Two-stage Stochastic Optimization Research Articles

A two-stage collaborative optimization method for emergency repair station planning and recovery strategy of cyber-physical-transportation networks

In this paper, a two-stage collaborative stochastic optimization method for pre-disaster emergency repair station planning and post-disaster recovery strategy of cyber-physical-transportation networks is proposed in this paper to enhance the resilience under extreme scenarios. In the first stage, based on the principle of hierarchical portioning, the responsible area for the emergency repair station is divided by the community theory, and the emergency repair station planning model is developed. In the second stage, the time sequence of repairing the faulty components is first established based on the graph theory. Then, based on the spatial distribution of adjacent time-series faulty components on the transportation network, a faulty component restoration time model is established to accurately reflect the faulty restoration time. On this basis, the functional coupling effects of economic dispatch control and substation automation system failures on load recovery are analyzed to establish a load recovery model. The case studies of the IEEE RTS-79 test system demonstrate that, compared with traditional independent recovery model, the model proposed in this paper performs better in enhancing resilience under extreme scenarios.

A Revisit to Sunk Cost Fallacy for Two-Stage Stochastic Binary Decision Making

This paper undertakes a revisit of the sunk cost fallacy, which refers to the tendency of people to persist investing resources into something, even if it is destined to have no good outcome. We emphasize that the utilities associated with different alternatives are not static for decision makers, which is exactly opposite to the traditional perspective. This paper argues that the utility of an option may change due to the choice of another option, suggesting that decisions considered irrational by the traditional analytical method, i.e., sunk cost fallacy, may be rational. We propose a novel analytical method for decision making with sunk cost when considering the utility change and validate the effectiveness of this method through mathematical modeling and computational experiments. This paper mathematically describes such decision-making problems, analyzing the impact of changes in the utilities across different alternatives on decision making with a real-world example. Furthermore, we develop a two-stage stochastic optimization model for such decision-making problems and employ the sample average approximation (SAA) method to solve them. The results from computational experiments indicate that some decisions traditionally considered irrational are, in fact, rational when the utility of an option changes as a result of choosing another option. This paper, therefore, highlights the significance of incorporating utility changes into the decision-making process and stands as a valuable addition to the literature, offering a refreshed and effective decision-making method for improved decision making.

Designing an optimal multi-energy system with fast charging and hydrogen refueling station under uncertainties

As the demand for battery-powered and hydrogen-based electric vehicles continues to rise, the need for integrated charging and refueling systems with energy systems becomes more critical. Therefore, it is crucial to consider the stations serving diverse vehicle users during the system design process. This paper presents an optimal sizing strategy for a multi-energy system that covers clean energy sources and two energy storage technologies under the presence of a fast-charging station and a hydrogen refueling station for various types of car users. For the considered system, the optimal sizing problem aims to minimize the loss of power supply probability (LPSP) and annualized total life cycle cost (TLCC) through a two-stage stochastic programming-based multi-objective optimization approach. Using a scenario-based approach, the study considers the uncertainties associated with renewables, homeowner’s vehicle demands, and energy demands from homes and stations. Different cases are produced to evaluate the impacts of the different stations with the availability of various electric vehicle profiles under different levels of reliability. The TLCC decreases if LPSP increases in each case. Also, the cost of installing a fast charging station is significantly lower than that of a hydrogen refueling station for all LPSP limits. Moreover, the overall cost of the system design is at its lowest when homeowners’ vehicles provide vehicle-to-grid services. It is worth noting that when the station loads which come from the customers’ vehicle increase by 10%, the TLCC increases by 1.19%. Additionally, the revenue from hydrogen refueling station is approximately 70% less than the fast charging station.

Stochastic optimization scheduling of wind-photovoltaic-storage-hydrogen-water system in independent island zero-carbon microgrid

A stochastic optimal scheduling strategy is being proposed for a wind-photovoltaic-storage-hydrogen-water system within a zero-carbon microgrid for independent islands. This strategy is aimed at addressing the challenge of sustainable energy resource supply prevalent in traditional islands, while also considering the demand response of seawater desalination. In order to harness the clean, low-carbon, and high-efficiency characteristics of hydrogen, an electrolytic water hydrogen production system is integrated into the power system to utilize wind power generation. The desalination equipment is regarded as a controllable load within the system. It is combined with a demand response strategy to maximize the full utilization of the desalination equipment for consuming wind power on the island. Furthermore, a two-stage stochastic optimization method is being employed to handle uncertainties related to wind and photovoltaic generation. The objective is to minimize the system’s operating cost. To solve this optimization problem, the commercial solver CPLEX is being utilized. Through comparative analysis across different scenarios, simulation results indicate that the proposed strategy effectively dissipates uncertainties in wind scenarios while concurrently reducing the overall system operation cost. This approach comprehensively considers the system’s economics, thus demonstrating its potential to efficiently promote wind power consumption.

