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By clustering multiple microgrids (MGs), a multi-microgrid (MMG) system plays a significant role in integrating a large amount of renewable generation. However, the large-scale utilization of renewable energy also brings uncertainties to MMG energy management. In this work, a novel comprehensive bi-level MMG energy management model considering uncertainties of renewable energy is first developed which includes unit commitment (UC) problem and other models in the lower MG level. Then considering the non-convexity of the bi-level problem, a decomposition method called analytical target cascading (ATC) is employed to deal with it by decentralization. With regard to the energy management of each individual MG, a two-stage distributionally robust model is developed which describes the uncertainties from probability distribution of renewable generation with a metric-based ambiguity set. Moreover, an efficient solution scheme based on column and constraint generation (CCG) algorithm and a decomposition method without duality is designed to handle the MG energy management problem. Finally, we implement extensive experiments to corroborate the effectiveness of the proposed approach. Particularly, the optimal solution from the proposed method can attain 2.98% less cost compared with that from robust optimization (RO) method and achieve 86.17% less energy trading amount than that in independent mode. [ABSTRACT FROM AUTHOR]

Integrative analysis of data from multiple sources is critical to making generalizable discoveries. Associations consistently observed across multiple source populations are more likely to be generalized to target populations with possible distributional shifts. In this article, we model the heterogeneous multi-source data with multiple high-dimensional regressions and make inferences for the maximin effect (Meinshausen and Bühlmann, AoS, 43(4), 1801–1830). The maximin effect provides a measure of stable associations across multi-source data. A significant maximin effect indicates that a variable has commonly shared effects across multiple source populations, and these shared effects may be generalized to a broader set of target populations. There are challenges associated with inferring maximin effects because its point estimator can have a nonstandard limiting distribution. We devise a novel sampling method to construct valid confidence intervals for maximin effects. The proposed confidence interval attains a parametric length. This sampling procedure and the related theoretical analysis are of independent interest for solving other nonstandard inference problems. Using genetic data on yeast growth in multiple environments, we demonstrate that the genetic variants with significant maximin effects have generalizable effects under new environments. The proposed method is implemented in the R package MaximinInfer available from CRAN. for this article are available online. [ABSTRACT FROM AUTHOR]

This article introduces a Wasserstein distance‐based distributionally robust optimization model to address the transmission expansion planning considering renewable‐based microgrids (MGs) under the impact of uncertainties. The primary objective of the presented methodology is to devise a robust expansion strategy that accounts for both long‐term uncertainty and short‐term variability over the planning horizon from the perspective of a central planner. In this framework, the central planner fosters the construction of appropriate transmission lines and the deployment of optimal MG‐based generating units among profit‐driven private investors. The Wasserstein distance uncertainty set is used to characterize the long‐term uncertainty associated with future load demand. Short‐term uncertainties, stemming from variations in load demands and production levels of stochastic units, are modeled through operating conditions. To ensure the tractability of the proposed planning model, the authors introduce a decomposition framework embedded with a modified application of Bender's method. To validate the efficiency and highlight the potential benefits of the proposed expansion planning methodology, two case studies based on simplified IEEE 6‐bus and IEEE 118‐bus systems are included. These case studies assess the effectiveness of the presented approach, its ability to navigate uncertainties, and its capacity to effectively optimize expansion decisions. [ABSTRACT FROM AUTHOR]

This paper is devoted to reduce the conservativeness of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservativeness of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent systems of inequalities for GDRCs under second order moment information with a separable convex distance function and a special jointly convex function, respectively. We also obtain the deterministic equivalent inequality for GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. Numerical tests of a portfolio optimization problem are given to show the effectiveness of our method and the results demonstrate that the globalized distributionally robust solution is non-conservative and flexible compared to the distributionally robust solution. [ABSTRACT FROM AUTHOR]

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified. [ABSTRACT FROM AUTHOR]

Problem definition: In this paper, we study the newsvendor problem under some distributional ambiguity sets and explore their relations. Additionally, we explore the benefits of implementing this robust solution in the feature-based newsvendor problem. Methodology and results: We propose a new type of discrepancy-based ambiguity set, the JW ambiguity set, and analyze it within the framework of first-order stochastic dominance. We show that the distributionally robust optimization (DRO) problem with this ambiguity set admits a closed-form solution for the newsvendor loss. This result also implies that the newsvendor problem under the well-known infinity-Wasserstein ambiguity set and Lévy ball ambiguity set admit closed-form inventory levels as a by-product. In the application of feature-based newsvendor, we adopt general kernel methods to estimate the conditional demand distribution and apply our proposed DRO solutions to account for the estimation error. Managerial implications: The closed-form solutions enable an efficient computation of optimal inventory levels. In addition, we explore the property of optimal robust inventory levels with respect to the nonrobust version via concepts of perceived critical ratio and mean repulsion. The results of numerical experiments and the case study indicate that the proposed model outperforms other state-of-the-art approaches, particularly in environments where demand is influenced by covariates and difficult to estimate. Funding: X. Li is supported by the Singapore Ministry of Education [Tier 1 Grant 23-0619-P0001, 24-0500-A0001] and National Natural Science Foundation of China [Grant 72331004]. L. Zhang is partially supported by the National Natural Science Foundation of China [Grants 72171156 and 72231002] and the Hong Kong Research Grants Council [Grant 16212419]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0159. [ABSTRACT FROM AUTHOR]

