Chance Constraints Research Articles

On data-driven chance constraint learning for mixed-integer optimization problems

When dealing with optimization problems, decision-makers often face high levels of uncertainty associated with partial information, unknown parameters, or complex relationships between these and the problem decision variables. In this work, we develop a novel Chance Constraint Learning (CCL) methodology with a focus on mixed-integer linear optimization problems, which combines and extends ideas from the literature on chance constraint and constraint learning. While constraint learning aims to model the functional relationship between straight-forward and non-tractable decision variables through the embedding of predictive models within the optimization problem, chance constraints set a probabilistic confidence level for a single or a set of constraints to be fulfilled. One of the main issues when establishing a learned constraint arises when we need to set further bounds for its response variable: the fulfillment of these is directly related to the accuracy of the predictive model and its probabilistic behavior. Therefore, the proposed CCL makes use of piece-wise linearizable machine learning models to estimate conditional quantiles of learned variables, providing a data-driven solution for chance constraints and adding probabilistic guarantees over constraints for learned variables. Open access software has been developed for use by practitioners. Furthermore, the benefits of CCL have been tested in two real-world case studies, demonstrating how robustness is added to optimal solutions when probabilistic bounds are established for learned constraints.

Modeling a hydrogen-based sustainable multi-carrier energy system using a multi-objective optimization considering embedded joint chance constraints

Hydrogen-based power generation could increase the sustainability and efficiency of future energy systems due to the higher storing capabilities resulting from the higher gravimetric density of hydrogen energy. This work proposes a hydrogen-based multi-carrier energy system (HMES) comprising renewable energy resource (RES), electricity, and hydrogen markets as input energy carriers and power, cooling, and heating as demands. Thermal water storage and hydrogen tank storage are considered to improve energy efficiency. This system uses the full capacity of intermittent wind and solar energy to increase RES utilization and decrease carbon emissions simultaneously. The environmental and economic goals of the proposed HMES are satisfied through the weighted-sum multi-objective method. The trade-off solution between operation and emission costs is ultimately obtained through the max-min fuzzy method. Chance constraint programming (CCP), which offers decision-makers a variety of risk-taker strategies, is utilized as a tool to manage the risk associated with stochastic optimization. The confidence level of the CCP determines the risk tolerated by the decision-maker, improving the optimization process by creating a more realistic approach to risk management in real-world models. The results show the benefits of hydrogen tank by reducing operation costs by 4.5% and importance of CCP leading to operation costs reduction.

Stochastic planning for low-carbon building integrated energy system considering electric-heat-V2G coupling

The low-carbon building has been proposed to mitigate the climate change caused by environmental problems and realize carbon neutrality in urban areas. In addition, the integrated energy system (IES) has been developed to reduce renewable energy curtailment in the power distribution system and improve energy efficiency due to the independent operation of traditional energy systems. In this paper, we propose a stochastic planning method for low-carbon building IES, in which the Vehicle to Grid (V2G) is also considered to further increase the flexibility of low-carbon buildings. The proposed planning method optimizes the investment and operation costs, and CO2 emission of the building IES, to achieve the maximum benefit of the low-carbon building and help realize carbon neutrality. By considering the uncertainty of distributed renewable energy, multi-energy load fluctuation and the random behavior of EV users, a two-stage stochastic programming model is formulated with chance constraints, in which the heuristic moment matching scenario generation (HMMSG) and sample average approximation (SAA) methods are applied to deal with the uncertainties. In the case study, a real IES office building in Shanghai, where photovoltaic (PV), energy storage system (ESS), fuel cell (FC), EV, etc. are included as planning options, is used as the test system to verify the effectiveness of the proposed planning method, and the functions of the ESS and EV in IES are analyzed in detail in different operation scenarios. The case study results show that the proposed planning scheme can reduce the total cost and carbon emission significantly.© 2017 Elsevier Inc. All rights reserved.

Joint chance-constrained multi-objective multi-commodity minimum cost network flow problem with copula theory

This study focuses on a specific problem in network optimization, namely the minimum cost multi-commodity network flow (MCNF) problem. The problem is complicated by the presence of uncertain parameters, including various types of costs associated with each arc in the network. The study presents a multi-objective approach to solving this problem, where the coefficients of the capacity constraints are modelled as random variables with a normal distribution, and the dependence between them is modelled using an Archimedean copula. The capacity constraints are presented as joint chance constraints, and a multi-objective problem is formulated to deal with the uncertainty. This uncertain multi-objective problem is then converted into a certain multi-objective problem using fuzzy programming. The resulting certain multi-objective MCNF problem is converted to a certain single objective problem using second-order cone programming (SOCP), which is solved using either piecewise tangent approximation or piecewise linear approximation methods. The proposed approaches are tested using numerical examples and experimental tests to demonstrate their effectiveness in solving large-scale network problems efficiently. The results show that the proposed approaches are a useful tool for solving uncertain multi-objective MCNF problems in real-world applications.

