Two-stage Stochastic Optimization Research Articles

Submodular reassignment problem for reallocating agents to tasks with synergy effects

We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given k submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s representation theorem for distributive lattices.

A multicountry, multicommodity stochastic game theory network model of competition for medical supplies inspired by the Covid-19 pandemic

In this paper, we construct the first stochastic Generalized Nash Equilibrium model for the study of competition among countries for limited supplies of medical items (PPEs, ventilators, etc.) in the disaster preparedness and response phases in the Covid-19 pandemic. The government of each country is faced with a two-stage stochastic optimization problem in which the first stage is prior to the pandemic declaration and the second stage is post the pandemic declaration. We provide the theoretical constructs, a qualitative analysis, and an algorithm, accompanied by convergence results. Both illustrative examples are presented as well as algorithmically solved numerical examples, inspired by the need for N95 masks and ventilators. The results reveal that, in addition to the preparedness of countries before the pandemic declaration, their ability to adapt to the conditions in different scenarios has a significant impact on their overall success in the management of the pandemic crisis. The framework can capture competition for other medical supplies, including Covid-19 vaccines and possible treatments, with modifications to handle perishability.

Retail supply chain network design with concurrent resilience capabilities

In this paper, we address the problem of designing/redesigning a resilient retail supply chain network under uncertainty. Considering pre- and post-disruption scenarios, this study proposes a two-stage stochastic optimization framework with multiple resilience strategies for designing a resilient retail supply chain. We consider facility fortification, inventory prepositioning, direct-to-store delivery, reserved capacity, inventory sharing and multiple set covering as our resilience strategies, covering all the three dimensions of resilience capabilities, namely, proactive, reactive, and network design quality. Taking into account the prominent role of retail outlets during Covid-19 epidemic outbreaks, through a real case study, we evaluate the impact of random and targeted disruptions on the performance of the retail supply chain network and provide managerial insights regarding the right balance between cost efficiency and resilience. Furthermore, we evaluate the effectiveness of these resilience capabilities in a number of simulated scenarios to show their applicability and importance under different disruptive situations. We show that when implemented concurrently, these resilience capabilities have a synergistic effect where the whole becomes larger than the sum of its parts. Subject to extensive stress testing, our results show that using any mixture of these resilience capabilities increases the retail network's resilience while considerably decreasing post-disruption costs.

Robust Management of Systemic Risks and Food-Water-Energy-Environmental Security: Two-Stage Strategic-Adaptive GLOBIOM Model

Critical imbalances and threshold exceedances can trigger a disruption in a network of interdependent systems. An insignificant-at-first-glance shock can induce systemic risks with cascading catastrophic impacts. Systemic risks challenge traditional risk assessment and management approaches. These risks are shaped by systemic interactions, risk exposures, and decisions of various agents. The paper discusses the need for the two-stage stochastic optimization (STO) approach that enables the design of a robust portfolio of precautionary strategic and operational adaptive decisions that makes the interdependent systems flexible and robust with respect to risks of all kinds. We established a connection between the robust quantile-based non-smooth estimation problem in statistics and the two-stage non-smooth STO problem of robust strategic–adaptive decision-making. The coexistence of complementary strategic and adaptive decisions induces systemic risk aversion in the form of Value-at-Risk (VaR) quantile-based risk constraints. The two-stage robust decision-making is implemented into a large-scale Global Biosphere Management (GLOBIOM) model, showing that robust management of systemic risks can be addressed by solving a system of probabilistic security equations. Selected numerical results emphasize that a robust combination of interdependent strategic and adaptive solutions presents qualitatively new policy recommendations, if compared to a traditional scenario-by-scenario decision-making analysis.

Two-Stage Chance-Constrained Stochastic Thermal Unit Commitment for Optimal Provision of Virtual Inertia in Wind-Storage Systems

The frequency security problem becomes a critical concern in power systems when the system inertia is lowered due to the high penetration of renewable energy sources (RESs). A wind-storage system (WSS) controlled by power electronics can provide the virtual inertia to guarantee the fast frequency response after a disturbance. However, the provision of virtual inertia might be affected by the variability of wind power generation. To address this concern, we propose a two-stage chance-constrained stochastic optimization (TSCCSO) model to find the optimal thermal unit commitment (i.e., economic operation) and the optimal placement of virtual inertia (i.e., frequency stability) in a power grid using representative power system operation scenarios. An enhanced bilinear Benders decomposition method is employed with strong valid cuts to effectively solve the proposed optimization model. Numerical results on a practical power system show the effectiveness of the proposed model and solution method.