This paper proposes a distributionally robust joint chance constrained (DRJCC) programming approach to optimize the service network design (SND) problem under demand uncertainty. The distributionally robust method does not need complete distribution information and utilizes restricted historical data knowledge, which is significant in scarce data situations. The joint consideration of chance constraints enables more effective control of event probability, by which network managers can realize the purpose of controlling the overall service level of multi-commodities in a service network. DRJCC optimization can also help decision-makers adjust the network's conservativeness, robustness, and service rates by setting the probability parameters of the chance constraints. We reformulate the DRJCC model by addressing the corresponding distributionally robust joint chance constraints with the worst-case Conditional Value-at-Risk method and Lagrange duality theory. The model is approximately reformulated as a mixed-integer linear program, which is easier to solve than the mixed-integer semi-definite programming model in existing literature. We also develop two benchmark approaches for comparison: Bonferroni inequality approximation and scenario-based stochastic program. Comparative numerical studies demonstrate the robustness and the validation of the proposed formulations. A case study is conducted to demonstrate the industrial performance of the uncertain SND under the DRJCC formulation. We explore the impact of the confidence level parameter on operational cost and real service level, reveal the general correlation between them. We also extract several risk-averse managerial insights for logistics fleet managers. [ABSTRACT FROM AUTHOR]

Distributionally robust optimization (DRO) is an increasingly popular approach for optimization under uncertainty when the probability distribution of the uncertain parameter is unknown. Well-explored DRO approaches in literature, such as Wasserstein DRO, do not make any specific assumptions on the nature of the candidate distributions considered in the ambiguity set. However, in many practical applications, the uncertain parameter may be sourced from a distribution that can be well modeled as a Gaussian Mixture Model (GMM) whose components represent the different subpopulations the uncertain parameter may belong to. In this work, we propose a new DRO method based on an ambiguity set constructed around a GMM. The proposed DRO approach is illustrated on a numerical example as well as a portfolio optimization case study for uncertainty sourced from various distributions. The results obtained from the proposed DRO approach are compared with those from Wasserstein DRO, and are shown to be superior in quality with respect to out-of-sample performance. [ABSTRACT FROM AUTHOR]

Pharmaceutical safety has received increasing attention from governments and corporations, and building a safe and effective pharmaceutical cold supply chain network has become an important issue. This paper proposes a two‐stage pharmaceutical cold supply chain network design problem considering drug safety, in which the drug demand, transportation costs, and drug safety risk costs are assumed to be random variables. Due to the presence of uncertainty and the fact that information about the distribution of uncertain parameters is often only partially known, a distributionally robust optimization method is used to handle the uncertainty. A two‐stage distributionally robust optimization model is constructed, in which the reliability of the estimation of the demand necessary to satisfy the entire cold chain network is ensured to be greater than a certain predetermined level by introducing an ambiguous joint chance constraint. The decision process for the problem can be divided into strategic and operational decisions with the goal of minimizing the total costs related to facility construction, the purchase of raw materials, drug production, transportation, and safety risks. By introducing an ambiguity set with mean and covariance information to describe the uncertain parameters, the two‐stage model is eventually reformulated as a standard second‐order cone program, thus making it computationally tractable. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed models and optimization methods. [ABSTRACT FROM AUTHOR]

The failures of high-consequence systems can cause serious harm to humans, including loss of human health, life security, finance, and even social chaos. To protect high-consequence systems, both optimal system design and maintenance activities contribute to improving system reliability and social safety. The existing works generally optimize these two problems sequentially and assume that the degradation process of components is precisely known. However, sequential optimization often results in significant losses due to redundancies, and such a presumption usually cannot be guaranteed in practice, due to limited historical data or a lack of expert knowledge, referred to as epistemic uncertainty. To fill this gap, in this article, we consider an integrated optimization of system design and maintenance scheduling for multi-state high-consequence systems in which the component's degradation is known with limited distributional information. To address this issue, we utilize the framework of distributionally robust optimization to provide a risk-averse decision to decision-makers even under the worst realizations of random parameters, and develop a two-stage integer distributionally robust model with moment-based ambiguity set to determine the system design and maintenance scheduling simultaneously. The proposed model can be converted to a tractable approximation as an integer linear stochastic programming problem. In order to solve large-scale problems, we develop a sample-based adaptive large neighborhood search algorithm to find the optimal system designs. In the numerical experiments, we present a case study on feedwater heating systems in nuclear power plants and demonstrate that an integrated optimization consideration creates significant benefits in profitability. We also present the out-of-sample performance of the distributionally robust design to avoid extreme risk. [ABSTRACT FROM AUTHOR]

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As a deep connection between agriculture and energy, the rural integrated energy system (RIES) is a micro-scale supply-distribution- storage-demand network, which provides an important means to realize the utilization of rural clean energy. This paper proposes a day-ahead scheduling model of the RIES to improve its economical effectiveness, where three energy carriers, namely, biogas, electric power, and heat, are integrated. To address the source and load uncertainties composed of photovoltaic power, power load, and heat load, this paper develops a constrained distributionally robust optimization (CDRO), which optimizes the cost expectation related to the extreme distribution to enhance the robustness, while limiting the loss of cost expectation in the historical distribution to ensure economical effectiveness. In addition, an ambiguous set of the source and load uncertainties incorporating 1-norm and infinity-norm constraints is established, which realizes a flexible adjustment for the conservativeness of CDRO. The distributionally robust dispatch is formulated as a deterministic programming in a two-stage solving framework, where the subproblem uploads its extreme probability distribution to the master problem, and these two problems are iteratively optimized until the convergence. Finally, the numerical simulations in a modern farm park prove the performance of the constructed dispatch model and the flexibility of CDRO in balancing the economical effectiveness and robustness of the dispatch. [ABSTRACT FROM AUTHOR]