A joint chance constrained optimization algorithm with robust reconstruction for multi-echelon and multi-period closed-loop manufacturing adjustable system design under multi-source uncertainty

The complex electronic assembly scheduling problem under uncertainty poses a new challenge. The joint chance constraint programming method based on robust reconstruction is used to hedge the endogenous and external uncertainty in the closed-loop manufacturing scheduling model of multi-echelon and multi-product electronic assembly process under multi-source uncertainty in this study. The numerical cases of electronic factory in assembly and closed-loop manufacturing mode are studied, the sensitivity of cost to different types of uncertain parameters is analyzed. The results show that different types of uncertainty have different effects on economic benefits, and endogenous uncertainty has a greater impact on manufacturing costs. Decision makers can adjust the adjustable parameters according to external policies and economic situation to control the conservatism and economic benefits of the scheme, and at the same time reduce the demand for uncertain data and improve the efficiency of solution. The model provides a new sustainable production method for electronic assembly manufacturing practitioners, and makes the best production decision when weighing economic benefits and risks.

Physician scheduling for emergency telemedicine across multiple facilities

Telemedicine has emerged as an effective means to connect health service providers with patients remotely. In the context of emergency care, telemedicine typically involves a telemedicine service hub (TSH) that matches remote physicians with patients in emergency wards. Effective operation of such systems requires careful and continuous coordination between physicians and local providers and as such, physician staffing and scheduling is a major managerial challenge that a TSH has to overcome. In this context, our paper studies a setting, where the TSH must respond to an arriving emergency case by assigning a physician within a considerably short time window. An emergency case at a facility can be matched only with a physician who is credentialed at that facility. Since care is urgent, queuing patients is not an option when there is no available on-shift physician. In such cases, the system must invoke the off-shift physicians, which is referred to as “blast.” The telemedicine company tries to avoid this option due to high costs. We propose a novel integer programming model for generating physician schedules with optimal mix of credentials and coverage across multiple hospitals. The proposed model aims to minimize total costs under a chance constraint that limits the blast probabilities and other tactical constraints that are unique to this setting. Two fast acting heuristic-based solution approaches are developed for real-life size problems and their computational performances were demonstrated via numerical analyses.

Coupling Fuzzy Bi-Level Chance Constraint Programming and Spatial Analysis for Urban Ecological Management

In this study, a fuzzy bi-level chance constraint programming (FBCP) model is developed for urban ecological management in Xiamen, China. FBCP has advantages in balancing trade-offs between multiple decision makers and can address fuzzy and stochastic uncertainty in ecosystem management. It also can reflect the impact of different violation risk levels and emission reduction measures on system benefit, ecosystem service value, and land resource allocation. Then, the conversion of land use and its effects at small regional extent (CLUE-S) model is employed to provide the spatial allocation of future land resources under different scenarios. Results reveal that (i) carbon fixation and climate regulation are the major contributors to the ecosystem service value, with a proportion of [15.4, 15.6]% and [18.5, 18.8]%, respectively; (ii) the main environmental problem in Xiamen is the water pollution caused by the excessive discharge of commercial and residential land, with COD and NH3-N account for [68.81, 69.33]% and [67.65, 68.20]% of the total discharge of the city, respectively; (iii) the violation risk p level is the most impact factor, and the schemes with high system benefit would face greater default risk and lower ecological quality; (iv) FBCP model considers the trade-off between economic benefit and ecological quality, while the fuzzy chance constraint programming (FCP) model achieves a high system benefit at the expense of the environment. These findings help decision makers to understand the impact of parameter uncertainty and pollutant discharge policies on system benefit, and adjust land-use patterns to weigh ecological environment protection with urban economic development.

Stochastic trajectory optimization for 6-DOF spacecraft autonomous rendezvous and docking with nonlinear chance constraints

This paper addresses the problem of stochastic trajectory planning for 6-DOF spacecraft close proximity in the presence of external disturbances and initial state uncertainties. Nonlinear chance constraints on collision avoidance, docking corridor, sensor field-of-view, and control magnitude are considered in the corresponding optimal control problem. Nominal trajectory along with affine feedback controller that restricts the dispersion caused by disturbances are optimized simultaneously. The nonlinear stochastic dynamics are transformed into equality constraints on the first two statistical moments of the random states by linearization and discretization, and further, the non-convex covariance update equation is relaxed to a semidefinite cone constraint. The nonlinear path chance constraints are first approximated by affine chance formations, and then are convexified by introducing auxiliary variables instead of direct linearization of the covariance matrix. Combined with the tail bound of sub-exponential random variables, the constraints on control input are replaced with conservative deterministic inequalities. The sub-optimal solution to the original problem is obtained by iteratively solving a series of semidefinite cone programs. The effectiveness of the proposed framework is verified using numerical simulations.

