Uncertainty Set Research Articles

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

Data-Driven Day-Ahead Dispatch Method for Grid-Tied Distributed Batteries Considering Conflict Between Service Interests

The rapid advancement of battery technology has drawn attention to the effective dispatch of distributed battery storage systems. Batteries offer significant benefits in flexible energy supply and grid support, but maximising their cost-effectiveness remains a challenge. A key issue is balancing conflicts between intentional network services, such as energy arbitrage to reduce the overall electricity costs, and unintentional services, like fault-induced unintentional islanding. This paper presents a novel dispatch methodology that addresses these conflicts by considering both energy arbitrage and unintentional islanding services. First, demand profiles are clustered to reduce uncertainty, and uncertainty sets for photovoltaic (PV) generation and demand are derived. The dispatch strategy is originally formulated as a robust optimal power flow problem, accounting for both economic benefits and risks from unresponsive islanding requests, alongside energy loss reduction to prevent a battery-induced artificial peak. Last, this paper updates the objective function for adapting possible long-run competition changes. The IEEE 33-bus system is utilised to validate the methodology. Case studies show that, by considering the reserve for possible islanding requests, a battery with limited capacity will start to discharge after a demand drop from the peak, leading to the profit dropping from USD 185/day (without reserving capacity) to USD 21/day. It also finds that low-resolution dynamic pricing would be more appropriate for accommodating battery systems. This finding offers valuable guidance for pricing strategies.

Low-carbon economic optimization strategy for industrial loads in parks considering source-load-price multivariate uncertainty

A low-carbon economic optimization method is proposed for industrial loads in parks considering multivariate uncertainty. Model of dynamic load node carbon intensity is proposed considering the uncertainty of clean energy and dynamic conventional units carbon intensity based on system currents. The robustness is improved based on multi-interval uncertainty set. The gap is filled by hybrid copula model with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) prices in the uncertainty model. The GARCH is used to built the marginal distribution function of prices, and a hybrid copula model is proposed to obtain the joint probability density function based on the weights calculated by Euclidean distance. The LNCI probability distributions of the regulation potential of loads are proposed with the virtual energy storage and the virtual power model to improve the descriptive ability of uncertainty based on the expansion and contraction functions. Finally, the typical scenarios are extracted based on Monte Carlo and Wasserstein measures. The Column sum Constraint Generation algorithm and strong duality theory are used to get the robust-stochastic optimization. The method proposed in the paper has a positive effect on enterprises to rationalize their production schedules, reduce electricity and carbon emissions costs and increase the revenue from participating in grid regulation.

A Tri-Level Transaction Method for Microgrid Clusters Considering Uncertainties and Dynamic Hydrogen Prices

The advancement of hydrogen technology and rising environmental concerns have shifted research toward renewable energy for green hydrogen production. This study introduces a novel tri-level transaction methodology for microgrid clusters, addressing uncertainties and price fluctuations in hydrogen. We establish a comprehensive microgrid topology with distributed power generation and hydrogen production facilities. A polygonal uncertainty set method quantifies wind and solar energy uncertainties, while an enhanced interval optimization technique refines the model. We integrate a sophisticated demand response model for hydrogen loading, capturing users’ behavior in response to price changes, thereby improving renewable energy utilization and supporting economically viable management practices. Additionally, we propose a tri-level game-theoretic framework for analyzing stakeholder interactions in microgrid clusters, incorporating supply–demand dynamics and a master–slave structure for microgrids and users. A distributed algorithm, “KKT & supply-demand ratio”, solves large-scale optimization problems by integrating Karush–Kuhn–Tucker conditions with a heuristic approach. Our simulations validate the methodology, demonstrating that accounting for uncertainties and dynamic hydrogen prices enhances renewable energy use and economic efficiency, optimizing social welfare for operators and economic benefits for microgrids and users.

Two-stage robust multimodal hub network design under budgeted demand uncertainty: A Benders decomposition approach and a case study

The widening use of hub networks in urban agglomeration freight systems requires several actual extensions in conventional hub network design problems. For this purpose, we introduce a two-stage robust multimodal hub network design problem for the urban agglomeration freight system by considering incomplete hub network topology, multiple transportation modes, travel time limit and discuss the uncertainty in the constructed network from the demand point of view. Particularly, we model the demand uncertainty for the considered problem in two different ways. The basic model supposes that interval-budgeted uncertainty set is adopted to characterize uncertain demand, while the expanded model additionally considers possible states of the uncertain demand and weights summation of performances over multiple uncertainty sets, namely state-wise budgeted uncertainty set. By using a min–max criterion, we develop the path-based mixed-integer programming formulations for the proposed problem, which can significantly decrease the number of required integer variables and constraints. To handle large-sized problems, we propose an improved Benders decomposition algorithm, in which the master problem is implemented in a branch-and-bound framework and the subproblem is optimality solved by a customized two-step strategy. In addition to evaluating on the standard CAB, TR and AP datasets, we conduct a real-world case study of the Beijing-Tianjin-Hebei urban agglomeration freight system to explore the effect of incorporating uncertainty and showcase the superior performance of the proposed methods.