We consider the two-machine permutation flow shop scheduling problem with uncertain job processing time, which is sampled from no specific distribution type. For the ease of discussion, an ambiguity set with a priori mean and support set information is constructed. We then introduce a distributionally robust optimization (DRO) perspective to handle the uncertainty. To the best of our knowledge, this is the first time that a DRO method is applied to this problem setting. Given that the original DRO model is nonlinear and intractable in nature, we first reformulate the inner maximization problem into a linear programming model with a fixed sequence, based on the duality theory and optimality conditions. By including the sequence decision, we further transform it into an equivalent mixed-integer linear programming (MILP) problem via incorporating the valid lower and upper bounds and McCormick inequalities. The obtained MILP could be solved to optimality with the off-the-shelf commercial solvers. In the numerical study, it is demonstrated that the DRO-based model could effectively solve the large scale instances with up to 100 jobs optimally within 30 s. Compared with the SLP, DRO model always triumphs on the worst-case indicator. And as the problem scale increases, the DRO model gradually outperforms the SLP in terms of the Up-90% and Up-75% indicators. Furthermore, the optimal sequence obtained by the deterministic model is less stable than the DRO model, which can enhance the robustness of the manufacturing system against processing uncertainty. [ABSTRACT FROM AUTHOR]

Copyright of Electric Power Engineering Technology is the property of Editorial Department of Electric Power Engineering Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Electric Power Automation Equipment / Dianli Zidonghua Shebei is the property of Electric Power Automation Equipment Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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. [ABSTRACT FROM AUTHOR]

Problem definition: We study supply chain contract design under uncertainty. In this problem, the retailer has full information about the demand distribution, whereas the supplier only has partial information drawn from historical demand realizations and contract terms. The supplier wants to optimize the contract terms, but she only has limited data on the true demand distribution. Methodology/results: We show that the classical approach for contract design is fragile in the small data regime by identifying cases where it incurs a large loss. We then show how to combine the historical demand and retailer data to improve the supplier's contract design. On top of this, we propose a robust contract design model where the uncertainty set requires little prior knowledge from the supplier. We show how to optimize the supplier's worst-case profit based on this uncertainty set. In the single-product case, the worst-case profit can be found with bisection search. In the multiproduct case, the worst-case profit can be found with a cutting plane algorithm. Managerial implications: Our framework demonstrates the importance of combining the demand and retailer information into the supplier's contract design problem. We also demonstrate the advantage of our robust model by comparing it against classical data-driven approaches. This comparison sheds light on the value of information from interactions between agents in a game-theoretic setting and suggests that such information should be utilized in data-driven decision making. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0325. [ABSTRACT FROM AUTHOR]

Wind power is a major source of green energy production. However, the energy generation of wind power is highly affected by uncertainty. Here, we consider the problem of designing the cable network that interconnects the turbines to the substation in wind farms, aiming to minimize both the infrastructure cost and the cost of the energy losses during the wind farm's lifetime. Nonetheless, the energy losses depend on wind direction and speed, which are rarely known with certainty in real situations. Hence, the design of the network should consider these losses as uncertain parameters. We assume that the exact probability distribution of these parameters is unknown but belongs to an ambiguity set and propose a distributionally robust two-stage mixed integer model. The model is solved using a decomposition algorithm. Three enhancements are proposed given the computational difficulty in solving real problem instances. Computational results are reported based on real data. [ABSTRACT FROM AUTHOR]

Abstract This article introduces a Wasserstein distance‐based distributionally robust optimization model to address the transmission expansion planning considering renewable‐based microgrids (MGs) under the impact of uncertainties. The primary objective of the presented methodology is to devise a robust expansion strategy that accounts for both long‐term uncertainty and short‐term variability over the planning horizon from the perspective of a central planner. In this framework, the central planner fosters the construction of appropriate transmission lines and the deployment of optimal MG‐based generating units among profit‐driven private investors. The Wasserstein distance uncertainty set is used to characterize the long‐term uncertainty associated with future load demand. Short‐term uncertainties, stemming from variations in load demands and production levels of stochastic units, are modeled through operating conditions. To ensure the tractability of the proposed planning model, the authors introduce a decomposition framework embedded with a modified application of Bender's method. To validate the efficiency and highlight the potential benefits of the proposed expansion planning methodology, two case studies based on simplified IEEE 6‐bus and IEEE 118‐bus systems are included. These case studies assess the effectiveness of the presented approach, its ability to navigate uncertainties, and its capacity to effectively optimize expansion decisions.

The multi-uncertainty of source and load poses significant challenges to the optimal scheduling of 'source-load-storage' integrated microgrid. However, a limitation of the traditional optimization model is its one-sidedness and use of a single time scale, which can result in suboptimal scheduling outcomes. Striking a balance between reliability and economy presents a considerable obstacle, as does coordinating the relationship between uncertainty analysis methods and varying time scales. Based on the data-driven multi-discrete scene distribution robust method, a two-stage distributed robust day-ahead optimal scheduling model of microgrid is proposed, which is solved by column and constraint generation algorithm. By combining the improved distributed robust optimization uncertainty method with a multi-time scale scheduling strategy and model predictive control theory, the accuracy of the scheduling can be enhanced. This is achieved through the gradual refinement of the scheduling time scale and reduction of the prediction period length. The day-ahead-day multi-time scale rolling optimization scheduling model is established to minimize the generation cost and adjustment cost, while also exhibiting a high degree of resilience to system uncertainties. Combined with the simulation analysis of the example, the proposed model has demonstrated advantages in incorporating new energy sources, reducing operating costs, and balancing considerations of safety and economy.

