Chance-constrained Optimization Research Articles

The Value of Drilling—A Chance-Constrained Optimization Approach

Managing uncertainty is a core challenge in mine planning. Mine planners often represent various planning variables, such as equipment performance and geological parameters, as random variables due to inherent uncertainties. This paper looks at geological uncertainty and its impact on mine planning. Some traditional approaches to manage this uncertainty include using conditional simulations or mathematical programming in the planning process. Drilling additional holes, despite its cost, is a common method to reduce uncertainty using additional samples to reduce deposit variance. In this paper, we first outline an ore blending optimization model which uses chance-constrained programming to manage property limit risk when selecting the order of ore feed into a processing facility. In coal mining, in tactical planning horizons, the order of coal seam removal is usually predetermined, allowing a blending model to ensure optimal feed properties. Using chance-constrained programming allows us to blend the uncertainties from geological models to maximize plant output while adhering to property constraints. We use the chance-constrained blending model to determine the value of additional information from infill drilling. The model prioritizes drilling locations that reduce uncertainty and improve blending outcomes. A case study on a coking coal mine in Queensland, Australia, demonstrates the model’s application, highlighting significant improvements in blending by reducing the variance of high-quality blocks. The study concludes that targeting high-quality blocks for variance reduction can better accommodate lower-quality material, offering a more valuable approach than the traditional focus of reducing uncertainty in low-quality blocks. This approach provides insights for improving mine planning strategies and showcases the potential of chance constraints in optimizing ore blending under uncertainty.

Closed-loop supply chain network design with price-greenness-sensitive demand: A distributionally robust chance-constrained optimization approach

In response to the government’s heightened focus on recycling and remanufacturing, as well as the growing awareness among consumers about environmental security, manufacturing companies are currently required to establish efficient closed-loop supply chain networks in order to improve their socialreputation and competitive advantage. This study investigates the optimization of a Closed-Loop Supply Chain (CLSC) network that involves multiple products, multiple periods, and uncertain returns, which also considers the influence of many factors, such as carbon cap-and-trade policy, raw part procurement discounts, and facility capacity constraints, on the supply chain. Simultaneously, customer demand is sensitive to both product pricing and product greenness, and product greenness can be improved by investing in emission reduction technologies. To address the uncertainty in the returns, we propose a two-stage distributionally robust chance-constrained optimization model, which is transformed into a mixed integer linear programming model. To efficiently address the complex problem, we designan improved Benders decomposition (IBD) algorithm. The experimental results confirm that the IBD algorithm has significant advantages when compared to the Benders decomposition algorithm. Additionally, this study conducted a sensitivity analysis on key parameters and proposed operation suggestions of practical importance.

Scenario-based robust reachability analysis for networked airport delay dynamics

This work introduces a novel scheme to represent delay dynamics in airport networks. A key feature of the proposed scheme is that uncertainty affecting network operation is accounted for via a robust reachability analysis approach. In particular, the dynamical system is adopted to describe the propagation of outbound and inbound delays between airport nodes. A probabilistic reach set for this networked system is formulated as a chance-constrained optimization program. Then, we resort to a data-driven randomized computation method called scenariooptimization , where the chance constraint is replaced by a set of scenario-based deterministic constraints. Besides, the robustness of the reach set is explicitly derived through a posterior algorithm. Moreover, we conduct the extensive experiments on a dataset of real-world airport delays in China and provide some interesting results. Additionally, the case study at a refined scenario level shows the potential of the proposed approach in application.

