Stochastic Robust Optimization Research Articles

Hybrid Stochastic-Robust Service Restoration for Wind Power Penetrated Distribution Systems Considering Subsequent Random Contingencies

Service restoration (SR) is essential to recovering distribution systems with outages when extreme weather conditions occur. However, the occurrence of long-duration extreme weather events may cause a restored system to become susceptible to subsequent random contingencies. To address this challenge, this paper proposes a microgrid-based SR approach for recovering critical loads in wind power penetrated distribution systems incorporating the impacts from subsequent random contingencies. A hybrid stochastic-robust optimization model is developed considering both the subsequent random contingencies that may occur and the uncertainty of wind power generation. The expected load shedding is minimized by locating mobile emergency generators, dynamically adjusting microgrid (MG) topologies, and dispatching power sources. To ensure the tractability of the optimization, the original problem is reformulated and decomposed into a master problem and several robust optimization subproblems based on the column-and- constraint generation approach. In addition, the master problem, which is in the format of stochastic optimization is solved by an improved progressive hedging algorithm to accelerate optimization. The proposed method is validated on the modified IEEE distribution systems and the simulation results show the effectiveness of the proposed SR method in reducing security violation risk and reducing the computational complexity.

Coupling Two-Stage Stochastic Robust Programming with Improved Export Coefficient for Water Allocation among Industrial Sectors

Water scarcity and water pollution are essential factors limiting coordinated regional development, especially in water-deprived regions. Industrial restructuring is an effective water management solution to alleviate water scarcity and mitigate water pollution. However, due to widely existing inexact parameter information in the water resource management system, it is challenging to allocate water resources among industrial sectors. To address these problems, an export coefficient coupled with a two-stage stochastic robust programming method (EC-TSRP) was developed through integrating an export coefficient model (ECM), two-stage stochastic programming (TSP) and robust optimization. The proposed EC-TSRP model could effectively deal with the multiple uncertainties expressed as stochastic and the intervals with fluctuation ranges, and enhance the robustness of optimal plans for supporting water resource allocation among industrial sectors under complex uncertainties. It was then applied to Bayan Nur City, in arid north-west China. The optimization alternatives indicate that wheat, sheep and services would be the most sensitive sectors among all industrial sectors, when non-point source (NPS) pollution exports are restricted. In addition, comparing the EC-TSRP results with the deterministic model, the reliability of the system could be improved significantly, while the value of the objective function would be decreased slightly. The simulation results were also compared with the historical data from 2012 to 2016. Although the total revenue of Bayan Nur City would decrease by 1.52%, the pollutant loads of total nitrogen, total phosphorus and chemical oxygen demand (TN, TP and COD) would decrease by 14.5%, 7.75% and 2.07%, respectively, and total water allocation also would decrease from 4.6 billion m3 to 4.23 billion m3.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

Preallocation of Electric Buses for Resilient Restoration of Distribution Network: A Data-Driven Robust Stochastic Optimization Method

In recent years, severe hurricanes have occurred frequently, posing a huge challenge to the distribution network (DN) operation. Electric buses (EBs) possess large-capacity batteries and are widely used in public transit under normal circumstances. Before a hurricane occurs, some idle EBs can be preallocated to different charging stations equipped with vehicle-to-grid and served as sources for emergency power supply. This article proposes a prehurricane EB preallocation method to assist the resilience enhancement of a fragile DN. A two-stage data-driven robust stochastic programming technique is applied to build this model. Different scenarios are generated, and the corresponding posthurricane restoration processes are considered in the determination of the preallocation strategy. Also, the uncertainties of hurricane-induced physical damages are considered in modeling, which is described as a strengthened confidence set. The established model aims at exploring the optimal preallocation strategy of EBs with minimum load losses under the worst-case distribution. A column-and-constraint generation algorithm is then used to solve this proposed model. The proposed method is tested on a modified IEEE 33-bus system with 50 EBs. The results indicate that preallocating EBs to charging stations can improve the resilience of the posthurricane DN effectively.

Viable closed-loop supply chain network by considering robustness and risk as a circular economy.