In this paper, we propose a general distributionally robust regression model based on distributionally robust optimization theory. The proposed model has a piecewise linear loss function and elastic net penalty term, and it generalizes many other regression models. We prove the piecewise linear property of the optimal solutions to this model, which enables us to develop a solution path algorithm for the hyperparameter tuning. A Doubly regularized Least Absolute Deviations (DrLAD) regression model is proposed based on this framework, and a solution path algorithm is developed to speed up the tuning of two hyperparameters in this model. Numerical experiments are implemented to validate the performance of this model and the computational efficiency of the solution path algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Carbon emission schemes are widely adopted and implemented in practice to control greenhouse gas emissions. However, a firm restricted by carbon emission schemes with only partial demand information may face greater challenges. In this paper, we investigate an ordering and pricing newsvendor problem with partial demand information, which considers carbon emissions generated from the processes of ordering and disposing of the excess inventory. We propose three distributionally robust models to maximize the worst-case profit under different carbon emission regulations: carbon tax, cap-and-trade and carbon cap, respectively. We derive the implicit solution under the carbon tax scheme and design heuristic algorithms based on Lagrangian relaxation under the cap-and-trade and carbon cap regulations. Numerical analysis indicates: (i) all three regulations are effective in reducing carbon emissions; (ii) the implementation of carbon schemes leads to higher optimal pricing and lower optimal ordering quantity, both carbon tax and carbon cap regulations will always lead to lower profit, while the retailer may benefit from the cap-and-trade regulation by selling out the carbon credit; (iii) the proposed approach performs well under different distribution specific scenarios. [ABSTRACT FROM AUTHOR]

In recent years, the penetration of solar and wind power has rapidly increased to construct renewable energy-dominated power systems (RPSs). On this basis, the forecasting errors of renewable generation power have negative effects on the operation of the power system. However, traditional scheduling methods are overly dependent on the generation-side dispatchable resources and lack uncertainty modeling strategies, so they are inadequate to tackle this problem. In this case, it is necessary to enhance the flexibility of the RPS by both mining the load-side dispatchable resources and improving the decision-making model under uncertainty during the energy and reserve co-dispatch. In this paper, due to the great potential in facilitating the RPS regulation, the demand response (DR) model of fused magnesium load (FML) is first established to enable the deeper interaction between the load side and the whole RPS. Then, based on the principal component analysis and clustering algorithm, an improved typical scenario set generation method is proposed to obtain a much less conservative model of the spatiotemporally correlated uncertainty. On this basis, a two-stage distributionally robust optimization model of the energy and reserve co-dispatch is developed for the RPS considering the DR of FML. Finally, the proposed method is validated by numerical tests. The results show that the costs of day-ahead dispatch and re-dispatch are significantly decreased by using the improved typical scenario set and considering the DR of FML in regulation, which enhances the operation economy while maintaining the high reliability and safety of the RPS. [ABSTRACT FROM AUTHOR]

In this paper, we study the distributionally robust joint chance-constrained Markov decision process. Utilizing the logarithmic transformation technique, we derive its deterministic reformulation with bi-convex terms under the moment-based uncertainty set. To cope with the non-convexity and improve the robustness of the solution, we propose a dynamical neural network approach to solve the reformulated optimization problem. Numerical results on a machine replacement problem demonstrate the efficiency of the proposed dynamical neural network approach when compared with the sequential convex approximation approach. [ABSTRACT FROM AUTHOR]

A considerable penetration of rooftop PV generation and increasing demand for heating loads will enlarge the peak-to-valley difference, imposing a great challenge to the reliable operation of distribution systems under cold climates. The objective of this paper is to establish a distributionally robust demand response (DR) model for building energy systems for suppressing peak-to-valley load ratios by exploiting cooperative complementarity and flexible transformation characteris-tics of various household appliances. The thermodynamic effect of buildings is modeled for harvesting intermittent renewable energy sources (RESs) on the building roof in the form of thermal energy storages to reduce RES curtailments and eliminate thermal comfort violations in cold weather. Furthermore, the Wasserstein metric is adopted to develop the ambiguity set of the uncertainty probability distributions (PDs) of RESs, and thus, only historical data of RES output is needed rather than prior knowledge about the actual PDs. Finally, a computationally tractable mixed-integer linear programming reformulation is derived for the original distributionally robust optimization (DRO) model. The proposed DRO-based DR strategy was performed on multiple buildings over a 24 h scheduling horizon, and comparative studies have validated the effectiveness of the proposed strategy for building energy systems in reducing the peak/valley ratio and decreasing operation costs. [ABSTRACT FROM AUTHOR]

