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

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COMPUTERS & OPERATIONS RESEARCH 2025年 174卷

作者： Zhang, Haifeng Yang, Kai Dong, Jianjun Yang, Lixing Beijing Jiaotong Univ Sch Syst Sci Beijing 100044 Peoples R China Beijing Inst Technol Sch Management Beijing 100081 Peoples R China Nanjing Univ Sci & Technol Sch Sci Nanjing 210094 Peoples R China

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.

关键词： Urban agglomeration Hub network design two-stage robust optimization Demand uncertainty Benders decomposition

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General Polyhedral Approximation of two-stage robust linear programming for budgeted uncertainty

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COMPUTERS & OPERATIONS RESEARCH 2025年 179卷

作者： Grunau, Lukas Niemann, Tim Stiller, Sebastian Tech Univ Carolo Wilhelmina Braunschweig Inst Math Optimizat Univ Pl 2 D-38106 Braunschweig Germany Cluster Excellence SE 2 A Sustainable & Energy Eff Hermann-Blenk-Str 42 D-38108 Braunschweig Germany

We consider two-stage robust linear programs with uncertain righthand side. We develop a General Polyhedral Approximation (GPA), in which the uncertainty set U is substituted by a finite set of polytopes derived from the vertex set of an arbitrary polytope that dominates U. The union of the polytopes need not contain U. We analyze and computationally test the performance of GPA for the frequently used budgeted uncertainty set U (with m rows). For budgeted uncertainty affine policies are known to be best possible approximations (if coefficients in the constraints are nonnegative for the second-stage decision). In practice calculating affine policies typically requires inhibitive running times. Therefore an approximation of U by a single simplex has been proposed in the literature. GPA maintains the low practical running times of the simplex based approach while improving the quality of approximation by a constant factor. The generality of our method allows to use any polytope dominating U (including the simplex). We provide a family of polytopes that allows for a trade-off between running time and approximation factor. The previous simplex based approach reaches a threshold at Gamma>root m after which it is not better than a quasi nominal solution. Before this threshold, GPA significantly improves the approximation factor. After the threshold, it is the first fast method to outperform the quasi nominal solution. We exemplify the superiority of our method by a fundamental logistics problem, namely, the Transportation Location Problem, for which we also specifically adapt the method and show stronger results.

关键词： robust optimization two-stage robust optimization Linear programming Approximation algorithm Transportation location problem

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A two-stage robust optimal capacity configuration method for charging station integrated with photovoltaic and energy storage system considering vehicle-to-grid and uncertainty

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ENERGY 2025年 319卷

作者： Lin, Hao Liu, Shilin Liao, Shiwu Wang, Shinong Minist Educ Key Lab Adv Percept & Intelligent Control High End Wuhu 241000 Peoples R China Nanjing TODE Technol Co Ltd Nanjing 210000 Peoples R China

This paper proposes a novel capacity configuration method for charging station integrated with photovoltaic and energy storage system, considering vehicle-to-grid technology and the uncertainties of photovoltaic and electric vehicles' behavior. Firstly, the improved k-means method is used to cluster the electric vehicles participating in vehicle-to-grid, and then the corresponding scheduling model is established by considering the constraints of schedulable time, charging and discharging power, state of charge and so on. Secondly, for uncertain scenarios, using the predicted charging load obtained through Monte Carlo simulation as the basis for robust optimization, a two-stage robust optimal configuration model with the objective of minimizing the comprehensive cost of investment and operation is established, so as to improve the reliability of the solution by only solving for the worst-case scenario and its uncertainty problem. Finally, the column-and-constraint generation algorithm and Karush-Kuhn-Tucker conditions are used to solve the problem. The study results show that the configuration capacity of energy storage system and the composite cost of investment and operation can be effectively reduced when vehicle-to-grid is considered, meanwhile considering uncertainty can improve the ability of the charging station to resist risks.