Optimizing quasi-dynamic wireless charging for urban electric buses: A two-scenario mathematical framework with grid and PV-battery systems

This paper presents a mathematical optimization framework for the strategic placement of quasi-dynamic wireless charging (QWC) stations within road networks to address the charging needs of battery electric buses (BEBs). This study evaluates two scenarios for powering the buses. In the first scenario, a grid-connected system is considered. The optimization aims to minimize annual costs related to capital, operation, and energy losses of the electric bus fleet. This involves determining the optimal locations for QWC stations, the length of power transmitters, and the corresponding battery capacities for the BEBs. Using MATLAB-based optimization tools Casadi and Yalmip, with solvers Bonmin and Fmincon, the optimal configuration includes a 13 kWh battery capacity and a 300 m power transmitter distributed across five bus stop areas. The second scenario employs a chance-constrained optimization approach for an isolated solar photovoltaic (PV) and battery energy storage system (BESS). This system is designed to reliably meet the BEBs' energy requirements throughout the day, considering different seasonal data (winter, summer, all seasons/year-round). The optimization results for the PV and BESS capacities vary with the seasons: 394.247 kW and 2012.6 kWh using summer data, 1762.1 kW and 2738.2 kWh using winter data, and 1610.8 kW and 2741.9 kWh using year-round data. Additionally, the paper examines the impact of varying bus fleet sizes on the optimal battery size and power transmitter combination using a real-world example of the bus route between Khalifa City and Abu Dhabi Downtown in the UAE. The findings suggest that larger batteries with fewer or no charging stations are more economical for smaller fleets. Conversely, as the fleet size increases, a combination of smaller battery sizes and a greater number (and length) of QWC (power transmitters) becomes more cost-effective. This research offers significant insights into the efficient deployment of QWC stations and the integration of renewable energy and energy storage for sustainable urban electric bus networks. The proposed optimization models provide a systematic approach to designing and operating charging infrastructure, contributing to sustainable urban transportation systems. Moreover, the study highlights the influence of seasonal data on PV system sizing and costs.

Sample-Based Continuous Approximate Method for Constructing Interval Neural Network.

In safety-critical engineering applications, such as robust prediction against adversarial noise, it is necessary to quantify neural networks' uncertainty. Interval neural networks (INNs) are effective models for uncertainty quantification, giving an interval of predictions instead of a single value for a corresponding input. This article formulates the problem of training an INN as a chance-constrained optimization problem. The optimal solution of the formulated chance-constrained optimization naturally forms an INN that gives the tightest interval of predictions with a required confidence level. Since the chance-constrained optimization problem is intractable, a sample-based continuous approximate method is used to obtain approximate solutions to the chance-constrained optimization problem. We prove the uniform convergence of the approximation, showing that it gives the optimal INN consistently with the original ones. Additionally, we investigate the reliability of the approximation with finite samples, giving the probability bound for violation with finite samples. Through a numerical example and an application case study of anomaly detection in wind power data, we evaluate the effectiveness of the proposed INN against existing approaches, including Bayesian neural networks, highlighting its capability to significantly improve the performance of applying INNs for regression and unsupervised anomaly detection.

Bridging chance-constrained and stochastic optimization for risk estimation of virtual energy hubs dominated by hybrid vehicles under diverse uncertainties: To improve economic sustainability

Virtual energy hubs (VRHBs) combine dispersed energy resources to enhance the efficient use of various energy sources, grid flexibility, and renewable energy adoption. This paper investigates the adoption of VRHBs with power-to-gas (P2G) facilities and hybrid electric-compressed natural gas vehicles (HNGEVs) in the presence of renewable resources to effectively tackle carbon emissions and increase the flexibility of VRHBs. The proposed P2G facility is constructed from carbon capture to collect carbon pollution from fossil fuel resources and HNGEVs to recycle into synthesis gas in a methanation plant. Furthermore, due to severe uncertainties due to renewable resources and energy consumption intensity, the proposed framework utilizes stochastic-driven chance-constrained optimization, which enables decision-makers in VRHBs to account for the probabilistic nature of uncertainties. Chance-constrained optimization explicitly considers uncertainty through probabilistic constraints, providing a more robust and reliable solution compared to deterministic methods. By modeling uncertainties using chance-constrained programming, VRHBs can manage the economic risks under unexpected worst realizations in uncertainties. Based on the results, the P2G facility leads to a significant reduction in the daily emissions by 63.06 % and 42.77 % observed under no-risk and full-risk conditions, respectively. In addition, the suggested chance-constrained method resulted in a major decrease of 69 % in overall operational costs.