Under the dual pressure of the fossil energy crisis and the global low carbon emission target, the utilization of various types of renewable energy has received widespread attention worldwide. In view of this, this study develops a geothermal-hydrogen integrated energy system (integrating geothermal power, power to hydrogen, and energy storage), which can generate "near-zero" CO2 emissions and obtain additional benefits by selling hydrogen. However, random fluctuations in market electricity prices and heat demand severely interfere with the economic, environmental, and reliable operation of the system, hindering its sustainability operation and development. Therefore, this paper proposes a two-stage distributionally robust optimization model considering multiple uncertainties. Specifically, a moment-based ambiguity set is constructed to portray the uncertain distribution of forecasting errors in heat demand and market electricity prices. In the first stage, the daily operating profit of the system is maximized through electricity, hydrogen, and heat scheduling decisions according to the forecast information. In the second stage, the operations of flexibility resources are adjusted from their schedules in the first stage to resist the interference of the "worst-case" distribution in the ambiguity set. Finally, the model is transformed into a mixed-integer linear programming for solution feasibility. Simulations verify that: (1) integrating geothermal and hydrogen energy improves the economic and environmental benefits of system operation; and (2) the model keeps low computational complexity and decision conservativeness, which can realize the benefits of cost-saving, profit improvement, emission reduction, and interference resistance. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies a distributionally robust multi-item newsvendor problem, where the demand distribution is unknown but specified with a general event-wise ambiguity set. Using the event-wise affine decision rules, we can obtain a conservative approximation formulation of the problem, which can typically be further reformulated as a linear program. In order to efficiently solve the resulting large-scale linear program, we develop a column generation-based decomposition scheme and speed up the computational efficiency by exploiting a special column selection strategy and stopping early based on a Karush-Kuhn-Tucker condition test. Focusing on the Wasserstein ambiguity set and the event-wise mean absolute deviation set, a computational study demonstrates both the computational efficiency of the proposed algorithm, which significantly outperforms a commercial solver and a Benders decomposition method, and the out-of-sample superiority of distributionally robust solutions relative to their sample average approximation counterparts. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [492997-2016, RGPIN-2016-05208], the National Natural Science Foundation of China [71972012], Alliance de recherche numérique du Canada, and Canada Research Chairs [CRC-2018-00105]. It was also supported by Groupe d'études et de recherche en analyse des décisions (GERAD). Finally, this research was enabled in part by support provided by Digital Research Alliance of Canada (https://alliancecan.ca/en). Supplemental Material: The software that supports the findings of this study is available within the paper and its supplemental information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0010) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0010). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

This paper presents the first comprehensive study of a data-driven formulation of the distributionally robust second order stochastic dominance constrained problem (DRSSDCP) that hinges on using a type-1 Wasserstein ambiguity set. It is, furthermore, for the first time shown to be axiomatically motivated in an environment with distribution ambiguity. We formulate the DRSSDCP as a multistage robust optimization problem and further propose a tractable conservative approximation that exploits finite adaptability and a scenario-based lower bounding problem. We then propose the first exact optimization algorithm for this DRSSDCP. We illustrate how the data-driven DRSSDCP can be applied in practice on resource-allocation problems with both synthetic and real data. Our empirical results show that, with a proper adjustment of the size of the Wasserstein ball, DRSSDCP can reach acceptable out-of-sample feasibility yet still generating strictly better performance than what is achieved by the reference strategy. Optimization with stochastic dominance constraints has recently received an increasing amount of attention in the quantitative risk management literature. Instead of requiring that the probabilistic description of the uncertain parameters be exactly known, this paper presents a comprehensive study of a data-driven formulation of the distributionally robust second order stochastic dominance constrained problem (DRSSDCP) that hinges on using a type-1 Wasserstein ambiguity set. This formulation allows us to identify solutions with finite sample guarantees and solutions that are asymptotically consistent when observations are independent and identically distributed. It is, furthermore, shown to be axiomatically motivated in an environment with distribution ambiguity. Leveraging recent results in the field of robust optimization, we further formulate the DRSSDCP as a multistage robust optimization problem and further propose a tractable conservative approximation that exploits finite adaptability and a scenario-based lower bounding problem, both of which can reduce to linear programs under mild conditions. We then propose, to the best of our knowledge, the first exact optimization algorithm for this DRSSDCP, the efficiency of which is confirmed by our numerical results. Finally, we illustrate how the data-driven DRSSDCP can be applied in practice on resource-allocation problems with both synthetic and real data. Our empirical results show that, with a proper adjustment of the size of the Wasserstein ball, DRSSDCP can reach "acceptable" out-of-sample feasibility yet still generating strictly better performance than what is achieved by the reference strategy. Funding: This research was partially supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2016-05208], the Canada Research Chair program [Grant 950-230057], and Groupe d'études et de recherche en analyse des décisions, and it was enabled in part by support provided by Calcul Quebec (https://www.calculquebec.ca/en/) and the Digital Research Alliance of Canada (https://www.alliancecan.ca/). Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2387. [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]

This paper considers facility location, inventory pre‐positioning and vehicle routing as strategic and operational decisions corresponding to preparedness and response phases in disaster relief logistics planning. For balancing surpluses and shortages, an effective lateral transshipment strategy is proposed to evenly distribute the relief resources between warehouses after the disaster occurs. To handle ambiguity in the probability distribution of demand, we develop a risk‐averse two‐stage distributionally robust optimization (DRO) model for the disaster relief logistics planning problem, which specifies the worst‐case mean‐conditional value‐at‐risk (CVaR) as a risk measure. For computationally tractability, we transform the robust counterpart into its equivalent linear mixed‐integer programming model under the discrepancy‐based ambiguity set centered at the nominal (empirical) distributions on the observed demand from the historical data. We verify the effectiveness of the proposed DRO model and the value of lateral transshipment strategy by an illustrative small‐scale example. The numerical results show that the proposed DRO model has advantage on avoiding over‐conservative solutions compared to the classic robust optimization model. We also illustrate the applicability of the proposed DRO model by a real‐world case study of hurricanes in the southeastern United States. The computational results demonstrate that the proposed DRO model has superior out‐of‐sample performance and can mitigate the adverse effects of Optimizers' Curse compared with the traditional stochastic programming model. [ABSTRACT FROM AUTHOR]