关键词： Electric vehicle charging station Capacity configuration Vehicle-to-grid Uncertainty two-stage robust optimization

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Study on two-stage robust optimal configuration of integrated energy system considering CCUS and electric hydrogen production

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ENERGY CONVERSION AND MANAGEMENT 2025年 324卷

作者： Qi, Ze Zhao, Huiru Zhang, Yuanyuan Guo, Sen North China Elect Power Univ Sch Econ & Management Beijing 102206 Peoples R China

Under the background of complementary and cascade utilization of multiple types of energy, how to carry out the capacity planning of economic, efficient, and low-carbon integrated energy system to meet the diversified needs of cooling, heating, electricity, and hydrogen loads is a challenging problem. This study conducted a twostage robust optimization (RO) configuration of integrated energy system considering Carbon Capture Utilization and Storage (CCUS) and electric hydrogen production. In order to adapt the configuration scheme to the potential extreme operational scenarios, the uncertainty sets of new energy and load are constructed based on Latin hypercubic sampling. The efficacy is validated through the design of several scenarios and a comparison with alternative methods. The results indicate: 1) Strengthening of carbon emission constraints facilitates the cleaner and electrified transformation of energy. In the shift from "low carbon" to "zero carbon" scenario, the proportion of new energy capacity in power generation equipment increases from 45.1% to 79.8%;the proportion of electric chillers in cooling equipment increases from 1.3% to 7.2%, and reaches 60% under "zero carbon" scenario. 2) With a 36% reduction in the investment cost of new energy, the spatiotemporal transfer value of energy is realized, which increases the investment capacity of energy storage by about 7 times. 3) The configuration of hydrogen production and storage equipment to meet hydrogen demand reduces costs by 19% compared to direct purchasing in "low carbon" scenario. 4) Compared to other methods, the proposed method in this study is able to reduce the solution time and configuration cost.

关键词： Integrated energy system two-stage robust optimization Capacity configuration Carbon costs Hydrogen energy

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Resilient Unit Commitment for Transmission Lines Hardening Under Endogenous Uncertainties

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IEEE TRANSACTIONS ON POWER SYSTEMS 2025年第1期40卷 204-217页

作者： Yang, Xiang Liu, Xinghua Zhao, Tianyang Xiao, Gaoxi Fan, Bangji Liu, Shengwei Wang, Peng Xian Univ Technol Sch Elect Engn Xian 710048 Peoples R China Xi An Jiao Tong Univ Sch Elect Engn Xian 710049 Peoples R China Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore Jinan Univ Energy Elect Res Ctr Zhuhai 519070 Peoples R China

This paper proposes a joint strategy of pre-disaster hardening for transmission lines and real-time dispatching for units to address the failures with endogenous uncertainties caused by hurricanes on transmission lines. Specifically, we delve into the resilient unit commitment (UC) problem, which is formulated as a two-stage robust optimization (TRO) problem with endogenous uncertainty. In the first stage, the generator states, outputs, reserves, and line reinforcement are pre-scheduled to minimize defense costs. The second stage adopts a recourse approach to quantify the worst-case failure scenario of the lines, reflected as the costs of load shedding and generator curtailment. The operating status of transmission lines is depicted by an endogenous uncertainty set which is affected by the line hardening decision and unit reserve decision in the first stage. The TRO model with endogenous uncertainty is solved using a modified column-and-constraint generation (C&CG) algorithm. The proposed model is tested on two reliability test systems (RTS), including a modified IEEE RTS-79 and a two-area IEEE RTS-96 exposed to a hurricane. The results verify the effectiveness of the proposed method, compared with the popular two-stage robust optimization method with endogenous uncertainty.

关键词： Uncertainty Power transmission lines Costs Generators Power systems Hurricanes Disasters Unit commitment transmission lines hardening two-stage robust optimization endogenous uncertainty

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Design of robust Var Reserve Contract for Enhancing Reactive Power Ancillary Service Market Efficiency

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CSEE Journal of Power and Energy Systems 2024年第2期10卷 767-771页

作者： Yunyang Zou Yan Xu School of Electrical and Electronic Engineering Nanyang Technological UniversitySingapore 639798Singapore

In a deregulated Var market, market power issue is more serious than in an energy market since reactive power cannot be transmitted over long distances. This letter designs a multi-timescale Var market framework, where market power that may arise in the hourly-ahead Var support service market due to system configuration deficiency and market structure flaws can be eliminated by day-ahead contract-based Var reserve service market. Settlement of day-ahead Var reserve contract is formulated as a two-stage robust optimization (TSRO) model considering worst case of uncertainty realization and potential market power that may arise in hourly-ahead market. TSRO with integer recourses is then solved by a new column and constraint generation algorithm. Results show a robust Var reserve contract can fully eliminate market power, and prevent suppliers from manipulating market prices.

