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This paper presents a novel moving horizon estimation approach based on distributionally robust optimisation to tackle the state estimation problem of non-linear systems with missing noise distribution information. The proposed method adopts a fuzzy set to mitigate the impact of uncertainties on state estimation. Specifically, the method derives an empirical distribution within the prediction window using a priori data and constructs a fuzzy sphere set using the Wasserstein metric with the empirical distribution as the sphere centre. This enables the estimation of the state sequence under the worst probability distribution of the fuzzy set. To demonstrate the effectiveness of the proposed method, a simple simulation example is conducted to compare its performance with that of traditional moving horizon estimation. The results provide evidence of the feasibility and superiority of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper builds a bridge between two area in optimization and machine learning by establishing a general connection between Wasserstein distributional robustness and variation regularization. It helps to demystify the empirical success of Wasserstein distributionally robust optimization and devise new regularization schemes for machine learning. Wasserstein distributionally robust optimization (DRO) is an approach to optimization under uncertainty in which the decision maker hedges against a set of probability distributions, specified by a Wasserstein ball, for the uncertain parameters. This approach facilitates robust machine learning, resulting in models that sustain good performance when the data are to some extent different from the training data. This robustness is related to the well-studied effect of regularization. The connection between Wasserstein DRO and regularization has been established in several settings. However, existing results often require restrictive assumptions, such as smoothness or convexity, that are not satisfied by many important problems. In this paper, we develop a general theory for the variation regularization effect of the Wasserstein DRO—a new form of regularization that generalizes total-variation regularization, Lipschitz regularization, and gradient regularization. Our results cover possibly nonconvex and nonsmooth losses and losses on non-Euclidean spaces and highlight the bias-variation tradeoff intrinsic in the Wasserstein DRO, which balances between the empirical mean of the loss and the variation of the loss. Example applications include multi-item newsvendor, linear prediction, neural networks, manifold learning, and intensity estimation for Poisson processes. We also use our theory of variation regularization to derive new generalization guarantees for adversarial robust learning. Funding: X. Chen is supported by the National Science Foundation [Grant IIS-1845444]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2383. [ABSTRACT FROM AUTHOR]

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings $ ({\rm DRCCPs}) $ (DRCCPs) with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a data-driven distributionally robust two-stage stochastic linear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stage stochastic programming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stage stochastic programming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models. [ABSTRACT FROM AUTHOR]

Wasserstein distributionally robust optimization is a recent emerging modeling paradigm for decision making under data uncertainty. Because of its computational tractability and interpretability, it has achieved great empirical successes across several application domains in operations research, computer science, engineering, and business analytics. Despite its recent empirical success, existing performance guarantees for generic problems are not yet satisfactory. In this paper, we develop the first finite-sample guarantee without suffering from the curse of dimensionality, which describes how the out-of-sample performance of a robust solution depends on the sample size, dimension of the uncertainty, and the complexity of the loss function class in a nearly optimal way. Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative because of the curse of dimensionality or plausible only in large sample asymptotics. In this paper, we develop a nonasymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipschitz and gradient regularization problems. To the best of our knowledge, this gives the first finite-sample guarantee for generic Wasserstein DRO problems without suffering from the curse of dimensionality. Our results highlight that Wasserstein DRO, with a properly chosen radius, balances between the empirical mean of the loss and the variation of the loss, measured by the Lipschitz norm or the gradient norm of the loss. Our analysis is based on two novel methodological developments that are of independent interest: (1) a new concentration inequality controlling the decay rate of large deviation probabilities by the variation of the loss and (2) a localized Rademacher complexity theory based on the variation of the loss. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2326. [ABSTRACT FROM AUTHOR]