A two-stage stochastic optimization framework for retail supply chain modeling with contemporaneous resilient strategies

A two-stage stochastic optimization framework for retail supply chain modeling with contemporaneous resilient strategies

Two-stage stochastic optimization of virtual power plant with wind power and uncertain demand

ABSTRACT The virtual power plant (VPP) has been recognized as an effective way to facilitate penetration of renewable and distributed energy resources in electricity markets. This paper introduces an adaptive curtailment strategy for a VPP comprising a wind power plant and uncertain demand, and explores the economic advantage of adaptively adjusting wind power curtailment. A two-stage stochastic model is proposed to deal with the uncertainties in wind power generation (WPG), load demand and market prices. In the model, the bid decision is made in the face of uncertainties in the first stage, while the control (curtailment) decision is made based on realized uncertain parameters in the second stage. This paper provides the closed-form optimal curtailment decision and characterizes the optimal bid decision. An efficient binary search algorithm is developed for optimizing the bid decision. By using a distribution-free approach, we show that as the prediction accuracy of WPG improves, the optimal bid decision converges toward the expected minimal power exchange, leading to a decrease in expected operational cost with diminishing marginal return. Numerical experiments based on real-world data demonstrate that compared with the existing greedy strategy and coordinated strategy, the proposed model can decrease the expected operational cost up to 16.9% and 11.0%, respectively.

An enhanced stochastic optimization for more flexibility on integrated energy system with flexible loads and a high penetration level of renewables

Full utilization of the potential regulation ability of flexible loads in an integrated energy system (IES) and expanding its structure for more flexibility via integrating auxiliary devices is the key to realizing its high-efficiency operation and high-proportion renewable consumption. The present paper builds the flexible regulation models of thermal and electrical loads and introduces them to the structure expansion planning of IES. A two-stage stochastic probability optimization method integrating the uncertain operation of introducing flexible loads is then proposed to balance the additional costs of device integration and the benefits of performance promotion. Herein, an enhanced sample average approximation based on a stochastic hierarchy scenario generation method is developed to solve the optimization. The regulation mechanisms of flexible loads and their influence on expansion planning are then analyzed by comparing the optimized results. The results show that the path to decreasing the cost via introducing flexible thermal and electrical loads into optimization is increasing renewable consumption and coordinating an increment in renewable consumption and average electrical efficiency, respectively. The regulation between them has synergism on cost reduction and saturation on increasing renewable consumption. The synergism can reduce the cost by 0.62 %. Besides, there is a synergism between electric boiler integration and flexible loads, which further reduces the total cost of the expanded IES by 24.86 %.

Techno-economic assessment of isolated micro-grids with second-life batteries: A reliability-oriented iterative design framework

Reusing lithium-ion batteries retired from electric vehicles (EVs) has received great attention as the performance of these batteries is still adequate for many stationary energy storage applications, such as micro-grids (MGs). To date, the economic and technical performance of second-life batteries (SLBs) in isolated MG systems remains obscure. Hence, this study focuses on the economic assessment of SLBs in an isolated 100% renewable MG, in order to identify a profitable application that promotes SLBs. The MG capacity design parameters are obtained by a two-stage chance-constrained stochastic optimization framework. To enhance the robustness of the MG design, a reliability-oriented iterative design process is proposed. This process involves continuous updates to the MG design, guided by validation results, until all specified requirements are satisfied. The lifetime of SLBs is determined using a semi-empirical degradation model seamlessly integrated into the validation process. Economic comparisons between SLBs and first-life batteries (FLBs) are conducted under identical design requirements, with consideration given to estimating the “price ceiling” and the impact of market factors. The results show that SLBs prove to be a more cost-effective option, especially when there is a demand for high reliability standards and the local electricity prices are comparatively lower.

