Ambiguity Set Research Articles

Optimal planning and operation of integrated energy systems in South Korea: Introducing a Novel ambiguity set based distributionally robust optimization

In this paper, we address the increasing focus on Renewable Energy Sources (RES) and energy policies in S. Korea, advocating for a shift from large, centralized power systems to decentralized Local Power Systems (LPS). This transition emphasizes the integration of Integrated Energy Systems (IES), which significantly enhances the target benefits for stakeholders in the power system. We have also adopted a Distributionally Robust Optimization (DRO) approach to account for the output uncertainty of RES and developed and applied new ambiguity sets tailored to each RES, enhancing the extensive robust design previously utilized with DRO. By applying our proposed ambiguity set-based DRO, we avoid excessive robustness and achieve superior optimal outcomes compared to the general DRO methodology, potentially making a significant contribution to business investment decisions. We have categorized the main stakeholders of the power system as Independent Systems Operators (ISO), Renewable Energy Providers (REP), Energy Exchange Aggregators (EXA), and End-users. The integration of IES leads to a benefit improvement rate of up to 34.8 %, 23.0 %, and 24.2 % for ISOs, REPs, and End-users, respectively, on a global scale. EXAs can secure an annual economic profit of up to $191,362.

Decision‐making for others: Ambiguity attitudes

AbstractWe investigated decision‐making for others in ambiguous settings. In an online survey, subjects were asked to make decisions for themselves, and for other people. In Experiment 1, ambiguity was conveyed in numerical ranges. In Experiment 2, we used verbal probabilities expressions to convey uncertainty. Decisions encompass three degrees of ambiguity (low, moderate, and high). Consistent with previous literature findings, our results showed no significant differences between self‐other decision‐making on ambiguity. We build on the existing literature on ambiguity attitudes, emphasizing the use of verbal probability expressions to measure ambiguity, and provide novel evidence into decision‐making for others.

Data‐driven inventory control involving fixed setup costs and discrete censored demand

AbstractWe investigate a data‐driven dynamic inventory control problem involving fixed setup costs and lost sales. Random demand arrivals stem from a demand distribution that is only known to come out of a vast ambiguity set. Lost sales and demand ambiguity would together complicate the problem through censoring, namely, the inability of the firm to observe the lost portion of the demand data. Our main policy idea advocates periodically ordering up to high levels for learning purposes and, in intervening periods, cleverly exploiting the information gained in learning periods. By regret, we mean the price paid for ambiguity in long‐run average performances. When demand has a finite support, we can accomplish a regret bound in the order of which almost matches a known lower bound as long as inventory costs are genuinely convex. Major policy adjustments are warranted for the more complex case involving an unbounded demand support. The resulting regret could range between and depending on the nature of moment‐related bounds that help characterize the degree of ambiguity. These are improvable to when distributions are light‐tailed. Our simulation demonstrates the merits of various policy ideas.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Distributionally robust optimal dispatching method for integrated energy system with concentrating solar power plant

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).

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization

We consider distributionally robust optimization (DRO) problems, reformulated as distributionally robust feasibility (DRF) problems, with multiple expectation constraints. We propose a generic stochastic first-order meta-algorithm, where the decision variables and uncertain distribution parameters are each updated separately by applying stochastic first-order methods. We then specialize our results to the case of using two specific versions of stochastic mirror descent (SMD): (i) a novel approximate version of SMD to update the decision variables, and (ii) the bandit mirror descent method to update the distribution parameters in the case of [Formula: see text]-divergence sets. For this specialization, we demonstrate that the total number of iterations is independent of the dimensions of the decision variables and distribution parameters. Moreover, the cost per iteration to update both sets of variables is nearly independent of the dimension of the distribution parameters, allowing for high-dimensional ambiguity sets. Furthermore, we show that the total number of iterations of our algorithm has a logarithmic dependence on the number of constraints. Experiments on logistic regression with fairness constraints, personalized parameter selection in a social network, and the multi-item newsvendor problem verify the theoretical results and show the usefulness of the algorithm, in particular when the dimension of the distribution parameters is large. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This work was supported by the National Science Foundation [Grant 2112533]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0167 .