Inspired by its success for their continuous counterparts, the standard approach to deal with mixed-integer recourse (MIR) models under distributional uncertainty is to use distribu- tionally robust optimization (DRO). We argue, however, that this modeling choice is not always justified since DRO techniques are generally computationally challenging when integer decision variables are involved. That is why we propose an alternative approach for dealing with distributional uncertainty for the special case of simple integer recourse (SIR) models, which is aimed at obtaining models with improved computational tractability. We show that such models can be obtained by pragmatically selecting the uncertainty set. Here, we consider uncertainty sets based on the Wasserstein distance and also on generalized moment conditions. We compare our approach with standard DRO both numerically and theoretically. An important side result of our analysis is the derivation of performance guarantees for convex approximations of SIR models. In contrast to the literature, these error bounds are not only valid for a continuous distribution but hold for any distribution. [ABSTRACT FROM AUTHOR]

We introduce a family of s-rectangular robust Markov decision processes (s-RMDPs) indexed with r in [1, infty ]. In each state, the ambiguity set of transition probability mass functions (pmfs) across actions equals a sublevel set of the ell rho -norm of a vector of distances from reference pmfs. Setting rho = infty recovers (s, a)-RMDPs. For any s-RMDP from this family, there is an (s, a)- RMDP whose robust optimal value is at least as good, and vice versa. This occurs because sand (s, a)-RMDPs can employ different ambiguity set radii, casting doubt on the belief that (s, a)- RMDPs are more conservative than s-RMDPs. If the distance is lower semicontinuous and convex, then, for any s-RMDP, there exists an (s, a)-RMDP with an identical robust optimal value. We also study data-driven s-RMDPs, where the reference pmf is constructed from state transition samples. If the distance satisfies a Pinsker-type inequality, the robust optimal and out-of-sample values both converge with sample-size to the true optimal. We derive rates of convergence and sample complexity when the distance satisfies a concentration inequality. Under this concentration inequality, the robust optimal value provides a probabilistic lower bound on the out-of-sample value. An artifact of the analyses behind these guarantees is the surprising conclusion that (s, a)-RMDPs might be the least conservative among all s-RMDPs within our family. The asymptotic and finite sample properties also extend for a class of nonrectangular RMDPs. [ABSTRACT FROM AUTHOR]

In this article, we present a Distributionally Robust Markov Decision Process (DRMDP) approach for addressing the dynamic epidemic control problem. The Susceptible-Exposed-Infectious-Recovered (SEIR) model is widely used to represent the stochastic spread of infectious diseases, such as COVID-19. Although the Markov Decision Process (MDP) offers a mathematical framework for identifying optimal actions, such as vaccination and transmission-reducing intervention, to combat disease spread calculated using the SEIR model. However, uncertainties in these scenarios demand a more robust approach that is less reliant on error-prone assumptions. The primary objective of our study is to introduce a new DRMDP framework that allows for an ambiguous distribution of transition dynamics. Specifically, we consider the worst-case distribution of these transition probabilities within a decision-dependent ambiguity set. To overcome the computational complexities associated with policy determination, we propose an efficient Real-Time Dynamic Programming (RTDP) algorithm that is capable of computing optimal policies based on the reformulated DRMDP model in an accurate, timely, and scalable manner. Comparative analysis against the classic MDP model demonstrates that the DRMDP achieves a lower proportion of infections and susceptibilities at a reduced cost. [ABSTRACT FROM AUTHOR]

The marine climate conditions are intricate and variable. In scenarios characterized by high proportions of wind and solar energy access, the uncertainty regarding the energy sources for island microgrid is significantly exacerbated, presenting challenges to both the economic viability and reliability of the capacity configuration for island microgrids. To address this issue, this paper proposes a distributionally robust optimization (DRO) method for island microgrids, considering extreme scenarios of wind and solar conditions. Firstly, to address the challenge of determining the probability distribution functions of wind and solar in complex island climates, a conditional generative adversarial network (CGAN) is employed to generate a scenario set for wind and solar conditions. Then, by combining k-means clustering with an extreme scenario selection method, typical scenarios and extreme scenarios are selected from the generated scenario set, forming the scenario set for the DRO model of island microgrids. On this basis, a DRO model based on multiple discrete scenarios is constructed with the objective of minimizing the sum of investment costs, operation and maintenance costs, fuel purchase costs, penalty costs of wind and solar curtailment, and penalty costs of load loss. The model is subjected to equipment operation and power balance constraints, and solved using the columns and constraints generation (CCG) algorithm. Finally, through typical examples, the effectiveness of this paper’s method in balancing the economic viability and robustness of the configuration scheme for the island microgrid, as well as reducing wind and solar curtailment and load loss, is verified.