Analysis of order-of-addition experiments

The order-of-addition (OofA) experiment involves arranging components in a specific order to optimize a certain objective, which is attracting a great deal of attention in many disciplines, especially in the areas of biochemistry, scheduling, and engineering. Recent studies have highlighted its significance, and notable works have aimed to address NP-hard OofA problems from a statistical perspective. However, solving OofA problems presents challenges due to their complex nature and the presence of uncertainty, such as scheduling problems with uncertain processing times. These uncertainties affect processing times, which are not known with certainty in advance. They introduce heteroscedasticity into OofA experiments, where different orders result in varying dispersions. To address these challenges, a unified framework is proposed to analyze scheduling problems without making specific assumptions about the distribution of these certainties. It encompasses model development and optimization, encapsulating existing homoscedastic studies (where different orders produce the same dispersion value) as a specific instance. For heteroscedastic cases, a dual response optimization within an uncertainty set is proposed, aiming to minimize the dispersion of response while keeping the location of response with a predefined target value. However, solving the proposed non-linear minimax optimization is rather challenging. An equivalent optimization formulation with low computational cost is proposed for solving such a challenging problem. Theoretical supports are established to ensure the tractability of the proposed method. Simulation studies are conducted to demonstrate the effectiveness of the proposed approach. With its solid theoretical support, ease of implementation, and ability to find an optimal order, the proposed approach offers a practical and competitive solution to solving general order-of-addition problems.

Collaborative supply chain network design under demand uncertainty: A robust optimization approach

This paper studies a collaborative robust supply chain network design (CRSCND) problem aimed at maximizing economic and social benefits by enabling enterprises to jointly address demand uncertainties. Through strategies including joint inventory replenishment, shared distribution centers (DCs), and pooled transportation resources, the CRSCND problem seeks to optimize plant and DC locations and the allocation of DCs to customers under a collaborative framework. To address this, we develop two robust optimization models incorporating a budget uncertainty set, each model representing a distinct risk-pooling policy. These models are then reformulated into solvable linear programming structures. Results from numerical experiments confirm the cost-reduction benefits of collaboration and robust optimization. Sensitivity analysis reveals that factors like violated probability and high demand volatility minimally impact cost savings enabled by collaboration and robustness. Moreover, each robust model shows distinct suitability depending on specific scenario parameters. Finally, we test three cost-saving allocation mechanisms, finding that only the Shapley value method yields best allocations in cases involving overlapping demand.

Deterministic scenarios guided [formula omitted]-Adaptability in multistage robust optimization for energy management and cleaning scheduling of heat transfer process

The heat transfer process is of paramount importance for energy management and heat recovery, but the inevitable uncertain fouling poses significant challenges to sustainable energy-saving efforts. This study aims to solve the energy management and cleaning scheduling problems under fouling uncertainty. To achieve this, a novel multistage Robust Optimization (RO) method based on the Deterministic Scenarios Guided K-Adaptability (DSGKA) strategy is proposed. The process scheduling problem is initially tackled through long-period Mixed Integer Optimal Control Problems (MIOCPs) with deterministic scenarios. Subsequently, considering the slow transition characteristics of energy efficiency degradation caused by fouling, the candidate paths generation algorithm guided by deterministic MIOCPs is developed. Additionally, the K-Adaptability method is employed to segment decision-dependent uncertainty sets into finite partitions, facilitating the resolution of the multistage RO problem through worst-case analysis. In theory, the rationality of the proposed guidance algorithm is established. Simulation results on a heat exchanger network are also provided to demonstrate the effectiveness of the DSGKA scheme. By transforming the original multistage RO problem into a finite-dimensional optimization problem solvable with intelligent heuristic algorithms, the proposed method adeptly manages nonlinearity in constraint equations, thereby improving its applicability for equipment maintenance scheduling across various slow transition industrial processes.