Optimal Prosumer Operation with Consideration for Bounded Rationality in Peer-to-Peer Energy Trading Systems

With the large-scale development of distributed energy on the demand side, the trend of “supply exceeding demand” has gradually become prominent, and regional peer-to-peer (P2P) energy trading has become an important measure to improve the local consumption of distributed energy. However, most existing studies usually assume that prosumers behave entirely rationally with the goal of maximum benefit, which has been proved to deviate from the observed actual behaviors. Aiming at the optimal energy of prosumers participating in the P2P market, a prospect theory-based two-stage stochastic optimization model considering the bounded rationality was proposed to accurately simulate the decision-making behavior. Then, a benefit maximization model for the energy trading service provider (ETSP) was constructed considering the power flow constraint to ensure the safe operation of the system. Finally, an improved R-ADMM algorithm considering timeout was proposed to solve the above model and improve the convergence speed. The effectiveness of the proposed model and algorithm was verified via simulation.

Optimal Operation and Market Integration of a Hybrid Farm with Green Hydrogen and Energy Storage: A Stochastic Approach Considering Wind and Electricity Price Uncertainties

In recent years, growing interest has emerged in investigating the integration of energy storage and green hydrogen production systems with renewable energy generators. These integrated systems address uncertainties related to renewable resource availability and electricity prices, mitigating profit loss caused by forecasting errors. This paper focuses on the operation of a hybrid farm (HF), combining an alkaline electrolyzer (AEL) and a battery energy storage system (BESS) with a wind turbine to form a comprehensive HF. The HF operates in both hydrogen and day-ahead electricity markets. A linear mathematical model is proposed to optimize energy management, considering electrolyzer operation at partial loads and accounting for degradation costs while maintaining a straightforward formulation for power system optimization. Day-ahead market scheduling and real-time operation are formulated as a progressive mixed-integer linear program (MILP), extended to address uncertainties in wind speed and electricity prices through a two-stage stochastic optimization model. A bootstrap sampling strategy is introduced to enhance the stochastic model’s performance using the same sampled data. Results demonstrate how the strategies outperform traditional Monte Carlo and deterministic approaches in handling uncertainties, increasing profits up to 4% per year. Additionally, a simulation framework has been developed for validating this approach and conducting different case studies.

Virtual Power Plant’s Optimal Scheduling Strategy in Day-Ahead and Balancing Markets Considering Reserve Provision Model of Energy Storage System

In recent years, with the rapid increase in renewable energy sources (RESs), a Virtual Power Plant (VPP) concept has been developed to integrate many small-scale RESs, energy storage systems (ESSs), and customers into a unified agent in the electricity market. Optimal coordination among resources within the VPP will overcome their disadvantages and enable them to participate in both energy and balancing markets. This study considers a VPP as an active agent in reserve provision with an upward reserve capacity contract pre-signed in the balancing capacity (BC) market. Based on the BC contract’s requirements and the forecasted data of RESs and demand, a two-stage stochastic optimization model is presented to determine the VPP’s optimal scheduling in the day-ahead (DA) and balancing energy (BE) markets. The probability of reserve activation in the BE market is considered in this model. The ESS’s reserve provision model is proposed so as not to affect its schedule in the DA market. The proposed optimal scheduling model is applied to a test VPP system; then, the effects of the BC contract and the probability of reserve activation on the VPP’s trading schedule are analyzed. The results show that the proposed model has practical significance.

A two-stage stochastic collaborative planning approach for data centers and distribution network incorporating demand response and multivariate uncertainties

As a critical infrastructure to support the development of digital economy, the scale of data center (DC) power consumption has been growing continuously in recent years. The plug-in of a large number of DCs to the distribution network (DN) presents both opportunities and challenges for the secure, stable, and sustainable operation of the system. In order to address the planning issues arising from the integration of DCs into the DN, this paper proposes a two-stage (planning stage and operation stage) collaborative planning method for DCs and the DN while considering the source-demand uncertainty. Firstly, spatial-temporal demand response characteristic of DCs is explicitly elaborated, and based on the energy consumption model, a two-stage stochastic planning model considering conditional value at risk (CVaR) for DCs and the DN is constructed using probabilistic stochastic simulation, which obeys low-carbon and security, and satisfies the source-network-load constraints of the system. The siting and sizing for DCs and renewable generation, feeder expansion for the DN are determined to minimize the investment cost during the planning stage, while the power output for the devices considering uncertainties from load and renewable energy are optimized to minimize the operation cost during the operation stage. Secondly, the L-shaped decomposition algorithm is proposed to solve the two-stage stochastic optimization problem by reconstructing the problem into a master problem and subproblems, where the dual multiplies are calculated to generate feasibility cuts and optimality cuts to achieve iterative solving. Finally, numerical case studies on a 33-bus distribution system demonstrate that the participation of DCs in demand response can achieve a significant reduction in the investment and operation cost of the DN, and further promote the consumption of distributed renewable energy sources.