A simulation-based decomposition approach for two-stage staffing optimization in call centers under arrival rate uncertainty

We study a solution approach for a staffing problem in multi-skill call centers. The objective is to find a minimal-cost staffing solution while meeting a target level for the quality of service to customers. We consider a common situation in which the arrival rates are unobserved random variables for which preliminary forecasts are available in a first stage when making the initial staffing decision. In a second stage, more accurate forecasts are obtained and the staffing may have to be modified at a cost, to meet the constraints. This leads to a challenging two-stage stochastic optimization problem in which the quantities involved in the (nonlinear) constraints can only be estimated via simulation, so several independent simulations are required for each first-level scenario. We propose a solution approach that combines sample average approximation with a decomposition method. We provide numerical illustrations to show the practical efficiency of our approach. The proposed method could be adapted to several other staffing problems with uncertain demand, e.g., in retail stores, restaurants, healthcare facilities, and other types of service systems.

Two-Stage Stochastic Optimization Via Primal-Dual Decomposition and Deep Unrolling

We consider a two-stage stochastic optimization problem, in which a long-term optimization variable is coupled with a set of short-term optimization variables in both objective and constraint functions. Despite that two-stage stochastic optimization plays a critical role in various engineering and scientific applications, there still lack efficient algorithms, especially when the long-term and short-term variables are coupled in the constraints. To overcome the challenge caused by tightly coupled stochastic constraints, we first establish a two-stage primal-dual decomposition (PDD) method to decompose the two-stage problem into a long-term problem and a family of short-term subproblems. Then we propose a PDD-based stochastic successive convex approximation (PDD-SSCA) algorithmic framework to find KKT solutions for two-stage stochastic optimization problems. At each iteration, PDD-SSCA first runs a short-term sub-algorithm to find stationary points of the short-term subproblems associated with a mini-batch of the state samples. Then it constructs a convex surrogate for the long-term problem based on the deep unrolling of the short-term sub-algorithm and the back propagation method. Finally, the optimal solution of the convex surrogate problem is solved to generate the next iterate. We establish the almost sure convergence of PDD-SSCA and customize the algorithmic framework to solve two important application problems. Simulations show that PDD-SSCA can achieve superior performance over existing solutions.

Cloud Electricity Storage for Multi-Service Battery Operation: Case of RTE

Problem definition: We study a cloud storage operator that provides shared storage service for electricity end-users using the residual part of a multi-service grid-scale battery primarily used for high-priority grid services. We design an optimal product with regard to pricing and customer portfolio. Academic/practical relevance: A framework and solution approach for assessing and operating such multi-service battery operations with stochastic services and different priority levels is an open problem. Methodology: We model the problem as a two-stage stochastic optimization between high-priority stochastic grid services and low-priority cloud storage for stochastic end-users. We also propose operational metrics of multiplexing gain and probability of blocking to assess the operation of the multi-service multi-user battery. To address the computational challenge of solving the stochastic optimization with a large number of end-users, we propose effective capacity as a convex approximation that allows an analytical solution. We then provide an empirical analysis based on real grid congestion data from RTE France and a large dataset of end-users' electricity consumption in California. Results: Our empirical analysis shows (i) our proposed effective capacity is a close approximation, (ii) battery operation and profit are sensitive to the cost of external resources, number of end-users, and the leasing price of the battery, and (iii) with only a slight discount of the leasing price (~1%), the profit of the third party from a stochastic residual battery can be the same as that from a deterministic one. Managerial implications: Cloud storage as a low-priority service can profitably exist alongside other high-priority battery services, making integration of more storage in the grid economically viable and allowing larger intermittent renewables, a key path towards reduced carbon emissions.

A two-stage stochastic optimization model for the retail multiskilled personnel scheduling problem: a k-chaining policy with k≥2.