The viable closed-loop supply chain network (VCLSCND) is a new concept that integrates sustainability, resiliency, and agility into a circular economy. We suggest a hybrid robust stochastic optimization by minimizing the weighted expected, maximum, and entropic value at risk (EVaR) of the cost function for this problem. This form considers robustness against demand disruption. Finally, CLSC components are located, and quantity flows are determined in the automotive industry. The results show that the VCLSCND cost is less than not considering viability and has a − 0.44% gap. We analyze essential parameters. By increasing the conservative coefficient, confidence level, and the scale of the main model, decreasing the allowed maximum energy, the cost function, time solution, and energy consumption grow. We suggested applying the Fix-and-Optimize algorithm for producing an upper bound for large-scale. As can be seen, the gap between this algorithm and the main problem for cost, energy, and time solution is approximately 6.10%, − 8.28%, and 75.01%.

COVID19 epidemic outbreak: operating rooms scheduling, specialty teams timetabling and emergency patients' assignment using the robust optimization approach.

The health care system is characterized by limited resources, including the physical facilities as well as skilled human resources. Due to the extensive fixed cost of medical facilities and the high specialization required by the medical staff, the problem of resource scarcity in a health care supply chain is much more acute than in other industries. In the pandemic of the Coronavirus, where medical services are the most important services in communities, and protective and preventive guidelines impose new restrictions on the system, the issue of resource allocation will be more complicated and significantly affect the efficiency of health care systems. In this paper, the problem of activating the operating rooms in hospitals, assigning active operating rooms to the COVID-19 and non-COVID-19 patients, assigning specialty teams to the operating rooms and assigning the elective and emergency patients to the specialty teams, and scheduling their operations is studied by considering the new constraints of protective and preventive guidelines of the Coronavirus. To address these issues, a mixed-integer mathematical programming model is proposed. Moreover, to consider the uncertainty in the surgery duration of elective and emergency patients, the stochastic robust optimization approach is utilized. The proposed model is applied for the planning of operating rooms in the cardiovascular department of a hospital in Iran, and the results highlight the role of proper management in supplying sufficient medical resources effectively to respond to patients and scheduled surgical team to overcome the pressure on hospital resources and medical staff results from pandemic conditions.

Optimal Scheduling of a Self-Healing Building Using Hybrid Stochastic-Robust Optimization Approach

This article provides a two-stage robust energy management method for a self-healing smart building that can handle contingencies that occur during real-time operation. Aside from an electrical link with the distribution network, the smart building is equipped with a diesel generator and photovoltaic solar power generating systems. The energy management system should be smart enough to plan different resources based on the situation. At first, bilevel programming identifies critical faults for affected components based on mean time to repair. After identifying major failures, the faults are described in operational scenarios, and a two-stage hybrid robust-stochastic programming technique is used to determine the bid/offer in day-ahead and real-time energy markets, in which stochastic programming is responsible for considering the uncertainty of faults, and the robust optimization approach is used to cope with the uncertainty of real-time market prices. After linearization, the final optimization is modeled as mixed-integer linear programming in the GAMS optimization package. For the studied smart building, the daily operational cost is expected to increase from $25.794 (for the deterministic case) to $ 28.097 (for the most conservative case) due to the uncertainty of real-time market prices. Due to power shortages caused by the failure of components, the total expected not-supplied load is 6.72 kW (2.53%). A comparison between naive and self-healing scheduling indicated that naive energy management will charge an additional $ 2.75 without considering the probability of components failures under the deterministic case.

Two-stage stochastic robust optimization model of microgrid day-ahead dispatching considering controllable air conditioning load

The wide application of wind power generation and photovoltaic (PV) power generation has greatly increased the uncertainty of power grid. Under the background of smart grid, peak shaving can be carried out with the help of users' equipment. This paper studies the cooling efficiency of air conditioning from the heat transfer process between room and outdoor, and establishes the air conditioning load control model. Based on this, a stochastic robust day-ahead dispatching model for microgrid is proposed, taking cost minimization as objective function. The day-ahead dispatching is considered as the first stage which considering the arrangement of electricity purchasing and generating of thermal power units. The real-time dispatching is regarded as the second stage. Based on the results of the first stage, real-time dispatching of the various situations of the renewable energy output can be carried out. The decision variables are air conditioning control and ESS charge and discharge arrangement. Then the results are fed back to the first stage, and the cost is minimized by iteration. In this paper, the double-layer problem in the second stage is transformed into a single-layer problem by mathematical method, and the Column-and-Constraint Generation (CCG) algorithm is used to solve the two-stage stochastic robust optimization model. Finally, based on the data of a microgrid in a central city, the effectiveness of the method is verified, and the economic superiority of the day-ahead dispatching model at stochastic and robust level is discussed.