Optimization of renewable energy-based integrated energy systems: A three-stage stochastic robust model

The integration of renewable energy into an integrated energy system (REIES) represents a promising approach to achieving clean and low-carbon energy consumption. However, the inherent uncertainty and variability of renewable energy sources and load demands present significant challenges to the operational efficiency and stability of REIES. To address these challenges, we propose a three-stage stochastic robust optimization (TRSRO) model. This model accounts for market electricity prices, source-load scenario probabilities, and output uncertainties to optimize system configurations in REIES. It also includes the optimization of power exchanges with the regional grid and strategic operational scheduling. Initial scenarios of uncertainty variables are generated using the spectral normalized conditional Wasserstein generative adversarial network (SNCWGAN), forming polyhedral uncertainty sets. Comprehensive norm constraints are then applied to capture probability distribution uncertainties within these sets. To improve the efficiency of solving the model, we introduce a two-layer column constraint generation algorithm, considering variable alternate iterations (TLCCG-VAI). The results demonstrate that the proposed method can achieve a 12.16 % reduction in total cost compared to the baseline method. Furthermore, the TLCCG-VAI algorithm reduces runtime by 82.97 % while maintaining the same level of solution accuracy.

Robust day-ahead scheduling of cooperative energy communities considering multiple aggregators

Future cities must play a vital role in reducing energy consumption and decarbonizing the electricity sector, thus evolving from passive structures towards more efficient smart cities. This transition can be facilitated by energy communities. This emerging paradigm consists of collectivizing a set of residential installations equipped with onsite renewable generators and storage assets (i.e., prosumers), which can eventually share resources to pursue collective welfare. This paper focuses on cooperative communities, where prosumers share resources without seeking selfish monetary counterparts. Despite their apparent advantages, energy management and scheduling of energy communities suppose a challenge for conventional tools due to the high level of uncertainty (especially due to intermittent renewable generation and random demand), and privacy concerns among prosumers. This paper addresses these issues. Specifically, a novel management structure based on multiple aggregators is proposed. This paradigm preserves users' confidential features while allowing them to extract the full potential of their assets. To efficiently manage the variety of assets available under uncertainty, an adaptive robust day-ahead scheduling model is developed, which casts as a solvable and portable Mixed Integer Linear Programming framework, which eases its implementation in real-world cases. The new proposal concerns uncertain generation and demand using a polyhedral representation of the uncertainty set. A case study is conducted to validate the developed model, showing promising results. Moreover, different results are obtained and analysed. Finally, it is worth remarking on how the level of robustness impacts the collective bill, incrementing it by 75 % when risk-averse conditions are assumed. In addition, the role of storage assets under pessimistic conditions is remarked, pointing out that these assets rule the scheduling plan of the community instead of renewable generators.

Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets

Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.

A distributionally robust optimization approach of multi-park integrated energy systems considering shared energy storage and Uncertainty of Demand Response

To enhance the economic efficiency and renewable energy integration capacity of multi-park integrated energy systems (MPIES) and address the issue of insufficient consideration of demand response uncertainty in existing studies, this paper proposes a distributionally robust optimization approach for multi-park integrated energy systems, considering shared energy storage and the uncertainty of demand response. First, models of the shared energy system and demand response are established. Based on these models, a deterministic optimization scheduling model for MPIES is developed, aiming to minimize system costs while considering constraints such as grid power balance. To address uncertainty in demand response, this paper employs Interval Type-2 Fuzzy Theory to construct uncertainty sets and considers system reliability constraints. The original optimization problem is then transformed into an equivalent robust counterpart model, and the optimal distributionally robust solution is obtained through parameter domain decomposition methods. Finally, the proposed method is validated using the IEEE 33-bus system. The results show that considering shared energy storage and demand response individually can reduce total system costs by 4.86 % and 26.46 %, respectively. After accounting for the uncertainty in demand response, the total system cost increases only slightly by 4.36 %, but this improves the system's robustness.

