Two-stage Stochastic Optimization Model Research Articles

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.

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.

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.

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.

Flexible storage yard management in container terminals under uncertainty

As the port bottleneck shifts from the shore side to the yard side, yard management is crucial to the operation efficiency of the container port. This paper studies the yard management problem in container terminals under the flexible storage strategy which does not fix the size of the storage templates in the yard. The problem considers the uncertainty of container demand and the flexibility of the block and jointly optimizes the storage space allocation, yard crane deployment, and container allocation problems. A two-stage stochastic optimization model under uncertainty is established to minimize the port operation cost. Two heuristic algorithms, a column generation-based heuristic algorithm and a tabu search algorithm, are proposed to solve the model. Numerical experiments validate the effectiveness of the proposed algorithms. Furthermore, some management implications for port operators are proposed according to the results of sensitivity analysis experiments.

Stochastic generation maintenance scheduling with inertia-dependent primary frequency regulation constraints

The problem of insufficient inertia and frequency security becomes a critical concern in generation maintenance scheduling due to the increasing penetration of renewable energy sources (RESs). To address the issue, this paper presents a stochastic generation maintenance scheduling with inertia-dependent primary frequency regulation constraints. First, inertia-dependent primary frequency regulation constraints consisting of the minimum inertia requirement, frequency nadir, and quasi-steady-state frequency deviation are formulated based on the dynamic frequency characteristic of power systems in response to large disturbances. Then, a two-stage stochastic optimization model for generation maintenance scheduling including frequency security constraints is proposed. Finally, a solution methodology based on Benders decomposition (BD) is used to promote the calculation speed of the proposed model. Numerical simulations demonstrate that the proposed generation maintenance scheduling could address the potential frequency security issues and promote integration of RESs associated with calculation speed.

Cooperative mechanisms for multi-energy complementarity in the electricity spot market

Promoting a diversified and sustainable energy mix in the electricity market necessitates the implementation of multi-energy complementarity. However, the absence of effective cooperative mechanisms among diverse power sources causes a significant challenge in maximizing the overall economic benefits of multi-energy complementarity and fostering individual cooperative willingness. This paper employs cooperative game theory to investigate the joint offering and operation of multiple complementary power sources in the electricity spot market, including wind, thermal, and pumped storage. A two-stage stochastic optimization model is established to co-optimize offering strategies at the day-ahead stage and operation policies at the balancing stage for the multi-energy alliance. The Shapley value is used to allocate profits fairly among various power sources. Simulation results show that two- and three-party alliances among wind, thermal, and pumped storage can lead to multiple-win situations in the spot market. The proposed model effectively couples the day-ahead and balancing markets, resulting in higher profits compared to the scenario without coupling. Both wind-thermal and wind-thermal-pumped storage alliances prove highly effective, with a nearly 100% reduction in imbalance power and approximately a 12% increase in wind power absorption. Relatively, the latter alliance performs slightly worse than the former. Furthermore, the wind-thermal-pumped storage alliance achieves a 4.48% carbon emission reduction, surpassing the wind-thermal and wind-pumped storage alliance. This study reveals the cooperation mechanism and its influencing factors among diverse power sources. It provides valuable decision support for stakeholders to achieve effective multi-energy complementarity, mitigate imbalance power, reduce carbon emissions, and increase renewable energy absorption.

Two-stage travel itinerary recommendation optimization model considering stochastic traffic time

In an increasingly competitive market, most online tour operators have launched personalized travel itinerary recommendation services. Different from previous studies, this research considers the travel itinerary planning problem under total time limit and uncertain traffic time. This problem requires two stages of decision-making: firstly, selecting the tourist attractions to visit from the candidate attractions based on maximizing the popularity utility of tourists; secondly, planning the tourist attraction visit sequence under stochastic traffic time to maximize the tourist activity utility. Therefore, a two-stage stochastic optimization model with chance constraint is constructed and then solved by the sample average approximation (SAA) method. In order to verify the effectiveness of our model, we introduced two benchmark models for comparative analysis. The results show that in the worst case, our model increases the reliability level of travel itineraries by almost 40% compared with the two benchmark models. In addition, case studies of two large cities, i.e., Nanjing and Beijing, indicate that a tour operator can optimize travel itinerary recommendations by improving tourists utility as well as their own profitability without much loss of reliability.