Coordinated and Priority‐Based Surgical Care: An Integrated Distributionally Robust Stochastic Optimization Approach

We study a coordinated clinic and surgery appointment scheduling problem for in‐advance scheduling of surgical patients. Our models seek to provide timely access to care by coordinating clinic and surgery appointments to ensure that patients can see a surgeon in the clinic and (if needed) schedule their surgery within a maximum wait time target based on patient classes. There are different types of uncertainty including the number of appointment requests, whether a patient requires surgery, and surgery durations. We develop an integrated multi‐stage stochastic and distributionally robust optimization (IMSDRO) approach to determine the optimal clinic and surgery dates for patients such that the access target constraints are satisfied, and the clinical and surgical overtimes are minimized. The IMSDRO approach synergizes multi‐stage stochastic optimization with distributionally robust optimization to simultaneously incorporate multiple types of uncertainties by including stochastic scenarios for appointment request arrivals and ambiguity sets for surgery durations. Several new transformations are introduced to turn the nonlinear model derived from the IMSDRO approach to a tractable one, and a constraint generation algorithm is developed to solve it efficiently. We propose a data‐driven rolling horizon procedure to facilitate implementation. We use case data to assess the performance of our policies. The results suggest that our policy can significantly improve surgical access delay times compared to the current practice. Our methodology is not limited to a particular setting and can be applied to other service industries where access delay matters.

A deep learning-based microgrid market modeling with planning assumptions

Microgrids (MGs) can be considered as one of the best solutions for the distribution grid's resiliency and reliability. The research study suggests a hybrid stochastic-robust optimization method for determining the optimum schedule of a MG within usual and resilient operation situations. A new optimization method is used to model the effects of main electrical network cost uncertainty on the optimum planning of MGs. A stochastic optimization process is modeled with other dominant uncertainties, such as wind turbine power, photovoltaic power, and reactive/active loads, through creating appropriate case studies for each. MG's resilient operation is improved further by implementing curtailable and shiftable demand response programs.The suggested method enhances the performance of MGs under uncertainties when the network is in its normal and resilient mode. In addition, the paper discusses different approaches for RO planning and evaluates the effectiveness of the suggested structure in a large-scale MG trial system.

Data-Driven Stochastic Robust Optimization for Industrial Energy System Considering Renewable Energy Penetration

Achieving carbon neutrality has been one of the main tasks these decades. In this study, renewable energies were introduced to reduce the greenhouse gas emissions of industrial energy systems. Considering solar heat and wind energy uncertainties, a data-driven stochastic robust optimization framework was proposed. Machine learning methods were applied to derive data information: a data mining method to classify the high-volume uncertain data; a kernel-based method to construct the uncertainty sets for each data class. The stochastic robust optimization model of the industrial energy system was developed as a bilevel optimization procedure: the outer level is a two-stage stochastic programming problem to optimize the expected objective value of different data clusters, and the robust optimization is nested internally to ensure robustness. A case study on the practical industrial energy system was performed, and the results are that the total annual cost is reduced by 1 507 730 $/a and 7.62% GHG emissions are decreased by introducing renewable energies; the proposed method is superior to the traditional ones in terms of PoR (2.91%) and robustness (99.79%). The results of multiobjective optimization considering economic and environmental revenues can provide multipreference schemes for decision-makers.

Narrative-based robust stochastic optimization

Many portfolio optimization techniques rely heavily on past data and modeling assumptions. In an uncertain and ambiguous world, these techniques are prone to amplify model misspecification and therefore have poor out of sample results. Robust optimization explicitly recognizes uncertainty in model specification and performs better out of sample. The Achilles’ heel of the method is the selection of the uncertainty set. In this paper we focus on the construction of the uncertainty set around the stochastic model specification. We propose to use narratives to select the elements in the uncertainty set to avoid using a logically inconsistent or too large uncertainty set. The narratives provide useful tools in a qualitative sense to the portfolio management process.