The penetration of wind turbines in the power grid is increasing rapidly. Still, the wind turbine output power has uncertainty, leading to poor grid reliability, affecting the grid's dispatching plan, and increasing the total cost. Thus, a distributionally robust optimization method for thermal power unit commitment considering the uncertainty of wind power is proposed. For this method, energy storage and interruptible load are added to simulate increasingly complex electricity consumption scenarios. Furthermore, the amount of load cutting reflects the satisfaction level of electricity consumption on the user side. Based on Wasserstein metric, an ambiguity set is established to reflect the probabilistic distribution information of the wind power uncertainty. An ambiguity set preprocessing method is proposed to depict the probability distribution of ambiguity set more clearly, to minimize the operation cost under the condition that the uncertainty of wind turbine output power obeys the extreme probabilistic distribution of the ambiguity set. The test case in a modified version of the IEEE 6‐bus system shows that the proposed method can flexibly adjust the robustness and economy of optimization decisions by controlling the sample size and the confidence of Wasserstein ambiguity set radius. In addition, the proposed ambiguity set preprocessing method can obtain more economical dispatching decisions with a smaller sample size. [ABSTRACT FROM AUTHOR]

Multi-energy virtual power plant (MEVPP) has attracted more and more attention due to its advantages in renewable energy consumption and carbon emission reduction. However, the characteristics of multi-energy coupling and the access of renewable energy may lead to some challenges in the operation of MEVPP. In this paper, a data-driven distributionally robust chance constraints optimization model (DD-DRCCO) is proposed for the dispatching of MEVPP. Firstly, the uncertainties of wind power and photovoltaic output forecasting errors are modeled as an ambiguity set based on the Wasserstein metric. Secondly, combined with the chance constraint, the expected probability of the inequality constraint with uncertain variables is limited to the lowest allowable confidence level to improve the reliability of the model. Thirdly, the forecast errors of wind power and photovoltaic are considered in the constraint conditions, so that the system can effectively resist the interference of uncertain output. Besides, based on the strong duality theory, the DD-DRCCO model is equivalent to a MILP problem which is easy to solve. Finally, simulations implemented on a typical MEVPP are delivered to show that our proposed model: 1) The model is data-driven, and the conservativeness is kept at a low level, and the solution time is about 7s~8s; 2) The MEVPP system can achieve a balance between economy and low-carbon, making the total operation cost reduced by 0.89% compared with no increase of electric boiler; 3) The CO2 emission during the operation of the MEVPP system was significantly reduced by about 87.33 kg. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model. [ABSTRACT FROM AUTHOR]

• A new distributionally robust mean-variance model with Wasserstein metric was developed. • The novel model was transformed into a tractable convex problem by using duality theory. • A nonparametric bootstrap method was introduced to estimate the radius of the Wasserstein ball. • The effects of the parameters and the advantages of the model were demonstrated by some experiments. The mean-variance model formulated by Markowitz for a single period serves as a fundamental method of modern portfolio selection. In this study, we consider a multi-period case with uncertainty that better matches the reality of the financial market. Using the Wasserstein metric to characterize the uncertainty of returns in each period, a new distributionally robust mean-variance model is proposed to solve multi-period portfolio selection problem. We further transform the developed model into a tractable convex problem using duality theory. We also apply a nonparametric bootstrap method and provide a specific algorithm to estimate the radius of the Wasserstein ball. The effects of the parameters on the corresponding strategy and evaluation criteria of portfolios are analyzed using in-sample data. The analysis indicate that the return and risk of our portfolio selections are relatively immune to parameter values. Finally, a series of out-of-sample experiments demonstrate that the proposed model is superior to some other models in terms of final wealth, standard deviation, and Sharpe ratio. [ABSTRACT FROM AUTHOR]

Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is a known true underlying probability distribution, one hedges against a chosen set of distributions. In this paper, we first point out that the set of distributions should be chosen to be appropriate for the application at hand and some of the choices that have been popular until recently are, for many applications, not good choices. We next consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from other popular choices of sets. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a strong duality reformulation of the corresponding DRSO problem and construct approximate worst-case distributions (or an exact worst-case distribution if it exists) explicitly via the first-order optimality conditions of the dual problem. Our contributions are fourfold. (i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function. (ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation. (iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization. (iv) Our strong duality result holds in a very general setting. As examples, we show that it can be applied to infinite dimensional process control and intensity estimation for point processes. [ABSTRACT FROM AUTHOR]

Concentrating solar power (CSP) plants have significant potential to complement the growing wind energy in power scheduling. This study examines an integrated energy system (IES) that incorporates a wind turbine (WT), CSP, and combined heat and power (CHP) to promote the utilization of renewable energy (RE), reduce fluctuations caused by uncertainty, and enhance the economic viability of the system. We propose a distributionally robust optimization (DRO) model for IES scheduling that considers the uncertainty of wind power by using an ambiguity set defined by the Wasserstein metric. Before the occurrence of uncertainties, the system determines the initial dispatching scheme based on the forecast data. In the second stage, the system aims to minimize the adjustment cost expectation under the worst distribution of the ambiguity set and adjusts the flexible resources in real time to offset the fluctuations caused by forecasting errors. The proposed DRO model is transformed into a conventional two-stage robust problem using strong dual theory and KKT conditions, and then solved with a modified column-and-constraint generation (C&CG) algorithm. The results of case studies show that the CSP plant enhances the system's flexibility and controllability through thermal energy storage (TES). [ABSTRACT FROM AUTHOR]

We propose a distributionally robust optimization formulation with a Wasserstein-based uncertainty set for selecting grouped variables under perturbations on the data for both linear regression and classification problems. The resulting model offers robustness explanations for grouped least absolute shrinkage and selection operator algorithms and highlights the connection between robustness and regularization. We prove probabilistic bounds on the out-of-sample loss and the estimation bias, and establish the grouping effect of our estimator, showing that coefficients in the same group converge to the same value as the sample correlation between covariates approaches 1. Based on this result, we propose to use the spectral clustering algorithm with the Gaussian similarity function to perform grouping on the predictors, which makes our approach applicable without knowing the grouping structure a priori. We compare our approach to an array of alternatives and provide extensive numerical results on both synthetic data and a real large dataset of surgery-related medical records, showing that our formulation produces an interpretable and parsimonious model that encourages sparsity at a group level and is able to achieve better prediction and estimation performance in the presence of outliers. [ABSTRACT FROM AUTHOR]

Hydro-wind-solar integrated operation is a promising way to balance the growing amount of variable renewable energy (RE) and enhance energy utilization efficiency. This study focuses on the short-term reliable-economic equilibrium operation of the hydro-wind-solar energy systems. A Distributionally Robust Optimization based on Wasserstein metric, which considers multiple power supply risks (output shortage, power curtailment and spilled water) and the economy (the water consumption and the stored energy) of hydropower operation simultaneously is proposed to address the above issue. The model is transformed into a mixed-integer linear programming framework using a reformation approach based on ϵ -constraint method, strong duality theory, and linearization technology. Case studies are conducted for a hydro-wind-solar energy system in Southwest China's Beipan River basin. Results indicate that: (1) with more available data, more information about RE uncertainties is involved in the decision-making process to hedge against RE variability's interference. (2) Decisions obtained from the proposed model overcome the issues of stochastic optimization's over-optimism and robust optimization's over-conservativeness. (3) Reliable-economic trade-off mechanisms are derived by imposing a decision-maker's risk attitude, the risks in dry and wet season typical cases decline from 10483.05 MW to 1606.39 MW and from 8394.15 MW to 118.08 MW by adjusting the economy. [ABSTRACT FROM AUTHOR]

