Scenario Tree Research Articles

Adaptive robust optimization for European strategic gas infrastructure planning

The European natural gas market is undergoing fundamental changes that foster uncertainty on both supply and demand sides. This uncertainty has elicited questions about the value of the Projects of Common Interest—strategic infrastructure investments supported by public funding from the European Union. This paper addresses this matter by deploying the adaptive robust optimization framework to consider long-term uncertainties in the gas infrastructure expansion planning problem. This framework confronts the drawbacks of mainstream methods of incorporating uncertainty in gas market models, such as stochastic scenario trees, in which the modeler predefines the probabilities and realization paths of unknown parameters. Our mathematical model endogenously identifies the unfortunate realizations of unknown parameters and suggests the optimal investment strategies to address them. We use this feature to assess which Projects of Common Interest are valuable in maintaining system resilience amid cold-winter demand spikes and supply shortages, while also considering budget constraints. The robust solutions indicate consistent preferences for specific projects. We find that real-world construction efforts have been focused on the most promising projects from a business perspective. However, we also find that the vast majority of projects are unlikely to be realized without financial support, even if they would serve as a hedge against stresses in the European gas system.

Modeling, Optimization, and Analysis of a Virtual Power Plant Demand Response Mechanism for the Internal Electricity Market Considering the Uncertainty of Renewable Energy Sources

The penetration of renewable energy sources (RESs) in the electrical power system has increased significantly over the past years due to increasing global concern about climate change. However, integrating RESs into the power market is highly problematic. The output of RESs such as wind turbines (WTs) and photovoltaics (PVs) is highly uncertain. Their correlation with load demand is not always guaranteed, which compromises system reliability. Distributed energy resources (DERs), especially demand response (DR) programs and energy storage systems (ESSs), are possible options to overcome these operational challenges under the virtual power plant (VPP) setting. This study investigates the impact of using a DR program and battery energy storage system (BESS) on the VPP’s internal electricity market, and also cost-minimization analysis from a utility viewpoint. Three different constrained optimal power flow (OPF) problems are solved such as base case, DR case, and BESS case to determine total incurred costs, locational marginal prices (LMPs), and generator commitments. A scenario tree approach is used to model the uncertainties associated with WTs, PVs, and load demand. The proposed model is investigated on a 14-bus distribution system. The simulation results obtained demonstrate a favorable impact of DR and a BESS on renewable operational challenges.

Optimal long-term Tier 1 employee pension management with an application to Chinese urban areas

We formulate a stochastic optimization problem from the perspective of an investment committee responsible for Tier 1 social security pension policies and whose decisions are bound to have relevant economic and social consequences. The adopted modelling approach combines canonical multistage stochastic programming (MSP) with dynamic stochastic control (DSC): the first applies to the short-medium term, the second to the long-term. Through the combined framework, we are able to span a long planning horizon without jeopardizing the accuracy of scenario tree based medium-term planning. We apply this methodology to the Chinese pension system, which relies on two large reference areas for rural and urban populations. In this article, we concentrate on the ever-growing urban public pension system, which is facing significant challenges due to a declining workforce and a rapidly ageing population. This welfare area, originally conceived as a pay-as-you-go (PAYG) system, has undergone several recent reforms to enhance its long-term sustainability and reduce the interventions of the central government required to improve its funding condition. Among those relevant in our setting, is the reduction of policy constraints that until 2015 severely limited the possibility to invest in assets other than traditional, locally traded, long-term fixed income securities. We propose an optimization model in which the decisions of the investment management aim at significantly reducing central government interventions as a last resort liquidity provider and progressively improving the system funding condition. A rich set of computational and economic evidence is presented to validate the methodology and clarify its potential benefits to pension system efficiency.