Low carbon economic dispatch of biogas-wind-solar renewable energy system based on robust stochastic optimization

This paper proposes an economic dispatch based on robust stochastic optimization to reduce the operational difficulty of the integrated energy system under the low-carbon background. The concept of energy hub represents the coupling relationship among various energy sources, and the operational constraints of the gas-electric network architecture are considered. The optimization model of the integrated energy microgrid is established, which includes biogas-wind-solar energy on the basis of the realization of energy conversion, transfer, and storage. The system operation cost, waste treatment cost and carbon trading cost are taken as the optimization target, and the economic and environmental benefits are considered comprehensively. Lastly, renewable energy uncertainty is modeled using the robust stochastic optimization model of the Wasserstein ambiguity set based on 1-norm. The simulation calculation is carried out on TEST6 and 6-node natural gas systems, IEEE118 and 20-node natural gas systems. The results verify the rationality and effectiveness of the system.

A two-stage stochastic-robust optimization for a hybrid renewable energy CCHP system considering multiple scenario-interval uncertainties

The optimal device capacity can impose the hybrid combined cooling, heating and power (CCHP) system integrated with renewable energy on both the significant energy, economy, and environment benefits and the low dependence on the grid. However, the multiple uncertainties pose enormous challenges in the modeling framework and corresponding solving strategy. This paper proposes a two-stage multi-objective stochastic-robust hybrid optimization model for the optimal device capacity of a hybrid CCHP system. The multiple uncertainties are incorporated into this model, and they are modeled by integrating the stochastic hierarchy scenario-generation method into the robust box uncertainty set. The problems of multi-objective optimization and multi-stage coupling are handled by the ε-constraint method and column constraint generation, respectively. Eventually, the vital parameter influence of this model is analyzed by an industrial park case. The results show that the befitting scenario number can make scenario-based stochastic optimization save 97.71% of computation time and ensure the reliability of the optimized results. Moreover, the uncertain budget value significantly influences the successive device capacity while hardly impacting discrete device numbers. Besides, the installed capacities of the wind turbine, photovoltaic system and electrical energy storage considerably determine the economic performance of the hybrid system.

Integrating GIS in reorganizing blood supply network in a robust-stochastic approach by combating disruption damages

As supplying adequate blood in multiple countries has failed due to the Covid-19 pandemic, the importance of redesigning a sensible protective-resilience blood supply chain is underscored. The outbreak-as an extensive disruption-has caused a delay in ordering and delivering blood and its by-products, which leads to severe social and financial loss to healthcare organizations. This paper presents a robust multi-phase optimization approach to model a blood supply network ensuring blood is collected efficiently. We evaluate the effectiveness of the model using real-world data from two mechanisms. Firstly, a Geographic Information System (GIS)-based method is presented to find potential alternative locations for blood donation centers to maximize availability, accessibility, and proximity to blood donors. Then, a protective mathematical model is developed with the incorporation of (a) blood perishability, (b) efficient collation centers, (c) multiple-source of suppliers, (d) back-up centers, (e) capacity limitation, and (f) uncertain demand. Emergency back-up for laboratory centers to supplement and offset the processing plants against the possible disorders is applied in a two-stage stochastic robust optimization model to maximize the level of hospitals' coverage. The results highlight the fraction cost of considering back-up facilities in the total costs and provide more resilient decisions with lower risks by examining resource limitations.

Data-Driven-Based Stochastic Robust Optimization for a Virtual Power Plant With Multiple Uncertainties

Virtual power plant (VPP) has gradually become a key technology to the increasing penetration of renewable energy. The uncertainty and variability of renewable energy and load demand pose significant challenges to the efficiency and stability of VPP's operation. In this paper, a data-driven two-stage stochastic robust optimization (SRO) scheduling model is proposed for a VPP considering wind power, solar power, load demand, and market price uncertainties. The objective is to maximize the profit of the VPP in the energy market and minimize the total system cost of the VPP in the reserve market under the worst-case realization of the uncertainties. The Dirichlet process mixture model (DPMM) and variational inference algorithm are employed for constructing the data-driven uncertainty ambiguity set considering the correlations among multiple uncertainties. The tailored column-and-constraint generation algorithm is developed to solve the SRO model iteratively by reformulating the second stage with the application of the Karush-Kuhn-Tucker conditions. Results from a case study illustrate the effectiveness and superiority of the proposed model.