Optimal demand response for a virtual power plant with a hierarchical operation framework

There are significant uncertainties and variations in demand response under different operating conditions and incentives. These challenges hinder virtual power plants (VPPs) from fully leveraging the regulatory capabilities of flexible resources. To address this issue, this study introduces an optimal demand response (DR) approach for VPPs, featuring a hierarchical operational framework that accounts for the uncertainty in user participation in DR. First, a refined DR model is developed, incorporating response characteristic parameters based on the DR mechanism. Second, a two-stage distributionally robust optimization model is formulated within a market trading framework for energy scheduling of VPPs. This model aims to minimize operational costs while considering uncertainties in DR. Additionally, the ambiguity set for the uncertainty probability distributions is constructed using a data-driven method. Finally, the study presents an optimal DR disaggregation method that aims to maximize aggregator satisfaction and accounts for the impact of load rebound on the optimization results. Simulation results demonstrate that the proposed method significantly reduces the deviation in demand response by 40 % and enhances the profit of the VPP by 20.7 %.

Bi-level distributionally robust optimization model for low-carbon planning of integrated electricity and heat systems

Interdependence among diverse energy forms within integrated electricity and heat systems (IEHSs) holds the potential to harness cooperation effects. This paper proposes a bi-level optimization model for the low-carbon planning of IEHSs, which realizes the coordination between IEHSs and their upstream networks. The precise carbon emissions of IEHSs are traced through carbon emission flows of components, power distribution networks, and district heating networks. Moreover, the distributionally robust optimization approach is employed to account for the inherent uncertainty in renewable power outputs faced by IEHS planning, by incorporating the moment information into an ambiguity set. The bi-level distributionally robust planning model of IEHSs is initially reconfigured into a second-order cone structure, employing the strong duality theory, second-order cone duality theory, and linear decision rules. Subsequently, an iterative approach is introduced to attain convergence of the reformulated bi-level model. Numerical results demonstrate the effectiveness and superiority of the proposed approach for reducing carbon emissions and economic costs of IEHSs.

A computational efficient approach for distributionally robust unit commitment with enhanced disjointed layered ambiguity set

AbstractTo achieve the sustainable development of the society, renewable energy dominated power systems are gradually formed. However, the uncertainty of renewable power poses challenges for power system operations, such as balancing the load and generation in day‐ahead unit commitment problems. To address this issue, an enhanced disjointed layered ambiguity set is proposed in this paper to effectively capture the uncertainty of renewable generation forecast errors. On this basis, a two‐stage distributionally robust unit commitment optimization problem is formulated, which is computational challenge. To this end, a novel reformation approach and a worst‐case support set identification method are proposed to derive the optimal solution in a computationally efficient way. Simulation results demonstrate that the proposed ambiguity set brings lower operation costs of unit commitment while maintaining similar reliability compared to existing methods. In detail, the proposed ambiguity set achieves a reduction in operating costs of approximately 10.01%. The proposed novel reformation approach and solution method are also shown to have higher computation efficiency than the existing Benders decomposition method, resulting in a reduction in computation time to less than 3.78% of the original duration.

Worst-case Conditional Value at Risk for asset liability management: A framework for general loss functions

Asset–liability management (ALM) is a challenging task faced by pension funds due to the uncertain nature of future asset returns, employees’ wages, and interest rates. To address this challenge, this paper presents a new mathematical model that uses a Worst-case Conditional Value-at-Risk (WCVaR) constraint to ensure that, with high probability, the funding ratio remains above a regulator-mandated threshold under the worst-case density function that plausibly explains historical sample data. A tractable reformulation of this WCVaR constraint is developed based on the definition of the Worst-case Lower Partial Moment (WLPM) for a general loss function. Additionally, a data-driven moment-based ambiguity set is constructed to capture uncertainty in the moments of the density functions of random variables in the ALM problem. The proposed approach is evaluated using real-world data from the Canada Pension Plan (CPP) and is shown to outperform classical ALM models, based on either CVaR or WCVaR with fixed moments, on out-of-sample data. The proposed framework for handling correlated uncertainty using WCVaR with nonlinear loss functions can be used in other application areas.

Distributionally Robust Observable Strategic Queues

This paper presents an extension of Naor’s analysis on the join-or-balk problem in observable M/M/1 queues. Although all other Markovian assumptions still hold, we explore this problem assuming uncertain arrival rates under the distributionally robust settings. We first study the problem with the classical moment ambiguity set, where the support, mean, and mean-absolute deviation of the underlying distribution are known. Next, we extend the model to the data-driven setting, where decision makers only have access to a finite set of samples. We develop three optimal joining threshold strategies from the perspectives of an individual customer, a social optimizer, and a revenue maximizer such that their respective worst-case expected benefit rates are maximized. Finally, we compare our findings with Naor’s original results and the traditional sample average approximation scheme. Funding: This research was supported by the National Science Foundation [Grants 2342505 and 2343869].