A multistage distributionally robust optimization approach to water allocation under climate uncertainty

This paper investigates a Multistage Distributionally Robust Optimization (MDRO) approach to water allocation under climate uncertainty. The MDRO is formed by creating sets of conditional distributions (called conditional ambiguity sets) on a finite scenario tree. The distributions in the conditional ambiguity sets remain close to a nominal conditional distribution according a ϕ-divergence (e.g., Kullback-Leibler divergence, Hellinger distance, Burg entropy, etc.). The paper discusses a decomposition algorithm to solve the resulting MDRO with ϕ-divergences, which uses the dual formulation and solves only linear subproblems instead of convex ones. Some properties of the algorithm such as generating feasible policies and valid upper/lower bounds are established. The paper then applies the modeling and solution techniques to allocate water in a rapidly-developing area of Tucson, Arizona. Tucson, like many arid and semi-arid regions around the world, faces considerable uncertainty in its ability to provide water for its citizens in the future. The primary sources of uncertainty in the Tucson region include (1) unpredictable population growth, (2) the availability of water from the Colorado River, and (3) the effects of climate variability on water consumption. This paper integrates forecasts for all these sources of uncertainty into a single optimization model for robust and sustainable water allocation. Then, it uses this model to analyze the value of constructing additional treatment facilities to reduce future water shortages. The results indicate that the MDRO approach can be very valuable for water managers by providing insights to minimize their risks and help them plan for the future.

Operations Research Helps the Optimal Bidding of Virtual Power Plants

As distributed energy resources (DERs) continue to emerge, a new cloud-based information technology platform business model, the virtual power plant (VPP), is being introduced into the electricity market. The competitiveness of VPPs mainly depends on data analytics and operational technologies. Among the several operational problems, we focus on the optimal bidding decision problem in the day-ahead market. The bidding decision is a VPP’s commitment to supply the market with electricity from uncertain DERs, thereby affecting the VPP’s profits. Based on a collaboration with a VPP company in South Korea, H Energy Co. Ltd., we formulate a Markov decision process model for the problem and use a stochastic dynamic programming-based solution approach. This is the first study under the incentive-based market structure. To describe the uncertainty in the power supply from DERs, we build frameworks to generate scenario trees or lattices. Additionally, we apply heuristic techniques to reduce the computational burden. Through a pilot test based on real data, we verify the performance and practicality of our proposed model and solution approach. The case company has begun implementing the model and solution approach on its platform and has found that performance has improved after using advanced forecasting models for DERs.

Quantitative risk assessment of rabies being introduced into mainland France through worldwide noncommercial dog and cat movements.

France has been rabies-free among nonflying mammals since 2001. Despite this status, the rabies virus has been introduced several times through noncommercial pet movements, posing a threat of infection by this 100%-lethal zoonosis among local animal and human populations. To quantify the risk of rabies being introduced through worldwide noncommercial dog and cat movements, we performed a quantitative risk assessment using stochastic scenario tree modeling. The mean annual probability of at least one rabies introduction incident was 0.35 (median: 0.24, 90% prediction interval (PI) [0.04; 0.98]) and the mean annual number of rabies-infected pets introduced through pet movements was 0.96 (median: 0.27, 90% PI [0.04; 3.88]). These results highlight a nonnegligible, even high risk due to the associated consequences of such events. In alternative scenario testing, preventive anti-rabies vaccination proved to be an effective measure since removing the vaccination requirement led to a>15-fold increase in risk. The serological testing requirement had less of an effect (approximately two-fold increase when removed) and the posttest waiting period to ensure that antibodies were not linked to an infection had a negligible effect. Any change in pet owner compliance, especially regarding vaccination, could have a major impact on the risk. This study also shows that reinforced border control staff training could be more effective in reducing risk than more frequent checks. These results provide quantitative data for assessing the probability of the rabies virus entering France, and could help policymakers decrease this risk in rabies-free areas.

An equitable and accessible vaccine supply chain network in the epidemic outbreak of COVID-19 under uncertainty.

Vaccination is one of the most efficient ways to restrict and control the spread of epidemic outbreaks such as COVID-19. Due to the limited COVID-19 vaccine supply, an equitable and accessible plan should be prepared to cope with. This research focuses on designing a vaccine supply chain while aiming to achieve an equitable and accessible network. We present a novel mathematical formulation that helps to optimize vaccine distribution to inoculate people with various priority levels to achieve an equitable plan. The transshipment strategy is also incorporated into the model to enhance the accessibility of COVID-19 vaccine types between health facilities. The nature of COVID-19 is dynamic over time due to mutations, and the protection level of each vaccine type against this disease is not exact. Besides, complete information about the demand for different vaccine types is not available. Hence, we use Multi-Stage Stochastic Programming as a reliable strategy that is organized to manage stochastic data in a dynamic environment for the first time in the vaccine supply chain network. The scenarios in this approach are generated using a Monte Carlo simulation method, and then a forward scenario reduction technique is conducted to construct a suitable scenario tree. The practicality and capability of the model are shown in a real-life case of Iran. The results show that the performance of the Multi-Stage Stochastic Programming is significantly improved compared with the two-stage stochastic programming regarding the total cost of the vaccine supply chain and the number of the shortage units.