We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover's distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest and provide an overview of various settings leading to tractable formulations. Some of the arising side results, such as the mathematical programming expressions for robustified risk measures in a discrete space, are also of independent interest. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications, and present a numerical study for a novel risk-averse scheduling problem with controllable processing times. Summary of Contribution: In this study, we introduce a new class of optimization problems that simultaneously address distributional and decision-dependent uncertainty. We present a unified modeling framework along with a discussion on possible ways to specify the key model components, and discuss the main computational challenges in solving the complex problems of interest. Special care has been devoted to identifying the settings and problem classes where these challenges can be mitigated. In particular, we provide model reformulation results, including mathematical programming expressions for robustified risk measures, and describe how these results can be utilized to obtain tractable formulations for specific applied problems from the fields of humanitarian logistics and machine scheduling. Toward demonstrating the value of the modeling approach and investigating the performance of the proposed mixed-integer linear programming formulations, we conduct a computational study on a novel risk-averse machine scheduling problem with controllable processing times. We derive insights regarding the decision-making impact of our modeling approach and key parameter choices. [ABSTRACT FROM AUTHOR]

We study the distributionally robust linearized stable tail adjusted return ratio (DRLSTARR) portfolio optimization problem, in which the objective is to maximize the worst-case linearized stable tail adjusted return ratio (LSTARR) performance measure under data-driven Wasserstein ambiguity. We consider two types of imperfectly known uncertainties, named uncertain probabilities and continuum of realizations, associated with the losses of assets. We account for two typical combinatorial trading constraints, called buy-in threshold and diversification constraints, to reflect stock market restrictions. Leveraging conic duality theory to tackle the distributionally robust worst-case expectation, the proposed problems are reformulated into mixed-integer linear programming problems. We carry out a series of empirical tests to illustrate the scalability and effectiveness of the proposed solution framework, and to evaluate the performance of the DRLSTARR-constructed portfolios. The cross-validation results obtained using a rolling-horizon procedure show the superior out-of-sample performance of the DRLSTARR portfolios under an uncertain continuum of realizations. [ABSTRACT FROM AUTHOR]

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first-order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide an explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least-squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples. [ABSTRACT FROM AUTHOR]

We investigate a class of fractional distributionally robust optimization problems with uncertain probabilities. They consist in the maximization of ambiguous fractional functions representing reward-risk ratios and have a semi-infinite programming epigraphic formulation. We derive a new fully parameterized closed-form to compute a new bound on the size of the Wasserstein ambiguity ball. We design a data-driven reformulation and solution framework. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function. We design modular bisection algorithms which enjoy the finite convergence property. This class of problems has wide applicability in finance, and we specify new ambiguous portfolio optimization models for the Sharpe and Omega ratios. The computational study shows the applicability and scalability of the framework to solve quickly large, industry-relevant-size problems, which cannot be solved in one day with state-of-the-art mixed-integer nonlinear programming (MINLP) solvers. [ABSTRACT FROM AUTHOR]

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem. [ABSTRACT FROM AUTHOR]

A rise in traffic accidents has led to both traffic congestion and subsequent secondary accidents. Effectively addressing this issue requires rapid accident investigation and management. In this paper, we aim to improve the efficiency of traffic accident assessment and investigation with the aid of drone technologies. Our approach involves strategically pre-positioning drones, enabling traffic supervisory agencies to dispatch drones immediately upon receiving an accident report. Methodology-wise, we present a data-driven robust stochastic optimization (RSO) model, which encapsulates the uncertainty of traffic accidents within a scenario-wise Wasserstein ambiguity set. To the best of our knowledge, this is the first study that incorporates covariates, i.e., weather conditions, into the Wasserstein ambiguity set with the CVaR metric. We demonstrate that the proposed RSO model can be reformulated into a mixed-integer programming model, allowing an efficient solution approach. Via a real-world dataset of London traffic accidents, we validate the practical applicability of the RSO model. Across various parameter settings, our RSO model exhibits superior out-of-sample performance compared with various benchmark models. The numerical results yield valuable insights for traffic supervisory agencies. • We propose a drone pre-positioning problem for traffic accident rapid assessment. • We propose an RSO model that incorporates the uncertain traffic accident. • We demonstrate that the RSO model admits a tractable MILP reformulation. • A real world instance is used to verify the practicality of our model. [ABSTRACT FROM AUTHOR]