Global multi-sourcing network design with inventory planning under uncertainty

ABSTRACT In the current era of turbulence, demand uncertainty and significant supply delays have become recurrent challenges for every global supply chain, which are accentuated by geopolitical issues, regional conflicts, natural disasters, etc. Companies are keen to decide when and where nearshoring and offshoring their supply locations and to adapt their inventory control policies to cope with significant perturbations. To deal with these challenges, we investigate the integration of inventory planning at the strategic level through a novel setting of the location-inventory problem under uncertainty. This problem is characterised by inbound multi-sourcing, order splitting among multiple suppliers, and optimised inventory decisions from sourcing to customers under uncertain demand and lead time. A two-stage stochastic optimisation model is proposed and solved using a Benders decomposition algorithm, supported by a sample average approximation approach. The results show that both lead time and demand uncertainties impact strategic and tactical decisions of the global supply network. Specifically, multi-sourcing and inbound order splitting policies have proven beneficial when the network faces a high lead time uncertainty. Based on an illustrative case of a global automobile spare parts supply network under different supply-side scenarios, managerial insights are gained on the benefit of onshore, nearshore, and offshore decisions. When facing a high lead time uncertainty, as in the pandemic case or any geopolitical episode, our findings suggest the necessity to rely on nearshoring for the supply side.

A holistic two-stage decision-making methodology for passive and active building design strategies under uncertainty

In the last decade, many studies focused on finding optimal design solutions considering trade-offs between different aspects of building design. Accordingly, multi-objective optimization (MOO) approaches have been increasingly applied in the building industry. However, certain aspects must be deepened to ensure a more effective decision-making process in the early planning phase. On the one hand, uncertainties should be considered before making decisions to ensure the robustness of the optimal solutions; on the other hand, decisions are made at different times in building planning, and the sequential order of making decisions should be modeled. This paper presents a holistic two-stage multi-objective stochastic optimization (MOSO)-II framework to minimize the environmental impact of global warming potential (GWP) and cost throughout the entire life cycle of a building (phases of A1-A3, B4, B6 and C3-C4), considering passive and active design strategies in two consecutive stages, under uncertainty. Herein, individual/use and political/market uncertainties are considered. As a proof of concept, the proposed framework is applied in a case study for a generic zone in a multi-family terraced residential building type with solid brick construction. The advantages of the proposed framework are validated by comparing it with alternative single-stage MOSO frameworks. Results show that the proposed two-stage MOSO-II framework can deliver a smaller range of solutions with a better performance in terms of lower GWP and cost. It indicates that the proposed framework can effectively assist planners in decision-making by reducing the effort in choosing the proper solution. Secondly, the results also emphasize the importance of passive design strategies in sustainable building planning. In addition, the energy mix structure and cost of energy sources should be carefully adjusted in the future to promote a more ecologically sustainable building design.

Day-ahead bidding and dispatch strategy of novel battery charging and swapping station under multiple uncertainties

The novel battery charging and swapping station (NBCSS) has great operational flexibility due to its integration of wind power, photovoltaic power, gas turbine and energy storage. This paper presents a day-ahead bidding and dispatch strategy of NBCSS under multiple uncertainties, which consists of battery demand, market price, renewable energy and load. Firstly, the optimized backup method is used to address the uncertainty of battery demand. Then, a two-stage stochastic robust optimization model is further applied to address the uncertainties of market price, renewable energy and load. The model’s goal is to minimize the total operational cost in the worst-case scenario, which is limited by the corresponding uncertainty set. Especially, to control the conservatism of the model, the total adjustable budget and temporal adjustable budget of uncertainties are considered together. Next, the two-stage model is solved efficiently based on the strong duality theory and column-and-constraint generation (C&CG) method. Finally, case studies are given to verify the effectiveness and performance of the proposed model. The results demonstrate that the NBCSS can be efficiently deployed, enabling market arbitrage and reducing the overall cost. Additionally, the conservatism of the proposed model can be adjusted according to the budgets of uncertainties.