Frequency control in standalone renewable based-microgrids using steady state load shedding considering droop characteristic

This paper presents a new systematic scheme for designing optimized robust and efficient steady state load shedding (LS) in standalone inverter-based microgrids (IBMGs) considering uncertainties of the load and renewable generations. In order to incorporate different states of non-dispatchable renewable based-Distributed Energy Resources (DERs) and load demand fluctuations, the scenario tree model is implemented. To reduce the computational burden of the proposed model, a scenario-reduction technique using the Kantorovich distance (or Wasserstein distance) criteria is adopted to extract a sufficiently small number of scenarios. To reflect the system frequency in each opted scenario, a new power flow (PF) procedure, in which the system frequency turns up as a supplementary state variable is adopted. This procedure aims at minimizing the required shed loads to maintain the MG frequency over the critical frequency, when the inverters cannot tolerate any overload. According to the optimal curtailed load for different scenarios, a step-wise LS scheme is proposed for applying to an on-line operation manner. The performance of the proposed scheme is evaluated in two typical MV MG in standalone mode, a medium-scale 21 bus test system and a large-scale 118 bus test system, to ensure the efficiency and reliability of the proposed method in real conditions and systems. The obtained results evidence the effectiveness and the correctness of the presented LS scheme to diagnose power shortages and to maintain system frequency in the desirable range. Furthermore, the obtained results by proposed LS model have been compared with the conventional priority-based LS and the superiority of the proposed model over conventional models is proved.

The risks from climate change to sovereign debt.

The exposure of sovereigns to climate risks is priced and can affect credit ratings and debt servicing costs. I argue that the climate risks to fiscal stability are not receiving adequate attention and discuss how to remedy the situation. After providing evidence of divergent climate risks to advanced economies, I describe the transmission channels from climate change to public finance. Then, I suggest how integrated assessment models (IAMs) can be linked with stochastic debt sustainability analysis (DSA) to inform our understanding of climate risks to sovereign debt dynamics and assess the available fiscal space to finance climate policies. I argue for adopting the narrative scenario architecture developed within the IPCC to bring structure and transparency to the analysis. The analysis is complicated by deep uncertainty —risks, ambiguity, and mis-specifications— of climate change. Using scenario trees, narrative scenarios, and ensembles of models, respectively, we can deal with these three challenges. I illustrate using two prominent IAMs to generate the debt dynamics of a high-debt country under climate risks to economic growth and find adverse effects from as early as 2030. I conclude with the policy implications for fiscal stability authorities.Supplementary InformationThe online version contains supplementary material available at 10.1007/s10584-022-03373-4.

Output-based assessment of herd-level freedom from infection in endemic situations: Application of a Bayesian Hidden Markov model

Countries have implemented control programmes (CPs) for cattle diseases such as bovine viral diarrhoea virus (BVDV) that are tailored to each country-specific situation. Practical methods are needed to assess the output of these CPs in terms of the confidence of freedom from infection that is achieved. As part of the STOC free project, a Bayesian Hidden Markov model was developed, called STOC free model, to estimate the probability of infection at herd-level. In the current study, the STOC free model was applied to BVDV field data in four study regions, from CPs based on ear notch samples. The aim of this study was to estimate the probability of herd-level freedom from BVDV in regions that are not (yet) free. We additionally evaluated the sensitivity of the parameter estimates and predicted probabilities of freedom to the prior distributions for the different model parameters. First, default priors were used in the model to enable comparison of model outputs between study regions. Thereafter, country-specific priors based on expert opinion or historical data were used in the model, to study the influence of the priors on the results and to obtain country-specific estimates.The STOC free model calculates a posterior value for the model parameters (e.g. herd-level test sensitivity and specificity, probability of introduction of infection) and a predicted probability of infection. The probability of freedom from infection was computed as one minus the probability of infection. For dairy herds that were considered free from infection within their own CP, the predicted probabilities of freedom were very high for all study regions ranging from 0.98 to 1.00, regardless of the use of default or country-specific priors. The priors did have more influence on two of the model parameters, herd-level sensitivity and the probability of remaining infected, due to the low prevalence and incidence of BVDV in the study regions. The advantage of STOC free model compared to scenario tree modelling, the reference method, is that actual data from the CP can be used and estimates are easily updated when new data becomes available.