In this letter, we propose a tractable formulation and an efficient solution method for the Wasserstein-metric-based distributionally robust unit commitment (DRUC-dW) problem. First, a distance-based data aggregation method is introduced to hedge against the dimensionality issue arising from a huge volume of data. Then, we propose a novel cutting plane algorithm to solve the DRUC-dW problem much more efficiently than state-of-the-art. The novel solution method is termed extremal distribution generation, which is an extension of the column-and-constraint generation method to the distributionally robust cases. The feasibility and cost efficiency of the model, and the efficiency of the solution method are numerically validated. [ABSTRACT FROM AUTHOR]

The multiple criteria sorting decision is a critical area of research in decision science. However, accurately capturing the decision maker's cognition for each criterion using precise values poses significant challenges. To address this issue, our study investigates an uncertain multiple criteria sorting problem with indeterminate values for each criterion, which closely aligns with real-life decision-making processes. We adopt stochastic programming (SP) and distributionally robust optimization (DRO) to deal with the associated uncertainties. We formulate the preference learning process as a binary classification problem, where a piecewise-linear function is employed to measure the preference. Specifically, we develop a distributionally robust support vector machine model that incorporates the Wasserstein metric. By leveraging duality theory, the resulting DRO-based model can be transformed into a tractable convex problem. Then, we introduce an incremental projected subgradient algorithm to solve the tractable model, which can effectively utilize the separable structure. Finally, the efficiency of the proposed method is demonstrated through numerical experiments. The comparative evolution of the marginal value functions exhibits a similar tendency, highlighting the advantage and robustness of our approach. • The multiple criteria sorting problem with uncertain criteria is considered. • A distributionally robust support vector machine model is developed. • The tractable reformulation of the model under Wasserstein metric is discussed. • An incremental projected subgradient algorithm is introduced. • Some experiments are conducted to demonstrate the advantages. [ABSTRACT FROM AUTHOR]

With the increasing global attention to environmental protection and sustainable development, governments around the world attach great importance to the low-carbon reform of rural energy and waste management. The interaction between waste-to-energy technology (W2E) and rural multi-energy system has emerged as one of the important issues that must be considered. Based on this, this paper combines two W2E technologies, biomass biogas power generation and garbage incineration power generation, with photovoltaic and energy storage systems. A distributionally robust optimization (DRO) model for the waste-disposal-facilities-involved rural multi-energy system (W-RMES) is proposed. An ambiguity set of the photovoltaic output probability distribution is constructed by the Wasserstein metric. The minimum operation cost of W-RMES is solved under the condition that the prediction error of photovoltaic output obeys the worst probability distribution of the ambiguity set. Finally, through simulation verification, the rational utilization of waste can effectively realize the coordination and optimization between the waste disposal and multi-energy supply of electricity, heat, and biogas. Compared with the conventional rural energy system, W-RMES is more economical and eco-friendlier, which is reflected in the reduction of operation cost by >14.87 %, the removal of flue gas production by >12.89 % and the improvement of waste disposal capacity by >21.77 %. Besides, the model also realizes systemic coordination of robustness and economic efficiency. • Combined the rural multi-energy system with waste disposal facilities (W-RMES) • Considered two waste-to-energy technologies: anaerobic fermentation and incineration • The DRO method was used to deal with the uncertainty in W-RMES dispatching. • The W-RMES coordinated the waste disposal capacity and rural multi-energy supply. • The model realized systemic coordination of robustness and economic efficiency. [ABSTRACT FROM AUTHOR]