An operational scheduling framework for Electric Vehicle Battery Swapping Station under demand uncertainty

This paper proposes a novel mathematical optimization model aimed at incorporating demand response strategies in the operational scheduling problem of Electric Vehicle (EV) Battery Swapping Stations (BSS). The primary goal of the model is to minimize the operational costs while accounting for intricacies arising from diverse charging modes, variations in battery capacities, charger availability, different states of charges (SOC) of incoming discharged batteries, time-of-use power prices, power constraints and fluctuating customer demand patterns. Furthermore, the model takes into consideration the effect of faster charging modes on battery health to ensure a sustainable operation. The proposed model is further extended to incorporate the effect of swapping demand uncertainty on the operational decision-making of the BSS, and a two-stage stochastic optimization model with a recourse approach is proposed. The proposed stochastic optimization model is tested with relevant case studies, and the results demonstrate the efficacy of the model in providing a risk-neutral solution by determining the optimal demand satisfaction pattern and calculating the number of additional batteries needed to minimize demand shortfalls across all scenarios. The proposed scheduling framework holds the promise of improving both the efficiency of the BSS operations and customer satisfaction, paving the way for accelerated EV adoption.

Two-stage stochastic robust optimization scheduling of electric–thermal microgrid with solid electric thermal storage

During the periods of high heat load (HL) demand in winter, the increased heat output of combined heat and power units (CHPs) can significantly compress the electric power regulation range, thereby posing a threat to the safety of the power system. The solid electric thermal storage (SETS) can be employed as the regulating resource for both electric and thermal systems, expanding the dispatch space of microgrids to promote renewable energy consumption. In this paper, a two-stage stochastic robust optimization scheduling model of an electric–thermal microgrid with SETS is proposed, and the electric–thermal bi-directional regulation characteristics of SETS are considered. First, a SETS operation model based on the heat transfer characteristics of the homogeneous material-type thermal storage unit is established. Second, a regional thermal inertia model under a constant-flow variable-temperature system is established, which integrates HL at different locations to the heat source, avoiding the calculation of variation in water temperature, thus reducing the calculation difficulty. Finally, the two-stage stochastic robust optimization scheduling model of an electric–thermal microgrid with SETS is established. The model decouples the power and heat generation of CHP through the bi-directional regulation function of SETS. Case studies demonstrate the validity and effectiveness of the proposed scheduling model.

Constrained Online Two-stage Stochastic Optimization: Algorithm with (and without) Predictions

Constrained Online Two-stage Stochastic Optimization: Algorithm with (and without) Predictions

Multi-Objective Optimization of Car Sharing Points Under Uncertainty for Sustainable Transportation

As an important part of the sharing economy, the usage of car sharing increases world widely with the help of developments in the technology. Especially after COVID-19 the demand for private car ownership and car sharing systems increased tremendously. Therefore, its market share attracts new investors and causes existing service providers to enlarge their service area. In this article, a novel multi-objective location-dependent two-stage stochastic optimization model is proposed to determine the most appropriate locations for car sharing system and allocate the demand to these locations. The model is applied to determine the best locations among 15 candidates, and three objectives are considered, which are the minimization of total cost that comprises locating costs minus income from satisfying the demand, minimization of CO <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$_2$</tex></formula> emission occurs by the usage of car sharing system's cars and minimization of average unsatisfied demand. Both location-independent and location-dependent demands are taken into account. The proposed model delivers a more precise decision process framework for problems include stochasticity and multiobjectivity, and it easily can be implemented to any region, providing region sensitive parameters.

Electrolysis as a flexibility resource on energy islands: The case of the North Sea

Energy islands are meant to facilitate offshore sector integration by combining offshore wind energy with power-to-x technologies and storage. In this study, we investigate the operation of electrolysers on energy islands, assess their potential contribution to flexibility provision, and analyse different market integration strategies of the islands. For this purpose, a two-stage stochastic optimisation model is developed to determine the cost-efficient dispatch for an integrated day-ahead and balancing electricity market. For the market integration of the energy island, we align our approach to the current debate and compare the case of a single offshore bidding zone to a case where the energy island is integrated into a home market zone. We find that electrolysers on energy islands will run at low capacity factors and provide flexibility in 29–36% of their run time. In addition, offshore electrolysers produce more hydrogen when they are allocated to an offshore bidding zone, and thus earn higher profits. We conclude that combining offshore wind with electrolysers on an energy island relies on additional economic incentives if their main role is envisioned to be the delivery of balancing flexibility.