A Hybrid Stochastic-Robust Optimization Model for Allocating Battery Swapping Stations Considering Demand Uncertainty

A Hybrid Stochastic-Robust Optimization Model for Allocating Battery Swapping Stations Considering Demand Uncertainty

Sustainable, resilient and responsive mixed supply chain network design under hybrid uncertainty with considering COVID-19 pandemic disruption.

The occurrence of the COVID-19 pandemic is a disruption that has adversely affected many supply chains (SCs) around the world and further proved the necessity of combination and interaction of resilience and sustainability. In This paper, a multi-objective mixed-integer linear programming model is developed for responsive, resilient and sustainable mixed open and closed-loop supply chain network design (SCND) problem. The uncertainty of the problem is handled with a hybrid robust-stochastic optimization approach. A Lagrangian relaxation (LR) method and a constructive heuristic (CH) algorithm are developed for overcoming problem complexity and solving large-scale instances. In order to assess the performance of the mathematical model and solution methods, some test instances are generated. The computations showed that the model and the solution methods are efficient and can obtain high-quality solutions in suitable CPU times. Other analyses and computations are done based on a real case study in the tire industry. The results demonstrate that resilient strategies are so effective and can improve economic, environmental and social dimensions substantially. Research findings suggest that the proposed model can be used as an efficient tool for designing sustainable and resilient SCs and the related decision-makings. Also, our findings prove that resilience is necessary for continued SC sustainability. It is concluded that using proposed resilience strategies simultaneously brings the best outcome for SC objectives. Based on the sensitivity analyses, the responsiveness level significantly affects SC objectives, and managers should consider the trade-off between responsiveness and their objectives.

Price-Maker Bidding and Offering Strategies for Networked Microgrids in Day-Ahead Electricity Markets

This paper proposes a price-maker bidding and offering model for networked microgrids (NMG) in a pool-based day-ahead electricity market. The objective of this model is to maximize the net revenue of NMG by coordinating the joined individual microgrids to submit aggregated offers/bids to the market operator. A hybrid stochastic-robust optimization framework is developed to offset multiple associated uncertainties. The bidding and offering model is first formulated as a hard-to-solve mixed-integer nonlinear programming (MINLP) problem, which is later converted to its easy-to-solve mixed-integer linear programming (MILP) counterpart. To resolve privacy concerns of each microgrid and improve the scalability of the proposed bidding and offering model, a coordinated scheduling framework for NMG based on the Dantzig-Wolfe decomposition (DWD) method is proposed to obtain the global optimum. Numerical simulations with real-world measured data validate the effectiveness of the proposed price-maker bidding model, which is shown to outperform existing price-taker models.

Electrifying light-duty passenger transport for CO[formula omitted] emissions reduction: A stochastic-robust input–output linear programming model

Electrification of light-duty passenger vehicles has been widely considered as an important strategy for decarbonizing the transport sector. This study proposes a novel input–output linear programming model with stochastic-robust optimization methodology to identify the optimal pathways to reduce direct and embodied emissions from light-duty passenger transport under technology cost and emission intensity uncertainties. The model introduces detailed transport technologies into a well-known economic input–output model so that the carbon dioxide (CO2) emissions embodied in production inputs of vehicles in the entire economic system can be fully accounted for. We apply the model to a case study in China, which shows that tightening cumulative CO2 emissions from light-duty passenger transport by 30% of the base case requires large deployment of electric vehicles, especially plug-in hybrids and battery electric vehicles. The emission reductions mainly come from vehicle operation-related emissions. Nevertheless, the total reduction in operation-related emissions is offset by the increase in capital-related CO2 emissions by 13%–18%, depending on the variability in future emission intensity of electricity. Finally, we compare the data uncertainty handling performance of stochastic-robust model with a stochastic programming model and two deterministic optimization models under optimistic and pessimistic parameter settings. The comparison shows that the optimal technology portfolio generated from the proposed model is not only reliable in achieving the predefined emissions target, but is also able to reduce the total investment and operation costs in the light-duty passenger transport sector under data uncertainty.