Stochastic home care transportation with dynamically prioritized patients: An integrated facility location, fleet sizing, and routing approach

We study a home health care (HHC) problem that is characterized by prioritized patients and uncertain demands. In practice, HHC supply chain networks often struggle to meet high demand due to a shortage of service vehicles. Additionally, disruptions caused by natural calamities and pandemics (e.g., COVID-19) further compound these challenges, necessitating the consideration of real-life characteristics such as patient priorities, infrastructure locations, and transportation of medical supplies with uncertain demands. To formulate the problem, we propose a multi-depot and multi-period chance-constrained optimization model with precedence constraints, assuming that the demand quantities for medical supplies are random variables. Since patients’ medical conditions vary in severity, the priority of each patient is translated into a time-dependent potential healthcare cost that changes dynamically over the planning horizon. The solution to the proposed model determines the optimal locations for the base Mobile Health Facilities (MHFs) and the fleet size of HHC vehicles, and generates scheduling and routing plans to visit patients within specified time windows. We propose a unique three-phase solution approach, integrated with stochastic simulation, to address the problem. We then assess the robustness of the proposed model based on a realistic case of HHC service provision in Hong Kong and explore the optimal values for two model parameters, namely the Vehicle Threshold Index and the MHF Threshold Index. The performance evaluation tests show that the proposed solution method is efficient and effective for solving real-world problems.

Distributionally robust chance-constrained optimization with Gaussian mixture ambiguity set

Conventional chance-constrained programming methods suffer from the inexactness of the estimated probability distribution of the underlying uncertainty from data. To this end, a distributionally robust approach to the problem allows for a level of ambiguity considered around a reference distribution. In this work, we propose a novel formulation for the distributionally robust chance-constrained programming problem using an ambiguity set constructed from a variant of optimal transport distance that was developed for Gaussian Mixture Models. We show that for multimodal process uncertainty, our proposed method provides an effective way to incorporate statistical moment information into the ambiguity set construction step, thus leading to improved optimal solutions. We illustrate the performance of our method on a numerical example as well as a chemical process case study. We show that our proposed methodology leverages the multimodal characteristics from the uncertainty data to give superior performance over the traditional Wasserstein distance-based method.

Chance-Constrained Optimal Design of PV-Based Microgrids under Grid Blackout Uncertainties

A grid blackout is an intractable problem with serious economic consequences in many developing countries. Although it has been proven that microgrids (MGs) are capable of solving this problem, the uncertainties regarding when and for how long blackouts occur lead to extreme difficulties in the design and operation of the related MGs. This paper addresses the optimal design problem of the MGs considering the uncertainties of the blackout starting time and duration utilizing the kernel density estimator method. Additionally, uncertainties in solar irradiance and ambient temperature are also considered. For that, chance-constrained optimization is employed to design residential and industrial PV-based MGs. The proposed approach aims to minimize the expected value of the levelized cost of energy (LCOE), where the restriction of the annual total loss of power supply (TLPS) is addressed as a chance constraint. The results show that blackout uncertainties have a considerable effect on calculating the size of the MG’s components, especially the battery bank size. Additionally, it is proven that considering the uncertainties of the input parameters leads to an accurate estimation for the LCOE and increases the MG reliability level.

Reliability-based sizing of islanded multi-energy microgrid: a conic chance-constrained optimization approach

AbstractThis paper presents a methodology to optimally design a multi-energy microgrid with thermal and electric loads considering $$N-1$$ N - 1 and probabilistic regulation reserves. This methodology consists of a chance-constrained optimization that determines the optimal sizing of the microgrid. Microgrid operations are rigorously considered by modelling hourly thermal and electric demand patterns as well as technology production schedules over a year. Such schedules include both electric and thermal power balances, ramp constraints, $$N-1$$ N - 1 and regulation reserves, among others. To ensure a reliable microgrid design and operation, reserve constraints have been proposed to deal with both $$N-1$$ N - 1 generation contingencies, and forecast errors. $$N-1$$ N - 1 reserves guarantee that a sudden outage of any of the electric generators is strategically covered by the remaining generators in order to avoid load shedding. Additionally, nonzero-mean random forecast errors of electric load and solar production are addressed by a set of chance constraints able to schedule asymmetric up and down regulation reserves. Their levels are high enough to cover hourly random forecast errors (or intermittencies) with a high threshold probability. The proposed methodology results in a mixed integer second-order cone program. Results of microgrid designs with and without reliability reserves are carefully analysed and compared. Neglecting reliability constraints leads to a lower-cost design at the expense of exposing the microgrid to unsafe operation. Finally, sensitivity analysis to study the optimal portfolio sizing with respect to electric BESS investment cost, solar production forecast mean error, and random intermittency threshold probability are performed.