Wasserstein distributionally robust optimization for train operation and freight assignment in a metro-based underground logistics system

As a shared passenger-and-freight transportation mode, the metro-based underground logistics system (M-ULS) can significantly alleviate negative effects caused by ground transportation vehicles and improve urban sustainability. Against this background, this paper addresses dedicated freight train operation and freight assignment problems in the M-ULS without affecting passenger service during the off-peak period. More importantly, this paper proposes a flexible skip-stop strategy to reduce the increased operational cost of freight trains. To hedge against uncertainty, this paper formulates a distributionally robust chance-constrained programming model by limiting uncertain freight demands to a Wasserstein ambiguity set. Moreover, the distributionally robust optimization model for a Wasserstein ball with the 1-norm can be transformed into a mixed-integer linear program, which allows for an efficient solution via CPLEX, thereby enabling metro operators to use it. To implement this study in real-life applications, a case study is conducted based on the Beijing Subway Yizhuang Line. The computational results verify the superiority of the skip-stop strategy and distributionally robust optimization approach proposed in this study. Finally, some managerial suggestions are provided for the metro operators.

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.

Distributionally robust disaster relief planning under the Wasserstein set

We study a two-stage natural disaster management problem modeled as a stochastic program, where the first stage consists of a facility location problem, deciding where to open facilities and pre-allocate resources such as medical and food kits, and the second stage is a fixed-charge transportation problem, routing resources to affected areas after observing a disaster. Our model has binary variables present in both stages. Due to the lack of data, classical stochastic programming approaches may be ill-suited, and we propose a two-stage distributionally robust formulation with a Wasserstein ambiguity set, where we consider distributions consistent with historical data and a tunable parameter to control the level of risk aversion. We develop a tailored column-and-constraint generation (CCG) algorithm to solve an extensive reformulation, where scenarios are iteratively generated. We handle the presence of binary variables in the second stage by leveraging the structure of our support set and second-stage problem, and provide conditions under which the optimal value of the latter is concave with respect to the intensity of the disaster, leading to an efficient scenario generation procedure. We also show that our results extend to the case where the second stage is a fixed-charge network flow problem. We perform extensive computational experiments demonstrating the computational advantage of our method over classical CCG implementations on synthetic instances, and illustrate the benefits of our approach on a popular case study from the literature of hurricane threats on the Gulf of Mexico in the United States.

Modeling a robust 2-service community healthcare network design problem with central hospital congestion

We address a more realistic variant of the community healthcare network design problem, considering a 2-service network framework and hospital congestion structures. Our objective is to minimise the total expected cost through strategic location of central hospitals and allocation of community health stations. Notably, each community health station is allocated to exactly two central hospitals. We also account for uncertainties in resident medical demands, which are challenging to predict accurately due to their volatility. To tackle this, we propose an adaptive distributionally robust optimisation approach that confines uncertain variables to an ambiguity set reflecting demand distribution characteristics. To enhance tractability, we reformulate the distributionally robust model as a mixed-integer linear programme within the specified ambiguity set. Numerical experiments on a real case demonstrate the validity and superiority of the proposed model. Finally, the impact of uncertainty is discussed in depth and some managerial insights for healthcare managers are summarised.

A risk-averse distributionally robust optimisation approach for drone-supported relief facility location problem

Drone-supported last-mile humanitarian logistics applications play a crucial role in the rapid and efficient delivery of essential relief items, such as medicine, blood, and vaccines, during disaster and emergency situations. This paper explores a novel drone-supported relief facility location problem (DSRFLP) aimed at establishing an effective drone-supported last-mile humanitarian logistics system. The problem involves making joint decisions for both pre-disaster and post-disaster phases while considering the characteristics of drone-based delivery operations and uncertain demands. In the pre-disaster phase, we make the decisions regarding the locations of drone-supported relief facilities, inventory prepositioning of relief items, assignment of drones to the opened facilities, and allocation of drones to disaster demand sites. In the post-disaster phase, we make decisions related to delivery quantities. To tackle the challenge of incomplete demand distribution information in chaotic disaster environments, we establish a distributionally robust optimisation (DRO) model to handle the uncertainty of demands. This model adopts worst-case mean Conditional Value-at-Risk as the risk measurement, reflecting the risk-averse attitude of humanitarian organisers. In this paper, three ambiguity sets (box, ellipsoidal, and polyhedral) are considered to describe the ambiguity distributions of demands. To overcome the computational challenge, we reformulate the DRO model under three ambiguity sets into two mixed-integer linear programming models and one second-order cone programming model, which can be efficiently solved by off-the-shelf solvers. Furthermore, we validate our proposed DRO model through a small-scale example and a large-scale case study based on the Lushan earthquake in China. The computational outcomes underscore the superior performance of the proposed DRO model to mitigate the impact arising from incomplete probability distributions. We propose managerial implications and insights to support the decision-making of humanitarian organisations based on the experimental results. Finally, we propose two extended models to incorporate multiple relief items and equity constraints in priority settings and conduct numerical experiments to adapt to various real-world disaster scenarios.