The interconnection of multiple regional integrated energy systems (RIES) can effectively enhance the low-carbon and flexible operation capabilities of RIES, but the uncertainty and multi-energy interaction of the system pose challenges to the stable operation of RIES. Therefore, a stochastic hierarchical optimization and revenue allocation approach is proposed to optimize the operational strategy of multi-RIES with multi-operator participation, multi-energy interactions, and multiple uncertainties. Firstly, agent-based modeling, Latin hypercube sampling and simultaneous backward reduction based on the Wasserstein metric are proposed to capture the power-side and load-side uncertainties. Secondly, based on the Nash bargaining game and the alternating direction method of multipliers algorithm, multi-RIES peer-to-peer transactions of electricity, thermal, and natural gas are optimized through energy cooperation and scheduling strategies. Subsequently, a Nash-Harsanyi bargaining game revenue allocation method that considers both fairness and renewable energy accommodation is proposed to ensure the stable operation of the alliance. Finally, simulations are conducted on a multi-RIES in northern China, demonstrating that the proposed model and approach can achieve inter-grid flexible resource complementarity, improve RIES economics, increase local renewable energy accommodation rate, and reduce carbon emissions. • Multi-RIES multi-energy internal scheduling and P2P transaction co-optimization. • Wasserstein metric is introduced into the SBR model to capture stochastic scenarios. • A multi-RIES stochastic hierarchical optimization model based on cooperative games. • ADMM is used to solve distributed stochastic hierarchical optimization model. • A revenue allocation method considering equity & renewable energy development is proposed. [ABSTRACT FROM AUTHOR]

There are numerous industrial settings in which a decision maker must decide whether to enter into long-term contracts to guarantee price (and hence cash flow) stability or to participate in more volatile spot markets. In this paper, we investigate a data-driven distributionally robust optimization (DRO) approach aimed at balancing this tradeoff. Unlike traditional risk-neutral stochastic optimization models that assume the underlying probability distribution generating the data is known, DRO models assume the distribution belongs to a family of possible distributions, thus providing a degree of immunization against unseen and potential worst-case outcomes. We compare and contrast the performance of a risk-neutral model, conditional value-at-risk formulation, and a Wasserstein distributionally robust model to demonstrate the potential benefits of a DRO approach for an "elasticity-aware" price-taking decision maker. • Investigate tradeoffs between long-term supply contracts vs. spot market sales. • Data-driven DRO approach exploiting Wasserstein ambiguity sets. • Compare risk-neutral, conditional value-at-risk, and data-driven DRO approaches. • Case studies in power portfolio optimization for PJM electricity market. [ABSTRACT FROM AUTHOR]

Natural disasters are highly unpredictable, with varying degrees of magnitude, and thus require a reliable and robust humanitarian relief network. Faced with the adverse effects of disasters, we advocate taking pre-disaster preventive actions, e.g., road link strengthening, to mitigate post-disaster disruptions to road networks. In this paper, we study a highly integrated humanitarian relief network design problem, in which the network strengthening plan and inventory pre-positioning scheme are optimized cooperatively. The proposed problem is formulated as a two-stage distributionally robust optimization (DRO) model, in which the first and second stages correspond to pre- and post-disaster relief operations, respectively. By leveraging prior data and the Wasserstein metric, a tailored ambiguity set is constructed to capture both node- and link-wise uncertainties. We demonstrate that the two-stage DRO model over the Wasserstein ambiguity set has an equivalent reformulation that can be solved via the L-shaped method with bilinear subproblems. Based on observations of the problem structure and the spirit of separation optimization, both exact and heuristic approaches are proposed to address the bilinear subproblems. Via the case study of the Yushu earthquake, we highlight the value of jointly optimizing network strengthening and inventory pre-positioning decisions. Specifically, with the same investment budget, integrating network strengthening into inventory pre-positioning can easily achieve an approximately 60% reduction in shortage penalties. Furthermore, our method can produce reliable solutions, based on which we explore several managerial implications. • We propose an integrated humanitarian network design problem. • The proposed problem is formulated as a two-stage distributionally robust model. • A Wasserstein ambiguity set is constructed to capture uncertainty. • We demonstrate that the two-stage Wasserstein DRO model can be solved by L-shaped. • A real-world instance was used to demonstrate the practicality of our model. [ABSTRACT FROM AUTHOR]