Considering an uncertain demand, this study evaluates the potential benefits of using a multiskilled workforce through a k-chaining policy with k≥2. For the service sector and, particularly for the retail industry, we initially propose a deterministic mixed-integer linear programming model that determines how many employees should be multiskilled, in which and how many departments they should be trained, and how their weekly working hours will be assigned. Then, the deterministic model is reformulated using a two-stage stochastic optimization (TSSO) model to explicitly incorporate the uncertain personnel demand. The methodology is tested for a case study using real and simulated data derived from a Chilean retail store. We also compare the TSSO approach solutions with the myopic approaches' solutions (i.e., zero and total multiskilling). The case study is oriented to answer two key questions: how much multiskilling to add and how to add it. Results show that TSSO approach solutions always report maximum reliability for all levels of demand variability considered. It was also observed that, for high levels of demand variability, a k-chaining policy with k≥2 is more cost-effective than a 2-chaining policy. Finally, to evaluate the conservatism level in the solutions reported by the TSSO approach, two truncation types in the probability density function (pdf) associated with the personnel demand were considered. Results show that, if the pdf is only truncated at zero (more conservative truncation) the levels of required multiskilling are higher than when the pdf is truncated at 5th and 95th percentiles (less conservative truncation).

A Bi-Level Model for the Bidding Strategy of an Inter-Regional Electricity Trading Aggregator Considering Reliability Requirements and Transmission Losses

In countries with large-scale power systems and nascent electricity markets, the inter-regional electricity trading aggregators (IRETAs) are in charge of participating in inter-regional electricity trading (IRET) on behalf of the generators and loads within the region except for operating the regional power systems (RPSs). As price makers in IRET and system operators of RPSs, IRETAs should make their bidding strategies considering both the RPSs’ reliability requirements and the IRET’s transmission losses to ensure both the safe operation of RPS and the profits in the IRET, which cannot be handled by the current methods. Therefore, this paper investigates an approach to help IRETAs make comprehensive bidding strategies in the IRET. Firstly, a hierarchical framework for an IRETA to participate in IRET is introduced. Then a bi-level model is proposed, in which the optimal operation of RPS and the market-clearing process of IRET are regarded as the upper and lower-level problems. Specifically, the upper-level problem is modeled as a two-stage stochastic optimization problem considering both the fault states and the reliability requirements of RPS. Moreover, the non-negligible losses from the inter-regional electricity transmission are internalized in the lower-level problem to consider their impacts on the IRETA’s bidding strategy-making. The proposed bi-level model is nonlinear and transformed into a linear single-level one through the Karush-Kuhn-Tucker (KKT) conditions and the dual theory. Finally, simulations demonstrate that the bidding strategy formed by the proposed method reduces the number of buses exceeding the limit of reliability requirement by over 50% and cuts down the transmission loss by over 40% compared with the former works.

Efficient Algorithms for Distributionally Robust Stochastic Optimization with Discrete Scenario Support

Recently, there has been a growing interest in distributionally robust optimization (DRO) as a principled approach to data-driven decision making. In this paper, we consider a distributionally robust two-stage stochastic optimization problem with discrete scenario support. While much research effort has been devoted to tractable reformulations for DRO problems, especially those with continuous scenario support, few efficient numerical algorithms were developed, and most of them can neither handle the non-smooth second-stage cost function nor the large number of scenarios $K$ effectively. We fill the gap by reformulating the DRO problem as a trilinear min-max-max saddle point problem and developing novel algorithms that can achieve an $\mathcal{O}(1/\epsilon)$ iteration complexity which only mildly depends on $K$. The major computations involved in each iteration of these algorithms can be conducted in parallel if necessary. Besides, for solving an important class of DRO problems with the Kantorovich ball ambiguity set, we propose a slight modification of our algorithms to avoid the expensive computation of the probability vector projection at the price of an $\mathcal{O}(\sqrt{K})$ times more iterations. Finally, preliminary numerical experiments are conducted to demonstrate the empirical advantages of the proposed algorithms.

Two-Stage Stochastic Optimization of Sodium-Sulfur Energy Storage Technology in Hybrid Renewable Power Systems

Two-Stage Stochastic Optimization of Sodium-Sulfur Energy Storage Technology in Hybrid Renewable Power Systems

A Decision Model for an Electricity Retailer With Energy Storage and Virtual Bidding Under Daily and Hourly CVaR Assessment

This paper presents a short-term decision-making model for an electricity retailer with battery energy storage system (BESS) and virtual bidding through a two-stage stochastic optimization framework. In the first stage, the retailer determines the amount of power to be purchased in the day-ahead wholesale market and the optimal incremental and decremental virtual bidding strategies. In the second stage, the optimal energy storage decisions and the retailer's involvement in the real-time market are determined. The proposed model minimizes the retailer's expected procurement cost and generates the optimal power and virtual bidding curves in the day-ahead market. Two types of Conditional Value at Risk (CVaR) are integrated in the proposed model to manage the retailer's hourly and daily risks, respectively. Case studies with real-world data are performed to verify the retailer's cost reduction obtained with the integration of BESS and virtual bidding and to study how the hourly and daily risk-management strategies affect the retailer's procurement cost distribution for different risk-aversion levels.