Interactive Multi-Modal Motion Planning With Branch Model Predictive Control

Motion planning for autonomous robots and vehicles in presence of uncontrolled agents remains a challenging problem as the reactive behaviors of the uncontrolled agents must be considered. Since the uncontrolled agents usually demonstrate multimodal reactive behavior, the motion planner needs to solve a continuous motion planning problem under these behaviors, which contains a discrete element. We propose a branch Model Predictive Control (MPC) framework that plans over feedback policies to leverage the reactive behavior of the uncontrolled agent. In particular, a scenario tree is constructed from a finite set of policies of the uncontrolled agent, and the branch MPC solves for a feedback policy in the form of a trajectory tree, which shares the same topology as the scenario tree. Moreover, coherent risk measures such as the Conditional Value at Risk (CVaR) are used as a tuning knob to adjust the tradeoff between performance and robustness. The proposed branch MPC framework is tested on an autonomous vehicle planning problem in simulation, and on an autonomous quadruped robot alongside an uncontrolled quadruped in experiments. The result demonstrates interesting human-like behaviors, achieving a balance between safety and performance.

Comparison of the confidence in freedom from infection based on different control programmes between EU member states: STOC free

The STOC free project constructed a generic framework that allows a standardised and harmonised description of different control programmes (CP) for cattle diseases. The STOC free model can be used to determine the confidence of freedom from infection that has been achieved in disease CPs, in support of an ongoing assessment of progress towards output-based standards as outlined in the EU Animal Health Law. With this information, and as required, further CP actions can be taken to mitigate the risks of persistence and (re-)introduction on the probability of freedom from infection. Bovine viral diarrhoea virus (BVDV) was chosen as the case disease because of the diversity in CPs in the six participating countries. A Bayesian hidden Markov model was considered the best modelling method. Detailed BVDV CP information was collected in the participating countries and the key aspects for inclusion in the STOC free model were identified. A first version of STOC free model was developed and tested on simulated data. The risk factors for BVDV infection that needed to be included in the model were defined and default values for these risk factors were quantified. A data collection tool was finalised with which the data for the STOC free model was collected. Subsequently, the developed model was tested and validated using real BVDV CP data from partner countries. Based on the feedback, the model was finalised and the report and corresponding computer code were made publicly available. There were roughly three different BVDV situations that occurred in the partner countries: 1. Endemic situation with a CP operating at herd level, 2. Endemic situation with a CP operating at animal level and 3. BVD free situation. The STOC free model is able to include herd level data only and animal level data has to be aggregated to herd level before the model can be applied. The STOC free model is not applicable for a country that is completely BVDV free given that it needs some infections to estimate its parameters and converge. In the latter situation, a scenario tree model could be a better suited tool, and this was evaluated in the Swedish case study. Further work is needed for generalisation of the method to other diseases and expansion of the method to include socioeconomic aspects of CPs.

An efficient, robust optimization model for the unit commitment considering renewable uncertainty and pumped-storage hydropower

This paper presents a new model for allocating pumped-storage hydropower units in the unit commitment program's next day's market. Because the market prices of the next day are highly volatile and have high uncertainty, the time series Autoregressive integral Moving Average (ARIMA) and Stationary Autoregressive integral Moving Average (SARIMA) have been used to predict the market price of the next day. A triple scenario tree is also used to cover electricity market uncertainties. The problem of allocating pumped-storage hydropower plants is planned as a mixed-integer linear programming (MILP) to maximize the power plant profit on a certain operating horizon. Because the variable nature of wind farms is also difficult to accurately estimate the output power of these power plants, the proposed method can be proposed on MNHPSO to consider wind power plant uncertainties with the presence of pumped-storage hydropower plant storage in the unit commitment program. The proposed strategy is simulated in MATLAB software. The simulation results show that in this paper while considering an accurate model for various uncertainties, in the unit commitment planning, it shows that SARIMA can predict the daily prices of the next day market with acceptable accuracy.