This paper proposed a Wasserstein metric-based distributionally robust approximate framework (WDRA), for unit commitment problem to manage the risk from uncertain wind power forecasted errors. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed framework minimizes the generating cost, start-up cost, shut-down cost, reserve cost, and the expected thermal generation adjusting cost under the worst-case distribution in the ambiguity set. The more historical available, the smaller the ambiguity set is and, hence, the less conservativeness the decision is. The size of the Wasserstein metric based robust counterpart (WDRC) model mainly depends on the size of sample set, which has a computation burden when more historical data are available. To overcome this drawback, this paper proposed an upper approximate of WDRC and verified the condition that makes the approximate model become exact. Comparisons with robust optimization and stochastic optimization illustrate that the proposed model can balance the economy and conservativeness effectively. Monte Carlo simulations on a modified IEEE-118 and a real 703-bus systems show that the proposed WDRA framework could reduce the computational time by several times when compared with WDRC. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a two-stage distributionally robust mixed binary problem (TSDR-MBP) in which the random parameters follow the worst-case distribution belonging to an uncertainty set of probability distributions. We present a decomposition algorithm, which utilizes a distribution separation procedure and parametric cuts within Benders' algorithm or the L-shaped method, to solve TSDR-MBPs with binary variables in the first stage and mixed binary programs in the second stage. We refer to this algorithm as a distributionally robust integer (DRI) L-shaped algorithm. Using a similar decomposition framework, we provide another algorithm to solve a TSDR linear problem where both stages have only continuous variables. We investigate conditions and the families of ambiguity sets for which our algorithms are finitely convergent. We present two examples of ambiguity set, defined using moment matching or the Kantorovich-Rubinstein distance (Wasserstein metric), which satisfy the foregoing conditions. We also present a cutting surface algorithm to solve TSDR-MBPs. We computationally evaluate the performance of the DRI L-shaped algorithm and the cutting surface algorithm in solving distributionally robust versions of a few instances from the stochastic integer programming library, in particular stochastic server location and stochastic multiple binary knapsack problem instances. We also discuss the usefulness of incorporating partial distribution information in two-stage stochastic optimization problems. [ABSTRACT FROM AUTHOR]

Portfolio optimization can lead to misspecified stock returns that follow a known distribution. To investigate tractable formulations of the portfolio selection problem, we study these problems with the ambiguity set defined by the Wasserstein metric. Robust optimization with Wasserstein models protects against ambiguity in the distribution when analyzing decisions. This study considers portfolio optimization using a robust mean absolute deviation model consistent with the Wasserstein metric. The core of our idea is to consider the sets of distributions that lie within a certain distance from an empirical distribution. However, since information in financial markets is often unclear, we extend this structure to the weighted mean absolute deviation model when the underlying probability distribution is not precisely known. We then construct a decomposition algorithm based on the Benders decomposition approach to solve such problems. For more efficient comparison, the acquired optimization programs are applied to real data. • We consider portfolio optimization models with robust mean absolute deviation. • The problems are presented when the underlying probability model is not fully known. • Our robust portfolio problems is based on the Wasserstein metric with ambiguity sets. • The Benders decomposition method adapts to provide an efficient solution strategy. [ABSTRACT FROM AUTHOR]