The coordinated optimization of maintenance and generation schedules of multiple offshore wind farms (OWFs) bring great benefits in reducing the maintenance and operation cost of OWF cluster. Firstly, a two-stage coordinated risk-averse distributionally robust optimization (CRDRO) model for maintenance and generation schedules of wind turbines (WTs) in OWF cluster is proposed. This model takes into account uncertainties related to the prediction errors in WTs’ maximum available active power outputs, the travel time of maintenance vessels, and the high operational cost risks associated with these uncertainties. A feasible route generating algorithm is designed to obtain all feasible maintenance combinations for WTs and the corresponding routes of maintenance vessels for the CRDRO model. The risk-averse term, which is a weighted sum of expected operation costs and GlueVaR value, is used to measure the tail risk of the worst probability distribution. To solve the two-stage CRDRO model, the risk-averse term is simplified into the form of expected value calculation, which can be further transformed into the tractable form by using the strong duality theorem. Finally, the two-stage CRDRO model is solved by the column-and-constraint generation algorithm, with the nonconvex sub-problem being addressed by the outer approximation algorithm. Case studies on an real-world cluster comprising three OWFs verify the correctness and efficiency of the proposed method, and demonstrate that the CRDRO model enhances the overall economic efficiency and reduces the high operational cost risk under extreme scenarios.

Pre-disaster evacuation network design is critical to improving evacuation efficiency and reducing the loss of life and property caused by disasters. However, evacuation network design is complicated by the uncertainty of pre-disaster evacuation demand and the heterogeneity of evacuees’ route choices when faced with a disaster scenario. Therefore, how to design efficient evacuation networks considering these uncertainties has become an important research topic. In this paper, we study the pre-disaster flood evacuation network design problem, define a bi-criteria generalized cost metric based on Evacuation Travel Time (ETT) and Evacuation Network Congestion Degree (ENCD), and develop a model that considers uncertain demand and evacuation behavioural choices to minimize the generalized cost in the evacuation network. For the uncertain behavioural choice, we propose an Expanded Stochastic User Equilibrium (ESUE) model based on the Random Regret Minimization (RRM) principle and Boundedly Rational User Equilibrium (BRUE) conditions to describe it. For the uncertain demand, we design a Distributionally Robust Optimization (DRO) method based on Mean Absolute Deviation (MAD) to deal with this uncertain model. This paper designs an Improved Method of Successive Averages Combined with Genetic Algorithm (IMSA-GA) to solve the model and uses the Nguyen-Dupuis test network to verify the applicability and feasibility of the model and algorithm. In addition, we also conduct an evacuation network design in the Hawkesbury-Nepean region of Australia. The larger the threshold value based on the regret utility ratio, the smaller the generalized cost value, which indicates the behavioural choices of the evacuees have an impact on the evacuation network design, and the generalized cost decreases by 6.12% when the threshold value is increased from 0.1 to 0.9. In addition, the generalized cost value obtained by IMSA-GA will be 0.0025 lower than Improved Method of Successive Averages Combined with Simulated Annealing (IMSA-SA) and 0.0005 lower than Improved Method of Successive Averages Combined with Taboo Search (IMSA-TS). The theoretical contribution of this paper is to find an approach to the uncertain evacuation network design problem, especially to improve the practicality of the model by fully considering evacuation behaviour. The practical contribution is that our findings can provide insights for government authorities to design efficient and reliable evacuation networks that balance evacuation travel time and evacuation network congestion. In the future, if evacuation behavioural choices can be analysed based on actual surveys, the estimation of behavioural parameters based on survey data will be more instructive for the evacuation process.

As the enthusiasm for electric vehicles passes the range anxiety and other tests, large-scale transportation electrification becomes a prominent topic in research and policy discussions. In consequence, the public attention has shifted upstream and holistically towards the integration of large-scale transportation electrification to power systems. This paper proposes a method to identify bottlenecks in power transmission systems to accommodate large-scale and stochastic electric vehicles charging demands. First, a distributionally robust chance-constrained direct current optimal power flow model is developed to quantify the impacts of stochastic electric vehicles charging demands. Subsequently, an agent-based model with the trip-chain method is applied to obtain the spatiotemporal distributions of electric vehicles charging demands for both light-duty electric vehicles and medium and heavy-duty electric vehicles. The first two moments of those distributions are extracted to build an ambiguity set of electric vehicles charging demands. Finally, a 121-bus synthetic transmission network for New York State power systems is used to validate the future transportation electrification in New York State from 2025 to 2035. Results show that the large-scale transportation electrification in New York State will account for approximately 13.33 % to 16.79 % of the total load demand by 2035. The transmission capacity is the major bottleneck in supporting New York State to achieve its transportation electrification. To resolve the bottlenecks, we explore some possible solutions, such as transmission capacity expansion and distributed energy resources investment. Wind power shows an advantage over solar energy in terms of total operational costs due to better peak alignment between wind power and electric vehicles charging demand.