Assessment of the Current Surveillance System for Human Leptospirosis in Ecuador by Decision Analytic Modeling.

Leptospirosis is a globally disseminated zoonotic disease with no national surveillance systems. On the other hand, surveillance is crucial for improving population health, and surveillance systems produce data that motivates action. Unfortunately, like many other countries, Ecuador put in place a monitoring system that has never been tested. The goal of this study was to use scenario tree modeling to assess the sensitivity of Ecuador's current national surveillance system to human leptospirosis as the basis for an economic assessment of the system. We created a decision-tree model to analyze the current system's sensitivity. The inputs were described as probabilities distributions, and the model assessed the program's sensitivity as an output. The model also considers the geographical and weather variations across Ecuador's three continental regions: Andean, Amazonia, and the Coast. Several data sources were used to create the model, including leptospirosis records from Ecuador's Ministry of Public Health, national and international literature, and expert elicitation, all of which were incorporated in a Bayesian framework. We were able to determine the most critical parameters influencing each scenario's output (CSU) sensitivity through sensitivity analysis. The Coast region had the best sensitivity scenario, with a median of 0.85% (IC 95% 0.41–0.99), followed by the Amazonia with a median of 0.54% (CI 95% 0.18–0.99) and the Andes with a median of 0.29% (CI 95% 0.02–0.89). As per the sensitivity study, the most influential criteria on the system's sensitivity were “Attendance or probability of going to a health center” and “probability of having symptoms,” notably for the Coast and Amazonia Regions.

Modeling and optimization of bioethanol production planning under hybrid uncertainty: A heuristic multi-stage stochastic programming approach

When making planning decisions, fluctuation in both product demand and yield for the bioethanol plant denotes an important factor to affect the results. In this paper, a mixed multi-stage stochastic optimization model is proposed to solve the production planning problem of a bioethanol plant with multiple raw materials simultaneously considering uncertain demand and yield. The multi-source uncertainty is described as a multi-layer hybrid scenario tree model according to the historical statistics. Due to the diversity of pretreatment approach in the bioethanol production process, the yield of saccharification liquid is different. At each node of the scenario tree, the uncertainty of demand is expressed by probability. The uncertain scenarios of the yield are integrated at each node of demand to form a mixed scenario tree. To improve the solution efficiency, a heuristic algorithm based on fast forward selection is designed to simplify the decomposition of the large-scale hybrid scenario tree in the mixed multi-stage stochastic programming problem. The effectiveness of the proposed model is verified by a practical case. The results show that the proposed approach performs better than the two-stage stochastic programming and the deterministic approaches, and the algorithm can significantly reduce the computational efforts.

Resource planning strategies for healthcare systems during a pandemic

We study resource planning strategies, including the integrated healthcare resources’ allocation and sharing as well as patients’ transfer, to improve the response of health systems to massive increases in demand during epidemics and pandemics. Our study considers various types of patients and resources to provide access to patient care with minimum capacity extension. Adding new resources takes time that most patients don't have during pandemics. The number of patients requiring scarce healthcare resources is uncertain and dependent on the speed of the pandemic's transmission through a region. We develop a multi-stage stochastic program to optimize various strategies for planning limited and necessary healthcare resources. We simulate uncertain parameters by deploying an agent-based continuous-time stochastic model, and then capture the uncertainty by a forward scenario tree construction approach. Finally, we propose a data-driven rolling horizon procedure to facilitate decision-making in real-time, which mitigates some critical limitations of stochastic programming approaches and makes the resulting strategies implementable in practice. We use two different case studies related to COVID-19 to examine our optimization and simulation tools by extensive computational results. The results highlight these strategies can significantly improve patient access to care during pandemics; their significance will vary under different situations. Our methodology is not limited to the presented setting and can be employed in other service industries where urgent access matters.