Optimal Analysis for Facility Configuration and Energy Management on Electric Light Commercial Vehicle Charging

Whilst the widespread adoption of electric vans is necessary to reduce carbon emissions, it is self-evident that adequate charging stations are a precondition. However, the investment case for basic charging stations without public subsidies is challenging. In the context of a London case study, four business models are compared, which integrate solar generation and new/second-life battery storage system with basic charging facilities. Considering uncertainties of electricity tariff and solar generation, the optimal infrastructure investment and operational planning has been formulated as a two-stage stochastic optimization model. Results show that: (i) in the integrated business models, the return on investment and charger installations could be increased by up to 5.39% and 17.06%, and the carbon intensity could be reduced by up to 8.13%; (ii) the nondiscriminatory grant annualized as 50£ is not sufficient, and a differentiated government subsidy policy may be conducive to achieving a positive return on investment, such as 50£ for fast chargers and 100£ for rapid chargers; (iii) in the integrated business models, fast chargers undertake more vehicle-to-grid electricity exchange with the pattern adoption rate increased by up to 52.38%, while rapid chargers mainly ensure the timely charging completion with the usage frequency increased by up to 2.82%.

An L-Shaped Method to Solve a Stochastic Blood Supply Chain Network Design Problem in a Natural Disaster

Level of blood and blood products in human body is very important. Therefor, managing supply of blood is critical issue in healthcare system specially when the system faced with high demand for the product. In natural disasters, demand for blood units increase sharply because of injuries. Hence, efficiency in blood supply chain management play a significant role in this situation in supplying blood for transfusion centers, it is vital to supply in right time to prevent from casualties. Present paper proposes an optimization model for designing blood supply chain network in case of an earthquake disaster. The proposed two-stage stochastic model is Programmed based on scenarios for earthquake in a populated mega-city. The designed network has three layers; first layer is donation areas, the second layer consists distribution centers and facilities and the last layer is transfusion centers. In proposed two-stage stochastic optimization model, decisions of locating permanent collection facilities and amount of each blood type pre-inventory are made in first stage and operation decisions that have dependent on possible scenarios are made in second stage. The model also considers the possibility of blood transfusion between different blood types and its convertibility to blood derivatives regarding medical requirements. In order to solve the proposed two-stage stochastic model, L-shaped algorithm, an efficient algorithm to solve scenario based stochastic models, has been used. In addition, application of the model and the algorithm tests with real data of likely earthquake in Tehran mega city (Densest city of Iran).

Bi-level two-stage stochastic optimization for wind energy transmission networks encompassing active distribution networks

The distribution network is active for the high integration rate of sustainable sources and provides excess electricity to meet transmission demand. This paper proposes a two-layer hierarchical transmission extension formula for coordinated distribution network. The upper layer of the distributed optimization framework relates to the goals and limits of the transmission level, while the lower layer corresponds to the distributed viewpoint. The approximation method is deployed to linearize the distribution network operation constraints, which ensures that the formulated two-layer problem can be handled by a dual-based method. Numerical studies conducted on two integrated transmission and distribution networks show that the proposed model based on hierarchical duality can reduce investment costs, provide operational economic benefits and accommodate renewable energy, with its effectiveness and high performance.

Assessing the performance of uncertainty-aware transactive controls for building thermal energy storage systems