关键词： Market efficiency market power reactive power ancillary service two-stage robust optimization Var reserve

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two-stage scheduling optimization model and benefit allocation strategy for virtual power plant clusters aggregated by multidimensional information indicators

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RENEWABLE ENERGY 2025年 240卷

作者： Ju, Liwei Lv, Shuoshuo Li, Yanbin Li, Yun Qi, Xin Li, Gen Zhang, Feng North China Elect Power Univ State Key Lab Alternate Elect Power Syst Renewable Beijing 102206 Peoples R China North China Elect Power Univ Beijing Key Lab New Energy & Low Carbon Dev Beijing 102206 Peoples R China Tech Univ Denmark DTU Dept Engn Technol & Didact DK-2750 Ballerup Denmark

With the large-scale integration of distributed energy resources into the distribution network, virtual power plant clusters (VPPs) control based on "intra-group autonomy and inter-group coordination" can reduce the difficulty of grid operation and control. Effective cluster partitioning is the key to realize the optimal operation of VPPs. Based on different types of distributed energy flexibility output models, this paper proposes the structuralfunctional aggregation strategy for VPPs. Additionally, A two-stage robust scheduling optimization model for VPPs is proposed. This model considers a multi-subject cooperative game for calculating the optimal operation strategy of the system. The solution algorithm is constructed through the integration of strong dyadic theory and the C&CG algorithm. Then, an improved Shapley-based benefit allocation strategy for VPPs by the cooperative game is proposed. Finally, an example analysis is carried out in Huangyuan County, Xining City, Qinghai Province, China. The results show that: (1) The proposed structure-function aggregation optimization strategy, the self-balancing indicator of the system increased by 24.65%, and the upward flexibility deficit and downward flexibility deficit decreased by 72.52 MW and 94.92 MW, compared to the case where the distributed energy sources are independently connected to the grid.(2) The proposed two-stage scheduling optimization model for VPPs reduces the required balancing power by 58.37 %. Compared to the non-considered robustness factor, the average energy supply cost of the proposed model is reduced by 12.02 & YEN;/MW. (3) Utilizing the proposed twolayer benefit allocation strategy, the profits obtained by VPPs for 45.49 %, 26.03 %, and 28.49 % of the total incremental gains. Non-Adjustable Generation Unit (Non-AGU) needs to make a profit of 29.09 & YEN;/MWh due to the uncertainty of output. Adjustable Generation Unit (AGU), Energy Storage Device (ESD) and Adjustable Load (AL) gain 6.24 & YE

关键词： Virtual power plant clusters Cluster aggregation optimization two-stage robust optimization two-layer benefit allocation

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Facility location and capacity planning for sampling and testing processes under demand uncertainty

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COMPUTERS & INDUSTRIAL ENGINEERING 2025年 204卷

作者： Zhou, Zhongbao Sun, Wenting Ren, Tiantian Zeng, Ximei Hunan Univ Sch Business Adm Changsha 410082 Peoples R China Hunan Univ Hunan Key Lab Emergency Management Intelligent Dec Changsha 410082 Peoples R China Xiangtan Univ Business Sch Xiangtan 411105 Peoples R China Changsha Univ Sci & Technol Sch Econ & Management Changsha 410076 Peoples R China