Distributionally robust chance-constrained optimization for the integrated berth allocation and quay crane assignment problem

In this paper, we are the first to propose a distributionally robust chance-constrained (DRCC) optimization model for the integrated berth allocation and quay crane assignment problem (BACAP) in container terminals. In contrast to the classical deterministic BACAP model, we consider the arrival time to be uncertain due to the frequent arrival delays in ports. We then impose a chance constraint that the service time must start after the uncertain arrival time with a probability of at least 1−ϵ, where 1−ϵ represents the target service level in container terminals (ϵ represents the target risk tolerance of the port manager). Under the moment-based ambiguity set, we reformulate the DRCC model into a mixed integer semi-definite programming (MISDP) model. Additionally, we develop an efficient decomposition branch-and-bound algorithm to solve the MISDP model and obtain the exact solution. Fortunately, a special case of the DRCC model arises when the mean and covariance utilized in the ambiguity set are precise, allowing for the transformation of the DRCC model into a mixed integer programming (MIP) model. This conversion significantly reduces the complexity of the problem. Impressively, the solving time of the MISDP model with the decomposition branch-and-bound algorithm is comparable to that of the transformed MIP model. The numerical results show that our model can achieve a schedule with high service at low cost. Meanwhile, we have made an intriguing discovery that the correlation between the target risk tolerance and the actual service level can be depicted as a staircase function regardless of the datasets. This finding offers crucial insights for port management, enabling them to strike a balance between the cost and actual service level by determining an appropriate target service level.

A chance-constrained optimization approach integrating project scheduling and material ordering to manage the uncertain material supply

To deal with the impact of uncertain material supply on the implementation of projects, we investigate a problem of concurrently finding a robust baseline schedule and a material ordering plan under different levels of uncertainty. Assuming the material warehouses are restricted, a chance-constrained model of the integrated resource-constrained project scheduling and material ordering problem with a confidence level (1−θ) is thus formulated. To tackle the difficult joint chance constraints, we firstly transform the chance-constrained model into a scenario-based integer programming model by the sample average approximation approach. Based on dominance relationship among scenarios, we then derive an exact branch-and-bound algorithm and a sampling-based genetic algorithm (SGA) to solve the model. For the branch-and-bound algorithm, a tailor-made branching scheme and four pruning rules are developed to accelerate the search process. For the SGA, two sampling methods and a local search procedure are presented to boost solution quality. Furthermore, extensive numerical experiments are carried out to test the performance of the proposed algorithms. Specifically, 4608 instances with four different confidence levels and six different sample sizes are generated. Then the Taguchi method is employed to calibrate the parameters of the SGA. Besides, the numerical results demonstrate that the proposed branch-and-bound algorithm can solve more instances with shorter CPU times than the CPLEX solver, and the solution quality of the proposed SGA is far better than that of two existing meta-heuristics and two variants of SGA. Finally, a sensitivity analysis for the key parameters of the model is conducted. By considering the uncertain material supply, limited warehouse capacity and various confidence levels, our research contributes to the effective decision on project scheduling and material ordering in real-world contexts.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Interval reservoir computing: theory and case studies