Evaluation of network expansion decisions for resilient interdependent critical infrastructures with different topologies

Resilient interdependent critical infrastructures (CIs) can better withstand cascading failures in disruptive events. This study proposes network expansion as a resilience improvement strategy for interdependent CIs and evaluates the influence of topology in interdependent network design for resilience optimization under disruption uncertainty. A resilience score consisting of network complexity and unmet demand metrics is introduced to quantify the resilience of expanded networks. Five synthetic interdependent network instances with random and hub-and-spoke (i.e., cluster) topologies are generated to represent CIs with heterogeneous node functions. Different network expansion opportunities are considered and critical node disruption scenarios are used to evaluate the impact of uncertain disruptions. We apply a two-stage stochastic multi-objective resilience optimization model to determine strategic investment decisions using the expected total cost and expected resilience score as competing objectives. Compromise solutions of expanded network designs are identified from Pareto optimal solutions and they are characterized according to their graph properties. The results show that expanded networks have improved resilience and the extent of improvement is affected by the network topology and type of disruption. Under critical node disruptions, a random network is more resilient than a hub-and-spoke structure due to its better connectivity. Characteristics of highly connected interdependent networks are high average node degree, high clustering coefficient, and low average shortest path length. Resilience improvement is more limited in expanded networks with a hub-and-spoke structure due to the negative impact of hub failures.

Generative adversarial network assisted stochastic photovoltaic system planning considering coordinated multi-timescale volt-var optimization in distribution grids

In this paper, a novel photovoltaic (PV) system planning framework is proposed for distribution grids. The main contribution of this work is that the volt-var control (VVC) capability of PV inverters is duly considered during the planning stage to reduce the expected power loss cost and meanwhile counteract uncertain voltage fluctuation and deviation caused by random PV production and load demand. Specifically, the proposed PV system planning framework is formulated as a two-stage stochastic optimization model, where the first stage is to determine the planning decisions of PV systems before the uncertainty realization, and after the uncertainty is observed, the second stage coordinates PV inverters and existing legacy VVC devices (capacitor banks) to minimize the power loss and voltage fluctuation in a multi-timescale manner. To solve the formulated complex optimization problem, the original model is decoupled and solved using an iterative solution method where commercial solvers can be directly used to obtain the optimal solutions. Besides, a data-driven based scenario selection method based on Generative Adversarial Network (GAN) is proposed to capture the uncertainties of PV production and load demand with the full diversity of behaviours. Finally, the proposed framework is tested on IEEE 37-node and 123-node test systems to demonstrate the effectiveness of the proposed method. This paper can provide important insights into investment strategies of renewable resources designed by power grid authorities, which can benefit from the proposed framework to maintain the reliable system operation and reduce the renewable energy curtailment.

A two-stage stochastic optimization model for port infrastructure planning

A two-stage stochastic optimization model for port infrastructure planning

Stochastic expansion planning of integrated energy system: A benders-based decomposition approach

This paper introduces a two-stage stochastic optimization model for the renewables-transmission expansion planning (RTEP) problem to obtain the optimal size and site of the renewable energy generating unit, as well as the investment decisions related to transmission lines and gas pipelines. The uncertainties associated with demand growth and stochastic renewable sources production levels are considered via a series of scenarios, and then, the scenario-reduction approach, i.e., the k-medoids method, is applied to reduce the computation burden. In addition, the gas flow-pressure difference steady-state formulations in the expansion planning model are reconstructed convexly using second-order cone constraints. A Benders-based decomposition method is adopted as a solution methodology in the co-planning problem. The proposed model was simulated on a modified IEEE 39-bus and Belgian gas-20 energy test network, and the simulation results validate the correctness and applicability of the presented approaches.

A two-stage stochastic optimization framework to allocate operating room capacity in publicly-funded hospitals under uncertainty.

Surgery demand is an uncertain parameter in addressing the problem of surgery block allocations, and its typical variability should be considered to ensure the feasibility of surgical planning. We develop two models, a stochastic recourse programming model and a two-stage stochastic optimization (SO) model with incorporated risk measure terms in the objective functions to determine a planning decision that is made to allocate surgical specialties to operating rooms (ORs). Our aim is to minimize the costs associated with postponements and unscheduled demands as well as the inefficient use of OR capacity. The results of these models are compared using a case of a real-life hospital to determine which model better copes with uncertainty. We propose a novel framework to transform the SO model based on its deterministic counterpart. Three SO models are proposed with respect to the variability and infeasibility of the measures of the objective function to encode the construction of the SO framework. The analysis of the experimental results demonstrates that the SO model offers better performance under a highly volatile demand environment than the recourse model. The originality of this work lies in its use of SO transformation framework and its development of stochastic models to address the problem of surgery capacity allocation based on a real case.

