involves computing centroids of distinctly separable clusters for the source using true labels, and for the target using confident predictions. We propose a novel technique for selecting confident predictions in the target domain. In the adaptation stage, we minimize compacting loss within the same cluster, separating loss across different clusters, inter-domain cluster discrepancy loss, and running combined loss to produce a domain-robust model. Experiments conducted in both cross-domain and cross-channel paradigms show the efficacy of the proposed method. Our method achieves superior performance compared to other state-of-the-art approaches in detecting ventricular ectopic beats (V), supraventricular ectopic beats (S), and fusion beats (F). Our method achieves an average improvement of 11.78% in overall accuracy over the non-domain-adaptive baseline method on the three test datasets.

A distributionally robust optimization approach for optimal load dispatch of energy hub considering multiple energy storage units and demand response programs

The coupling relationship among various energy sources has been consistently reinforced by the advancement of the energy Internet. In the context of modeling multi-energy systems, the energy hub (EH) represents a significant and valuable approach. To attain optimal operation of the EH, this study proposes an optimal load dispatch model for a system that incorporates a combined heat and power (CHP) unit, gas boiler (GB), electric chiller (EC), absorption chiller (AC), heat energy storage (HES) unit, and electrical energy storage (EES) unit. A two-stage distributionally robust optimization (DRO) method that is driven by data is employed to address the uncertainties of electricity prices. Additionally, the proposed two-stage DRO model is effectively solved through the utilization of the column and constraint generation (C&CG) algorithm. Three cases are discussed in the paper with various energy storage units and demand response (DR) settings. The simulation results show that by deploying energy storage units and participating in DR projects, the EH system can reduce the total costs by 2.56 % and 10 %, respectively. Meanwhile, simulation results reveal that the proposed data-driven DRO model can achieve a compromise between the economy and robustness of the scheduling model compared with stochastic optimization (SO) and robust optimization (RO) methods. The simulation results illustrate the effectiveness of the proposed model for ensuring the economical and efficient operation of the system in the face of electricity price uncertainties.

Distributionally robust optimization model considering deep peak shaving and uncertainty of renewable energy

In order to achieve the goal of carbon neutrality, the capacity of renewable power generation is continuously expanding while thermal power units are transitioning from main power source to auxiliary power source. To alleviate the peak shaving burden of thermal power units under the uncertainty of renewable energy and improve the absorption level of renewable energy, a two-stage distributionally robust optimization (DRO) model considering deep peak shaving and the uncertainty of renewable energy is proposed. The day-ahead unit commitment solutions are determined in the first stage, and the detailed scheduling strategies are obtained in the second stage. Column-and-constraint generation (C&CG) algorithm is applied to solve the model, and the master problem and subproblem are reformulated as duality-free mixed integer linear programming problems. The results show that the scheduling strategy obtained based on the model can alleviate the peak shaving burden brought by the uncertainty of renewable energy and reduce the abandonment rate of wind resources and solar resources, and the proposed DRO model provides a good trade-off between economy and robustness compared to stochastic optimization (SO) and robust optimization (RO).

A convex multi-objective distributionally robust optimization for embedded electricity and natural gas distribution networks under smart electric vehicle fleets

The indisputable environmental concerns have forced the imminent proliferation of renewable energy sources (RES) and electric vehicles (EV). However, the high penetration of such uncertain and variable sources, can pose significant challenges for maintaining supply–demand balance in electrical distribution networks (EDNs). To address these challenges, this paper presents a distributionally robust optimization (DRO) method for multi-objective scheduling in integrated electricity and natural gas distribution networks (IENGDNs). The proposed approach aims to minimize environmental-economic objectives while taking into account the high penetration of EVs and RESs. The impact of a smart EV charging strategy is evaluated to reduce operating costs and maximize the use of RESs. Additionally, demand response programs (DRPs) are used in the EDN to prevent overlapping of peak load hours between the EDN and natural gas distribution network (NGDN). Linepack technology is also used to store natural gas in NGDN pipelines, which increases the short-term flexibility of the entire IENGDNs. The proposed problem is mathematically structured as a second-order conical programming (SOCP) model to benefit from the reliable and efficient convex optimization solution. The simulations were conducted on a 123-EDN and a 40-NGDN systems. Different simulation cases show that the proposed economic-environmental framework can bring down the total emissions by 10.02%.