Energy storage systems provide a wide range of technological approaches to manage the balance between energy supply and demand in the electric grid. With the increasing uncertainty and variability that comes with wide-spread adoption of grid-scale and behind-the-meter renewable energy, it is imperative to develop stochastic operational planning and control approaches that can account for uncertainty in future conditions. Although, coordination of multiple thermal energy storage resources can support the transition to low carbon energy by enabling valuable system flexibility, few stochastic planning and control approaches have been developed for coordinating building-level thermal energy storage resources. In addition, there is also a need to analyze the potential benefits of an aggregator-level stochastic control framework versus applying stochastic planning and controls at each building individually. This work addresses these needs by developing an uncertainty-aware transactive control (UA-Tx) framework for an aggregator to coordinate the thermal energy storage (TES) assets of multiple buildings. A two-stage stochastic optimization framework is formulated for day-ahead energy procurement that considers uncertainty in building occupancy patterns, weather conditions, and real-time energy prices of the following day. In the second stage, possible recourse decisions through modifying TES operation are also considered. The dispatch of TES operational strategies is implemented through transactive controls, which use market mechanisms and customer preferences to achieve changes in building demand. During real-time operation, a local demand response aggregator determines the transactive clearing prices to dispatch the flexibility enabled by TES. Simulation case studies were conducted to demonstrate the capabilities of the uncertainty-aware aggregator control framework compared to the performance of applying intelligent controllers at each individual building. Up to 3.7% energy cost savings were observed for buildings under the UA-Tx aggregator control framework. Other potential benefits of the control approach are also discussed, along with anticipated future extensions.

Optimal bidding strategy for virtual power plants considering the feasible region of vehicle‐to‐grid

With the rapid development of vehicle-to-grid (V2G) technology, the use of the internet of things and advanced information technology to achieve the optimal dispatch of large-scale plug-in electric vehicles (PEVs) has become an important focus of scientific research and engineering practice worldwide. In order to achieve a balance between mass distributed energy resources and the secure and economic operation of the power grid, the authors propose an optimal market bidding strategy for a virtual power plant considering the feasible region of V2G. A detailed battery model considering the V2G mode of PEVs is established. A two-stage stochastic optimisation model for the virtual power plant considering massive volumes of PEVs is built, taking into account the day-ahead market revenue and the costs brought about by real-time market deviations. Three bidding strategies—PEVs charging immediately after arrival, without V2G, and considering V2G—are compared using case studies to demonstrate the efficiency and effectiveness of the proposed strategy. Through the complete interaction between PEVs and the virtual power plant, this V2G-based strategy can coordinate distributed energy resources, promote the accommodation of renewable energy, and significantly improve the economy of the virtual power plant operation.

A Two-Stage Stochastic Optimisation Methodology for the Operation of a Chlor-Alkali Electrolyser under Variable DAM and FCR Market Prices

The increased penetration of renewable energy sources in the electrical grid raises the need for more power system flexibility. One of the high potential groups to provide such flexibility is the industry. Incentives to do so are provided by variable pricing and remuneration of supplied ancillary services. The operational flexibility of a chlor-alkali electrolysis process shows opportunities in the current energy and ancillary services markets. A co-optimisation of operating the chlor-alkali process under an hourly variable priced electricity sourcing strategy and the delivery of Frequency Containment Reserve (FCR) is the core of this work. A short term price prediction for the Day-Ahead Market (DAM) and FCR market as input for a deterministic optimisation shows good results under standard DAM price patterns, but leaves room for improvement in case of price fluctuations, e.g., as caused by Renewable Energy Sources (RES). A two-stage stochastic optimisation is considered to cope with the uncertainties introduced by the exogenous parameters. An improvement of the stochastic solution over the deterministic Expected Value (EV) solution is shown.

Ensuring sustainability in the reverse supply chain in case of the ripple effect: A two-stage stochastic optimization model

The devastating impact of the ripple effect increases the importance of the reverse supply chain (RSC) design to ensure sustainability in the long-term. That being the case, in this study, a two-stage stochastic mixed-integer optimization model is proposed to design an RSC network under uncertainty sourcing from the ripple effect (i.e. external side of RSC) by considering the environmental and economic dimensions of sustainability. The environmental and economic disruptions of the ripple effect are represented by the increase in the carbon emission levels and the distance of roads, and the decrease in the capacity of facilities, respectively. Accordingly, a set of scenarios is considered based on the disruption levels (low- and high-impact) in case of the ripple effect. Furthermore, an α-reliability constraint is integrated into the model to further analyze the occurrence of scenarios. The model allows us to make integrated operational and strategic decisions by placing an emphasis on the carbon emission levels (i.e. environmental dimension) and the total cost (i.e. economic dimension). To obtain some remarkable insights, the proposed model is validated through computational experiments based on data extracted from a real case. The computational results show that the ripple effect increases the emission level and total cost up to 40%. For this reason, it suggested that the regulations regarding WEEE (Waste Electrical and Electronic Equipment) should be prepared by considering sustainability in the entire RSC network. Besides, it is realized that the centralized distribution strategy leads to a more resilient RSC network design.