A Bi-objective two-stage stochastic optimization model for sustainable reverse supply chain network design under carbon tax policy and government subsidy considering product quality

ABSTRACT There is a growing concern over environmental pollution resulting from the production. Moreover, the expansion of various industries has led to an increased demand for raw materials. To address these challenges, this paper aims to investigate the bi-objective optimization of a sustainable reverse supply chain network while considering two key sources of uncertainty in the returned product quality and the remanufacturing capacity. Additionally, the study considers the effects of carbon tax policies and government subsidies on remanufactured products, while also focusing on three important sustainability aspects - economic, social, and environmental. The study uses two quality thresholds at inspection centers to sort products, and the epsilon constraint and NSGA-II are applied to solve the model. Through numerical analysis, the research demonstrates that objective functions are sensitive to uncertain parameters and minimum acceptable quality levels. Furthermore, the study reveals that government subsidies can offset the negative effects of carbon tax policies.

Practicable robust stochastic optimization under divergence measures with an application to equitable humanitarian response planning.

We seek to provide practicable approximations of the two-stage robust stochastic optimization model when its ambiguity set is constructed with an f-divergence radius. These models are known to be numerically challenging to various degrees, depending on the choice of the f-divergence function. The numerical challenges are even more pronounced under mixed-integer first-stage decisions. In this paper, we propose novel divergence functions that produce practicable robust counterparts, while maintaining versatility in modeling diverse ambiguity aversions. Our functions yield robust counterparts that have comparable numerical difficulties to their nominal problems. We also propose ways to use our divergences to mimic existing f-divergences without affecting the practicability. We implement our models in a realistic location-allocation model for humanitarian operations in Brazil. Our humanitarian model optimizes an effectiveness-equity trade-off, defined with a new utility function and a Gini mean difference coefficient. With the case study, we showcase (1) the significant improvement in practicability of the robust stochastic optimization counterparts with our proposed divergence functions compared to existing f-divergences, (2) the greater equity of humanitarian response that the objective function enforces and (3) the greater robustness to variations in probability estimations of the resulting plans when ambiguity is considered.

Reliability-informed end-of-use decision making for product sustainability using two-stage stochastic optimization

The concept of circular economy has been diffused in recent decades to promote economic growth that does not add to the burden on natural resource extraction. Re-X options (e.g., reuse, repair, refurbish, remanufacture, recycle) have been gradually adopted in the product development process and optimized to reduce or eliminate waste and pollution. Although manufacturing incorporating Re-X options can be more environmentally friendly, it involves more sources of uncertainty than traditional manufacturing since the end-of-use products can be collected from multiple origins with various quantities and qualities, and the market demand for both new and remanufactured products cannot be forecasted perfectly. Thus, there is a need to optimize the Re-X policy to alleviate the negative impacts of the higher uncertainty. One option is using the reliability information of new products to estimate the end-of-use conditions and applying multi-stage stochastic optimization to capture multiple demand scenarios. This paper develops a two-stage stochastic optimization model to optimize the quality thresholds for reuse, recycling, and remanufacturing options. Our objective is to minimize the total cost, energy consumption, and environmental impact of producing and providing warranty service for a product family. The model employs reliability information of product components to estimate the warranty service cost and the end-of-use conditions of the returned resources. A case study on a general product family is implemented to illustrate the efficacy of the optimization model. Results show that the two-stage optimization can achieve cost and environmental impact reduction for a hybrid manufacturing and remanufacturing process.

Optimal ship lifetime fuel and power system selection under uncertainty

Ship designers face increasing pressure to comply with global emission reduction ambitions. Alternative fuels, potentially derived from bio-feedstock or renewable electricity, provide promising solutions to this problem. The main challenge is to identify a suitable ship power system, given not only uncertain emission requirements but also uncertain fuel and carbon emission prices. We develop a two-stage stochastic optimization model that explicitly considers uncertain fuel and carbon emission prices, as well as potential retrofits along the lifetime. The bi-objective setup of the model shows how the choice of optimal power system changes with reduced emission levels. Methanol and LNG configurations appear to be relatively robust initial choices due to their ability to run on fuel derived from different feedstocks, andtheir better retrofittability towards ammonia or hydrogen. From a policy perspective, our model provides insight into the effect ofthe different types ofcarbon pricing mechanismson a shipowner’s decisions.