Robust wind power capacity planning under fuel price uncertainty using conic duality theory and piecewise McCormick relaxation

Increasing the proportion of renewable energy sources (RESs) in power generation is crucial due to fossil fuel depletion and rising environmental pollution. In this regard, identifying the most lucrative sites and sizes for installing wind farms, as one of the fastest-growing types of RESs, is imperative. This paper presents a robust profit-oriented wind power capacity planning (WPCP) considering long- and short-term uncertainty. The model forms a two-stage min-max-min hierarchical structure. The first stage minimizes the investment cost plus maximum regret, while the second stage maximizes the profit under the worst-case uncertainty realization. Unlike the existing approaches where uncertainty in fossil fuel prices is neglected, we model fuel price uncertainty using both polyhedral and ellipsoidal uncertainty sets. In this respect, the third level is formulated as a bi-level program, with the upper level being the profit maximization and the lower level being the locational marginal price (LMP) calculation. In the case of the ellipsoidal set, the conic duality theory is employed to dualize the lower level. The piecewise McCormick relaxation (PMR) technique linearizes the bilinear terms. The nested column-and-constraint generation (NCCG) technique solves the formulated problem. A clarifying case study is employed to demonstrate the efficacy of the proposed model.

Liner fleet deployment and empty container repositioning under demand uncertainty: A robust optimization approach

This paper investigates a robust optimization problem concerning the integration of fleet deployment and empty container repositioning in a shipping line network, where a fleet of vessels is dispatched to transport both laden and empty containers, aiming to fulfill a predetermined set of requests over a defined time horizon. The sizes of customer demands are uncertain and are characterized by a budgeted uncertainty set. This study aims to ascertain the vessel types assigned to each shipping route, the routing of laden containers, and the repositioning of empty containers in a manner that minimizes the total cost. Simultaneously, it ensures the feasibility of all transportation plans for any realization of demand within the uncertainty set. We introduce a path-based two-stage robust formulation for addressing the problem. In the first stage, the assignment of vessel types to each shipping route is determined, and the second stage focuses on establishing the routing of laden containers and repositioning of empty containers under a worst-case scenario. We propose the Column-and-Constraint Generation algorithm for solving the proposed robust formulation. To address large-scale size instances, we propose an acceleration technique, i.e., the piece-wise affine policy, which reduces the dimensions of the uncertainty set while maintaining a bounded compromise in solution quality. Comprehensive numerical experiments derived from real-world industries, such as the Shanghai port and CMA CGM, are conducted to validate the proposed formulation and solution methodologies.

Two-stage robust optimization for optimal operation of hybrid hydrogen–electricity storage system considering ladder-type carbon trading

The prominent problems of renewable energy curtailment and its uncertainty have become a hot topic. To the end, with consideration of environmental friendliness, energy utilization efficiency and operation cost, this paper proposes a hybrid hydrogen–electricity storage system (HHES) operation framework comprising assorted types of coupling devices and carbon capture system (CCS) under ladder-type carbon trading (LCT) mechanism. By considering energy balance constraints, equipments’ operation constraints and erratic rates of RESs and multiple loads, a scheduling model with source-load uncertainties is developed with the goal of minimizing the system’s operation cost and carbon emission. Then, a two-stage tri-level robust model is built to find the optimal scheduling plan under multiple uncertainties given the various sources and loads. The first level is to make an optimal dispatch plan and the second level is to find the worst scenario under the given uncertainty set. The third level aims at enhancing renewable energy accommodation rate and reducing load shedding rate in the worst case. Since the model is formed as a tri-level mixed integer optimization problem, the Column and Constraint Generation (CCG) method is adopted to achieve the optimal result. Finally, numerical analyses are performed. The results show that the proposed approach realizes a flexible dispatch of multiple energy and a high utilization level of multiple storages. Furthermore, this paper studies the influence of uncertainty and uncertainty budget parameters with the comparison of the deterministic model and robust models with different uncertainties. The results show that the proposed HHES has strong robustness and can provide a good reference for energy dispatch.

Methodological Challenges in Aligning EPDs with Whole Life Carbon Limits for Buildings: A B2B Approach

Abstract The environmental performance from the materials used in buildings is pivotal in reducing greenhouse gas (GHG) emissions from the building sector; buildings are in the top three of the world’s most significant contributors of GHG emissions and are responsible for one-fifth of the overall resource consumption. Alongside multiple countries enforcing legal GHG limits and requiring Life Cycle Assessment (LCA) for new buildings, the availability of product-level environmental data, known as Type III Environmental Product Declarations (EPDs) has increased exponentially. EPDs were originally used for Business-to-Business purposes but are now the main data source for building-level LCAs. However, this often comes with a large set of uncertainties, as EPDs are still evolving as a documentation approach, and not always readily applicable in the whole life cycle approach. There are a multitude of complex areas to engage into, this study focuses on how use-stage modules are documented in EPDs, and how varied approaches create further complexity and perils in relation to their use in LCA and regulations, in the sense of, potential leading to high uncertainties and wrongful interpretations. The study aims to address the methodological gaps associated with the use of EPDs as data inputs in legally binding LCA requirements particularly concerning modules B1-5, which constitute the embodied part of the use-stage. The findings reveal a significant margin of error if EPDs are not correctly implemented, underscoring the importance of the Business-to-Business documentation approach.