[Display omitted] • A data-driven TSDRO model is developed for refinery planning under uncertainty. • The uncertainty set is constructed by the Wasserstein metric and RKDE approach. • The robust counterpart of the proposed model is reformulated by duality theory. • A real-world case study of an industrial refinery is presented. • The out-of-sample performance of different methods are analyzed. This work investigates the refinery planning problem under uncertainty in product prices. A novel data-driven Wasserstein distributionally robust optimization framework is proposed for handling uncertainties in the refinery-wide planning operations. A data-driven ambiguity set is constructed based on the Wasserstein metric to model the distributional uncertainty. The robust kernel density estimation technique is adopted to establish the support set to reduce the effect of the potential outliers. Based on the derived support set and ambiguity set, a data-driven two-stage distributionally robust optimization model for refinery planning is developed. Then, the robust counterpart of the proposed model is formulated to make the problem computationally tractable. Finally, a real-world case study on a petroleum refinery is presented to illustrate the effectiveness and applicability of the proposed framework. [ABSTRACT FROM AUTHOR]

• Constructed a wind-hydrogen-storage combined system with low-carbon characteristics. • Dispatching of wind-hydrogen-storage combined system based on DRO model. • A profit allocation model is proposed based on the bargaining game theory. Aiming at the coexistence of multiple players in the wind-hydrogen-storage combined system, a new profit allocation mechanism is proposed. The combination of multiple stakeholders such as wind power plant (WT), hydrogen energy system (HE), and energy storage system (ES) can achieve the purpose of promoting renewable energy consumption by using renewable energy to produce hydrogen, so as to improve overall system benefits. However, WT, HE, and ES belong to different stakeholders, and wind output is uncertain, which affects the efficient operation of the wind-hydrogen-storage combined system. Based on this, firstly, the Wasserstein metric is used to characterize the ambiguity set of the probability distribution of wind output forecast error, and a distributionally robust optimization model considering the uncertainty of wind output and demand response is constructed to maximize the benefits of the wind-hydrogen-storage combined system. Secondly, in order to balance the profits of multiple players in the combined system, a profit allocation model considering the real contribution of each player is proposed based on the Nash-Harsanyi bargaining game theory. Finally, the effectiveness of the proposed distributionally robust optimization operation model and profit allocation method are verified by simulation in a typical wind-hydrogen-storage combined system. [ABSTRACT FROM AUTHOR]

[Display omitted] • Power-to-gas facility is integrated into the trigeneration energy system. • Optimal operation of this integrated system is studied under multiple uncertainties. • A novel distributionally robust operation optimization model is proposed accordingly. • Economic and environmental benefits are improved by operating the integrated system. • Operation efficiency in various aspects are guaranteed by the proposed model. Power-to-gas is an emerging energy conversion technology. When integrating power-to-gas into the combined cooling, heating and power system, renewable generations can be further accommodated to synthesize natural gas, and additional revenues can be obtained by reutilizing and selling the synthesized gas. Therefore, it is necessary to address the optimal operation issue of the integrated system (Combined cooling, heating and power-Power-to-gas) for bringing the potential benefits, and thus promoting energy transition. This paper proposes a Wasserstein and multivariate linear affine based distributionally robust optimization model for the above issue considering multiple uncertainties. Specifically, the uncertain distribution of wind power and electric, thermal, cooling loads is modeled as an ambiguity set by applying the Wasserstein metric. Then, based on the ambiguity set, the proposed model with two-stage structure is established. In the first-stage, system operation cost (involving the energy exchange and carbon emission costs, etc.) is minimized under the forecast information. In the second stage, for resisting the interference of multiple uncertainties, the multivariate linear affine policy models are constructed for operation rescheduling under the worst-case distribution within the ambiguity set, which is capable of adjusting flexible resources according to various random factors simultaneously. Simulations are implemented and verify that: 1) both the economic and environmental benefits of system operation are improved by integrating power-to-gas; 2) the proposed model keeps both the conservativeness and computational complexity at low levels, and its solutions enable the effective system operation in terms of cost saving, emission reduction, uncertainty resistance and renewable energy accommodation. [ABSTRACT FROM AUTHOR]