A scenario-based distributed model predictive control approach for freeway networks

In this paper a scenario-based Distributed Model Predictive Control (DMPC) approach based on a reduced scenario tree is developed for large-scale freeway networks. In the new scenario-based DMPC approach, uncertainties in a large-scale freeway network are distinguished into two categories: global uncertainties for the overall network and local uncertainties applicable to subnetworks only. We propose to use a reduced scenario tree instead of using a complete scenario tree. A complete scenario tree is defined as a scenario tree consisting of global scenarios and all the combinations of the local scenarios for all subnetworks, while a reduced scenario tree is defined as a scenario tree consisting of global scenarios and a reduced local scenario tree in which local scenarios are combined within each subnetwork, not among subnetworks. Moreover, an expected-value setting and a min–max setting are considered for handling uncertainties in scenario-based DMPC. In the expected-value setting, the expected-value of the cost function values for all considered uncertainty scenarios is optimized by scenario-based DMPC. However, in the min–max setting, the worst-case of the cost function values for all considered uncertainty scenarios is optimized by scenario-based DMPC. The results for a numerical experiment show that the new scenario-based DMPC approach is effective in improving the control performance while at the same time satisfying the queue constraints in the presence of uncertainties. Additionally, the proposed approach results in a relatively low computational burden compared to the case with the complete scenario tree.

Scheduling cross-docking operations under uncertainty: A stochastic genetic algorithm based on scenarios tree

A cross-docking terminal enables consolidating and sorting fast-moving products along supply chain networks and reduces warehousing costs and transportation efforts. The target efficiency of such logistic systems results from synchronizing the physical and information flows while scheduling receiving, shipping and handling operations. Within the tight time-windows imposed by fast-moving products (e.g., perishables), a deterministic schedule hardly adheres to real-world environments because of the uncertainty in trucks arrivals. In this paper, a stochastic MILP model formulates the minimization of penalty costs from exceeding the time-windows under uncertain truck arrivals. Penalty costs are affected by products' perishability or the expected customer’ service level. A validating numerical example shows how to solve (1) dock-assignment, (2) while prioritizing the unloading tasks, and (3) loaded trucks departures with a small instance. A tailored stochastic genetic algorithm able to explore the uncertain scenarios tree and optimize cross-docking operations is then introduced to solve scaled up instances. The proposed genetic algorithm is tested on a real-world problem provided by a national delivery service network managing the truck-to-door assignment, the loading, unloading, and door-to-door handling operations of a fleet of 271 trucks within two working shifts. The obtained solution improves the deterministic schedule reducing the penalty costs of 60%. Such results underline the impact of unpredicted trucks’ delay and enable assessing the savings from increasing the number of doors at the cross-dock. • This research proposes a two-stage stochastic model for cross-docking with uncertain trucks arrival times. • The model minimizes penalty costs due to deliveries violating the retailers time-windows. • A tailored Stochastic Genetic Algorithm based on Scenario Tree (SGA-ST) is proposed to solve real-world instance. • SGA-ST outperforms the two-stage stochastic MILP formulation solved with commercial solvers. • The SGA-ST is implemented into a user-friendly decision-support tool to aid practitioners in cross-docking operational decisions.

Capacitated Stochastic Lot-sizing and Production Planning Problem Under Demand Uncertainty

This paper proposes two multi-period, multi-item capacitated stochastic lot-sizing problems under demand uncertainty. We model uncertainty via a scenario tree. The first model considers production, inventory, backlogging, line status, and worker group assignment decisions, where inventory and backlogging decisions have wait-and-see structure. The second model converts line status and worker group assignment decisions to the wait-and-see structure. Also, the second model enables us to take corrective extra-ordering decisions using scenario-based wait-and-see decisions. Numerical results compare the optimality and CPU time performances of two models and solution approaches using a data set inspired by a real-life electronics company.

A distributed scenario-based stochastic MPC for fault-tolerant microgrid energy management

This paper proposes a fault-tolerant energy management algorithm for microgrid systems composed of several agents. The method stems from the necessity to design an algorithm that takes explicitly into account the possibility of faults and their consequences to avoid solutions which are excessively conservative. A tree of possible fault scenarios is built in a completely distributed way by all the agents of the network; then the resulting optimization problem is solved through a distributed algorithm which not only does not require a high computational power for each agent, but keeps also private all local data and decision variables. The effectiveness of the proposed method is proved through simulation results.