A multi-objective robust dispatch strategy for renewable energy microgrids considering multiple uncertainties

The demand for low-carbon transformations and the uncertainty of renewable energy sources and loads present significant challenges for the optimal dispatch of microgrid. This study proposed a multi-objective robust dispatch strategy to reduce the risks associated with the uncertainty of renewable energy source output and loads while promoting low-carbon and economical microgrid operation. The economic emission dispatch problem for a microgrid was formulated as a multi-objective robust dual-layer optimization model. Consequently, a high-dimensional adjustable linear polyhedral uncertainty set was proposed to describe the uncertainty of renewable energy sources and loads. This study transformed the original model into an easy-to-solve single-layer second-order cone programming optimal power flow optimization model by employing second-order cone relaxation and duality transformation. Thereafter, a synthetic membership function was proposed to determine the optimal compromise solution. To determine the charging and discharging statuses of the battery storage system and the electricity traded between the microgrid and the external power grid, a battery storage system control strategy based on time-of-use electricity prices and real-time power flow calculations was proposed. Simulations conducted on a modified IEEE-30 bus system demonstrated that the proposed strategy effectively reduced the economic costs and carbon emissions of the microgrid by 8.23 % and 2.43 %, respectively.

Adaptive robust optimization framework for market-based wind power investment

Wind is distinguished by its eco-friendliness and sustainability, making it one of the most rapidly expanding forms of renewable energy sources (RESs). Hence, it is necessary to determine the most profitable plan for wind farm installation. This paper constructs a novel scheme for market-based wind power investment (WPI) problems using adaptive robust optimization (ARO). A tri-level robust WPI (RWPI) model is established, the first level of which is to minimize the investment cost plus the worst-case loss. In the second level, the worst-case loss (also known as the maximum regret) is identified by maximizing the minimum value of minus profit over the uncertainty sets. The third level maximizes the wind farm profit. Since the profit calculation requires the determination of the locational marginal price (LMP), the third level constitutes bi-level programming, with the upper level being the profit maximization and the lower level being the market clearing process. First, Karush-Kuhn-Tucker (KKT) conditions are applied to convert the bi-level model to a single-level model, resulting in an ARO with binary variables at the third level. Afterward, the nested column-and-constraint generation (NCCG) strategy is employed to solve the ARO with mixed-integer recourse. A case study is used to verify the scalability and practical applicability of the proposed model.

Optimizing sustainable multiple biomass-to-biofuel conversion network with integrated water resource management utilizing data-driven robust planning

Until now, the biofuel supply chain has lacked an integrated approach to address its fundamental challenges in real-world implementation. The first challenge is maintaining a continuous supply flow in the biofuel network, which can significantly impact economic outcomes and performance measures. The second challenge involves managing water resources for biofuel production to preserve freshwater reserves. The third challenge is managing operational risks, such as uncertainty in biofuel demand, which leads to conservative planning and underscores the need for realistic uncertain programming approaches. To address these challenges, the proposed study designs a biofuel conversion network that utilizes third-generation feedstocks and non-edible seed sources, sourcing necessary water from wastewater resources. This practical solution addresses water-energy relationship concerns, reducing dependency on limited freshwater supplies. To achieve a continuous and cost-effective feed supply, this study incorporates a pre-processing phase using a hybrid Multi-Criteria Decision-Making model. This approach ensures the feed supply remains consistent, economical, and sustainable. To manage the uncertainty in biofuel demand, we developed a robust optimization model with an intersected uncertainty set using a data-driven approach. This approach aims to reduce conservatism in the obtained solutions, making the model more practical for real-world implementation. The numerical results reveal some practical implementation for policymakers. The proposed robust optimization using the configured uncertainty set offers a tool to manage the conservatism of renewable energy demand uncertainty and operational risks more effectively than traditional approaches (0.914%). This formulation can help policymakers to maintain greater robustness against fluctuations in confidence levels. Additionally, the simultaneous use of third-generation and non-edible feedstocks creates a multi-biomass-to-biofuel conversion network, offering practitioners a practical solution that significantly reduces reliance on freshwater resources. Furthermore, policymakers can leverage the proposed pre-processing phase to promote economic and continuous supply feed based on geographical, resource accessibility, and infrastructure criteria.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.