During an outbreak, nucleic acid testing is essential for early infection detection and virus transmission control. In this study, we aim to location and capacity planning of testing facilities, balancing minimal costs with maximum population coverage during public health emergencies. We propose a novel two-stage robust optimization model that addresses uncertainties in sampling demand during an epidemic, with distinct phases for sampling and testing. Applying this model to medium and high-risk areas in Beijing during COVID-19, we use the column-and-constraint generation (C&CG) algorithm and compare its performance with three meta-heuristic algorithms: Differential Evolution (DE), Genetic Algorithm (GA), and Simulated Annealing (SA). Our findings reveal that the C&CG algorithm reduces sampling costs by 31.10% compared to DE, 26.77% to GA, and 21.17% to SA. It also lowers testing costs by 7.48%, 79.79%, and 60.63%, respectively, and achieves a higher completion rate for sampling and testing volumes, ranging from 93.12% to 100%. In addition, C&CG outperforms the other algorithms in handling large sample sizes by 43.98% to 61.84%. Despite its longer computational time, C&CG is more efficient in cost reduction and demand satisfaction. Furthermore, we analyze the impact of uncertainty set parameters, including a unified value of the demand risk parameter, and assess different cases. The corresponding location and capacity solutions can offer decision support for emergency agencies managing public health crises.

关键词： Facility location Capacity planning Demand uncertainty two-stage robust optimization Column-and-constraint generation algorithm

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Optimal coordinated design of building mix ratios and energy systems under uncertainty

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RENEWABLE ENERGY 2025年 246卷

作者： Xie, Yujie Li, Zhengrong Jiang, Yating Wang, Heyu Johnson, Brian C. Zhu, Han Tongji Univ Sch Mech Engn Shanghai 200092 Peoples R China Dais Corp 11552 Prosperous Dr Odessa FL 33556 USA

As renewable energy production surges, traditional load-leveling strategies for stabilizing terminal loads during planning have shown significant limitations, particularly in the face of uncertainties within integrated energy systems (IES) that can diminish energy efficiency. This study introduces a pioneering two-stage robustness model for IES configuration, offering a comprehensive analysis of how terminal load affects system performance by varying the mix of building types in a region. It uniquely assesses the simultaneous impact of thermal, cooling, and electricity storage under various uncertainties, shedding light on how regional building composition influences the economic viability of energy systems. Findings indicate that uncertainties in renewable energy and demand drive up investment costs, with each 6-unit increment in the robustness parameter increasing total costs by 3.7-4.9 %. The study identifies office-centric areas as optimal for IES implementation and residential areas as less favorable. It also suggests that with an increasing gas-to-electricity price ratio, there should be a strategic increase in energy storage installations, prioritizing thermal/cold energy storage (T/CES) given current pricing. The study concludes with regression models that facilitate expedited system design and feasibility assessments, providing valuable insights for optimizing IES under diverse uncertainties and identifying cost-effective scenarios.

关键词： Building mix ratio Multiple uncertainty Optimal configuration Integrated energy system two-stage robust optimization Dual variables

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Vulnerability Assessment and Reduction for Intermodal Freight Transportation Networks

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2024年 245卷

作者： Wei, Xiaoyang Wang, Mengtong Meng, Qiang ASTAR Inst High Performance Comp Singapore 138632 Singapore Natl Univ Singapore Dept Civil & Environm Engn Singapore 117576 Singapore

Intermodal freight transportation networks (IFTNs) are critical for global logistics, facilitating the efficient transportation of cargo within containers by integrating diverse transportation modes. However, such integration exposes IFTNs to a variety of disruptions, may leading to degradation in transportation capabilities. Thus, vulnerability assessment and reduction (VA&R) for IFTNs are of practical significance. Despite its importance, this research problem has received little attention in the literature. To fill the research gap, we propose the VA&R problem for IFTNs, considering uncertain disruptions that impact IFTN nodes and links, along with cascading failures and influence spreading. We further build models incorporating two-stage robust optimization to assess and reduce the impact of uncertain disruptions, and build exact solution methods for VA&R utilizing columnand-constraint generation and speedup strategies to identify non-conservative solutions. The effectiveness and efficiency of our modeling and solution methods are demonstrated by comparing them to benchmark modeling and solution methods, through numerical experiments for a real-world international IFTN. Furthermore, managerial insights and recommendations for IFTN operations are achieved based on comprehensive sensitivity analyses to address uncertain disruptions in IFTNs.

关键词： Intermodal freight transportation network Vulnerability assessment Vulnerability reduction Max-min optimization two-stage robust optimization Column-and-constraint generation

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