Information and communication technologies enable the transformation of traditional energy systems into cyber-physical energy systems (CPESs), but such systems have also become popular targets of cyberattacks. Currently, available methods for evaluating the impacts of cyberattacks suffer from limited resilience, efficacy, and practical value. To mitigate their potentially disastrous consequences, this study suggests a two-stage, discrepancy-based optimization approach that considers both preparatory actions and response measures, integrating concepts from social computing. The proposed Kullback-Leibler divergence-based, distributionally robust optimization (KDR) method has a hierarchical, two-stage objective function that incorporates the operating costs of both system infrastructures (e.g., energy resources, reserve capacity) and real-time response measures (e.g., load shedding, demand-side management, electric vehicle charging station management). By incorporating social computing principles, the optimization framework can also capture the social behavior and interactions of energy consumers in response to cyberattacks. The preparatory stage entails day-ahead operational decisions, leveraging insights from social computing to model and predict the behaviors of individuals and communities affected by potential cyberattacks. The mitigation stage generates responses designed to contain the consequences of the attack by directing and optimizing energy use from the demand side, taking into account the social context and preferences of energy consumers, to ensure resilient, economically efficient CPES operations. Our method can determine optimal schemes in both stages, accounting for the social dimensions of the problem. An original disaster mitigation model uses an abstract formulation to develop a risk-neutral model that characterizes cyberattacks through KDR, incorporating social computing techniques to enhance the understanding and response to cyber threats. This approach can mitigate the impacts more effectively than several existing methods, even with limited data availability. To extend this risk-neutral model, we incorporate conditional value at risk as an essential risk measure, capturing the uncertainty and diverse impact scenarios arising from social computing factors. The empirical results affirm that the KDR method, which is enriched with social computing considerations, produces resilient, economically efficient solutions for managing the impacts of cyberattacks on a CPES. By integrating social computing principles into the optimization framework, it becomes possible to better anticipate and address the social and behavioral aspects associated with cyberattacks on CPESs, ultimately improving the overall resilience and effectiveness of the system's response measures.

Abstract Virtual power plant (VPP) can aggregate distributed energy resources (DER) and controllable loads to participate in the bidding of electricity market and carbon trading market. However, the uncertainty of renewable power production brings high transaction risks to VPP. Given this background, this paper proposes a novel Wasserstein‐distance based two‐stage distributionally robust chance constrained (DRCC) bidding model for VPP participating in the electricity‐carbon coupled market. The uncertainties are modelled as an ambiguity set based on Wasserstein distance, in which the two‐sided chance constraints are guaranteed satisfied. In the first stage, VPP's revenue is maximized according to the forecast information. In the second stage, the re‐dispatch measures are determined to hedge against the perturbation of uncertainties under the worst‐case distribution within the ambiguity set. Finally, a reformulation approach based on strong duality theory and conditional value‐at‐risk (CVaR) approximation is proposed to transform the DRCC problem into a tractable mixed‐integer linear programming (MILP) framework. Case studies are carried out on the IEEE 30‐bus system to verify the effectiveness and efficiency of the proposed approach.

This paper presents a fully risk-aware model predictive control (MPC) framework for chance-constrained discrete-time linear control systems with process noise. Conditional value-at-risk (CVaR) as a popular coherent risk measure is incorporated in both the constraints and the cost function of the MPC framework. This allows the system to navigate the entire spectrum of risk assessments, from worst-case to risk-neutral scenarios, ensuring both constraint satisfaction and performance optimization in stochastic environments. The recursive feasibility and risk-aware exponential stability of the resulting risk-aware MPC are demonstrated through rigorous theoretical analysis by considering the disturbance feedback policy parameterization. In the end, two numerical examples are given to elucidate the efficacy of the proposed method.

A swift power system restoration in a post-blackout event is one the most important challenges faced by the transmission system (TS) operators (TSO), which is particularly essential in the presence of wind farms, as their potential can be great in a fast restoration. In this study, we propose a bi-level decentralized approach to examine the influence of natural gas network (NGN) constraints on the bulk power system restoration process, taking into account the concurrent effects of pump storage hydropowers (PSHs) and wind farms. At the upper level of the problem, the transmission system (TS) operator (TSO) submits the amount of natural gas fuel consumed by the gas-fired units (GFUs) to the NGN by observing the restoration, operational, and topological constraints. The objective of the TSO is to maximize load servicing in the power grid restoration process. The equilibrium of the proposed bi-level problem is calculated by the analytical target cascading (ACT) algorithm, preserving the privacy of both electricity and NGNs. In the proposed study, an investigation has been conducted into the impact of the gas storage system (GSS) and linepack technology on enhancing the restoration process. Moreover, a moment-based distributionally robust optimization (DRO) approach has been deployed to model the uncertain behavior of wind farms in the restoration process. The proposed approach comprehensively examines the effects of the decentralized interconnection between electricity and NGNs in the restoration process. This facet holds great significance for the advancement of future sustainable energy systems. The results show that ignoring the NGN model leads to 11.92% higher level of unserviced loads.

The carbon neutrality policy aiming for net zero carbon emissions has led to a significant increase in the use of renewable energy sources (RES) globally. However, due to their uncertain nature, RES can cause imbalances in power demand. Recently, Power-to-X (P2X) station technology has gained attention as a solution to the uncertainties of RES and as a means to enhance the capacity and efficiency of RES operations. P2X stations can be utilized when power demand imbalances occur due to the uncertain output of RES or when the power system cannot accommodate the power supply from RES due to various stability issues. Specifically, when supply disruptions occur in the power system due to RES uncertainties, P2X stations contribute to preventing RES curtailment by supplying power to electric vehicle (EV) fuel sources, producing heat using electric heat pumps (EHP), or producing hydrogen using electrolyzers (ELZ), thus improving the uncertain financial benefits for independent power producers (IPP). This paper proposes a mixed-integer linear programming (MILP) based chance-constrained two-stage stochastic optimization (CCTS) approach to address imbalances in power demand from RES and to enhance the profitability of IPP by finding the optimal planning and operational solutions for P2X stations. The proposed method provides hierarchical level results, demonstrating that economic benefits can increase by up to 60.2% with the application of P2X stations and that curtailed energy from RES can be reduced by up to 76.5%. The proposed methodology is also validated for its superior performance by being compared with both the non-linear stochastic chance constraint method and the stochastic method.