The time series data in many applications, for example, wind power and vehicle trajectory, show significant uncertainty. Using a single prediction value of wind power as feedback information for wind turbine control or unit commitment is not enough since the uncertainty of the prediction is not described. This paper addresses the uncertainty issue in time series data forecasting by proposing the novel interval reservoir computing method. The proposed interval reservoir computing can capture the underlying evolution of the stochastic dynamical system for time series data using the recurrent neural network (RNN). On the other hand, by formulating a chance-constrained optimization problem, interval reservoir computing outputs a set of parameters in the RNN, which maps to an interval of prediction values. The capacity of the interval is the smallest one satisfying the condition that the probability of having a prediction inside the interval is lower than the required level. The scenario approach solves the formulated chance-constrained optimization problem. We implemented an experimental data-based validation to evaluate the proposed method. The validation results show that the proposed interval reservoir computing can give a tight interval of time series data forecasting values for wind power and traffic trajectory. In addition, the confidence probability over the feasibility goes to 1 very quickly as the sample number increases.

Data-driven Wasserstein distributionally robust chance-constrained optimization for crude oil scheduling under uncertainty

Crude oil scheduling optimization is an effective method to enhance the economic benefits of oil refining. But uncertainties, including uncertain demands of crude distillation units (CDUs), might make the production plans made by the traditional deterministic optimization models infeasible. A data-driven Wasserstein distributionally robust chance-constrained (WDRCC) optimization approach is proposed in this paper to deal with demand uncertainty in crude oil scheduling. First, a new deterministic crude oil scheduling optimization model is developed as the basis of this approach. The Wasserstein distance is then used to build ambiguity sets from historical data to describe the possible realizations of probability distributions of uncertain demands. A cross-validation method is advanced to choose suitable radii for these ambiguity sets. The deterministic model is reformulated as a WDRCC optimization model for crude oil scheduling to guarantee the demand constraints hold with a desired high probability even in the worst situation in ambiguity sets. The proposed WDRCC model is transferred into an equivalent conditional value-at-risk (CVaR) representation and further derived as a mixed-integer nonlinear programming (MINLP) counterpart. Industrial case studies from a real-world refinery are conducted to show the effectiveness of the proposed method. Out-of-sample tests demonstrate that the solution of the WDRCC model is more robust than those of the deterministic model and the chance-constrained model.

Multi-Stage Decision Rules for Power Generation & Storage Investments With Performance Guarantees

We develop multi-stage linear decision rules (LDRs) for dynamic power system generation and energy storage investment planning under uncertainty and propose their chance-constrained optimization with performance guarantees. First, the optimized LDRs guarantee operational and carbon policy feasibility of the resulting dynamic investment plan even when the planning uncertainty distribution is ambiguous. Second, the optimized LDRs internalize the tolerance of the system planner towards the stochasticity (variance) of uncertain investment outcomes. They can eventually produce a quasi-deterministic investment plan, which is insensitive to uncertainty (as in deterministic planning) but robust to its realizations (as in stochastic planning). Last, we certify the performance of the optimized LDRs with the bound on their sub-optimality due to their linear functional form. Using this bound, we guarantee that the preference of LDRs over less restrictive – yet poorly scalable – scenario-based optimization does not lead to financial losses exceeding this bound. We use a testbed of the U.S. Southeast power system to reveal the trade-offs between the cost, stochasticity, and feasibility of LDR-based investments. We also conclude that the LDR sub-optimality depends on the amount of uncertainty and the tightness of chance constraints on operational, investment and policy variables.

Chance-Constrained Optimization of Storage and PFC Capacity for Railway Electrical Smart Grids Considering Uncertain Traction Load

Chance-Constrained Optimization of Storage and PFC Capacity for Railway Electrical Smart Grids Considering Uncertain Traction Load

Stochastic Operation of Multi-Terminal Soft Open Points in Distribution Networks with Distributionally Robust Chance-Constrained Optimization

Stochastic Operation of Multi-Terminal Soft Open Points in Distribution Networks with Distributionally Robust Chance-Constrained Optimization

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

Chance-Constrained Optimization for Flexibility Provision From the Local Energy Communities Using Logit-Based Regression

Chance-Constrained Optimization for Flexibility Provision From the Local Energy Communities Using Logit-Based Regression