Distributionally Robust Optimization for integrated energy system accounting for refinement utilization of hydrogen and ladder-type carbon trading mechanism

This study presents a hydrogen-IES with hydrogen refinement utilization under the framework of ladder-type carbon trading mechanism (LCTM). To navigate the uncertainties associated with RE in the proposed hydrogen-IES, we introduce a data-driven Distributionally Robust Optimization (DRO) method based on an innovative ambiguity set construction approach. This method leverages nonparametric Kernel Density Estimation (KDE) to fit the error probability distribution function and determine the predicted interval for wind turbine (WT) and photovoltaic (PV) output under a given confidence level. Subsequently, Latin hypercube sampling (LHS) and a reduction method based on probability distance are employed to generate original and typical scenarios, respectively. Ultimately, the construction of ambiguity sets imposes constraints on typical scenarios and probabilities using 1-norm and ∞-norm. Case study is conducted to validate the effectiveness of the proposed innovations. The results underscore the significant advantages of hydrogen refinement utilization within the proposed framework. Specifically, it demonstrates notable economic benefits, with costs lower by 11.98%, 2.46%, and 1.53% compared to scenarios omitting gas synthesis, hydrogen blending combustion in CHP and GB, and hydrogen storage, respectively. Moreover, hydrogen refinement exhibits superior performance in accommodating WT and PV, with ratios 25.29%, 0.43%, and 1.91% higher than alternative methods, with lower carbon emissions of 42.23%, 11.91%, and 6.13%. The LCTM can reduce more carbon emissions by 7.60% and 0.145% compared to scenarios without carbon trading mechanisms and with fixed-price trading mechanisms. The probability distribution derived from nonparametric KDE aligns closely with the true distribution, fostering a more objective analysis and mitigating the conservatism of the ambiguity set and optimization scheduling. Finally, sensitivity analyses are conducted in detail, including hydrogen blending ratio, basic price and growth ratio of LCTM.

Robust Drone Delivery with Weather Information

Problem definition: Drone delivery has recently garnered significant attention due to its potential for faster delivery at a lower cost than other delivery options. When scheduling drones from a depot for delivery to various destinations, the dispatcher must take into account the uncertain wind conditions, which affect the delivery times of drones to their destinations, leading to late deliveries. Methodology/results: To mitigate the risk of delivery delays caused by wind uncertainty, we propose a two-period drone scheduling model to robustly optimize the delivery schedule. In this framework, the scheduling decisions are made in the morning, with the provision for different delivery schedules in the afternoon that adapt to updated weather information available by midday. Our approach minimizes the essential riskiness index, which can simultaneously account for the probability of tardy delivery and the magnitude of lateness. Using wind observation data, we characterize the uncertain flight times via a cluster-wise ambiguity set, which has the benefit of tractability while avoiding overfitting the empirical distribution. A branch-and-cut (B&C) algorithm is developed for this adaptive distributionally framework to improve its scalability. Our adaptive distributionally robust model can effectively reduce lateness in out-of-sample tests compared with other classical models. The proposed B&C algorithm can solve instances to optimality within a shorter time frame than a general modeling toolbox. Managerial implications: Decision makers can use the adaptive robust model together with the cluster-wise ambiguity set to effectively reduce service lateness at customers for drone delivery systems. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72101049 and 72232001], the Natural Science Foundation of Liaoning Province [Grant 2023-BS-091], the Fundamental Research Funds for the Central Universities [Grant DUT23RC(3)045], and the Major Project of the National Social Science Foundation [Grant 22&ZD151]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2022.0339 .

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.