Frequency Constrained Scheduling Under Multiple Uncertainties via Data-Driven Distributionally Robust Chance-Constrained Approach

The declining system inertia in renewable-rich power systems raises a concern about the frequency stability problem. The wind farm equipped with the power electronic controller is capable of providing frequency support after a disturbance. However, both virtual inertia provision and wind power from wind farms are time-varying and uncertain. To account for this issue, we propose a data-driven distributionally robust (DR) chance-constrained approach for the frequency constrained scheduling problem, which simultaneously optimizes the unit commitment, generation dispatch, regulation reserves, and frequency responses. This approach explicitly considers frequency constraints and formulates virtual inertia uncertainty- and wind power uncertainty-related operational/frequency constraints as DR chance constraints under Wasserstein-metric ambiguity sets, which can limit the risk of constraint violations. Case studies demonstrate the effectiveness of the proposed approach and show that the proposed approach can achieve a desirable trade-off between operational cost and constraint violations.

Spatiotemporal learning-based stochastic MPC with applications in aero-engine control

To deal with the issue of model uncertainty, extensive stochastic MPC methods have been investigated, where the uncertainties are typically assumed to follow given statistical distributions. However, in practical scenarios, the statistical properties of model uncertainties may depend on certain hyperparameters varying both temporally and spatially. For instance, in the applications of aero-engine control, different types of uncertainties may occur under different operational conditions, such as flight altitudes and Mach numbers, which are dynamically changing. Traditionally, the stochastic MPC methods may not be able to handle these types of uncertainties directly. Therefore, we propose a spatiotemporal learning-based stochastic model predictive control algorithm to study the stochastic optimal control problem with dynamically changing uncertainties, by constructing spatiotemporal Gaussian processes (GPs) to approximate the uncertainties based on measurement data. Since the spatiotemporal GPs may be difficult to evaluate with long processing time series, we present a state–space representation of GPs to employ computationally efficient Kalman filtering. Then, we derive a computationally tractable control strategy by parameterizing the controller and reformulating the cost and chance constraints, and analyze the corresponding recursive feasibility and closed-loop stability. Finally, the proposed algorithm is applied to the compressor control of the aero-engine, and the comparisons with other MPC controllers are demonstrated to show the effectiveness of our methods.

Output‐feedback stochastic model predictive control of chance‐constrained nonlinear systems

AbstractThis study covers the output‐feedback model predictive control (MPC) of nonlinear systems subjected to stochastic disturbances and state chance constraints. The stochastic optimal control problem is solved in a stochastic dynamic programming fashion, and the output‐feedback control is performed with the extended Kalman filter. The information state is summarized as a dynamic Gaussian belief model. Thus, the stochastic Bellman equation is transformed into a deterministic equation using this model. The resulting constrained Bellman equation is solved using the proposed constrained, approximate dynamic programming algorithm. The algorithm is proved to have a Q‐superlinear local convergence rate. Numerical experiments show that the proposed algorithm can attain good control performance and reasonable chance‐constraint satisfaction and is computationally efficient owing to its dynamic programming structure.

An Integrated Risk Sensitive Crop Allocation Model with Pollination Intelligence Algorithm

A risk sensitive model for allocation of crops is considered in this work. The constructed model was designed to help farmers decision making process, thereby maximizing the use of agricultural land. Market price, cost of cultivation, yield of crops and climatic conditions were factors considered in the models. The theory of chance constraint programing was used to handle uncertainties that arise in crop planing. Data of known yield of crops were harvested and analyzed with the help of statistical tools. A class of Pollination Intelligence Algorithm was adopted to solve the model.

Chance‐constrained co‐expansion planning for power systems under decision‐dependent wind power uncertainty

AbstractVariability and uncertainty in wind resources pose significant challenges to the expansion planning of wind farms and associated flexible resources. In addition, the spatial smoothing effect, indicating the impact of wind farm scale on aggregated wind power prediction errors, further aggravates the challenge. This paper proposes a chance‐constrained co‐expansion planning method considering the spatial smoothing effect, where the expansion of wind farm capacity, batter energy storage capacity, and power transmission lines are co‐optimized. Specifically, a decision‐dependent uncertainty (DDU) model is established capturing the dependency of wind power uncertainties on wind farm expansion decisions under the spatial smoothing effect. Unlike traditional optimization diagram where decisions are made under only decision‐independent uncertainty (DIU) with fixe properties, properties of decision‐dependent uncertain parameters would be inversely altered by decisions. To effectively tackle the coupling relation between decisions and DDU, DDU‐based chance constraints are formulated in an analytical manner, where the decisions and decision‐dependent uncertain parameters are expressed in a closed form. Eventually, with piecewise linearization of the DDU model and the polynomial approximation of cumulative distribution function of uncertain parameters, the proposed chance‐constrained optimization model with DDU is converted into a mixed‐integer second‐order cone program (MISOCP). Case studies verify the effectiveness of the proposed method.

Model Predictive Control of Parabolic PDE Systems under Chance Constraints

Model predictive control (MPC) heavily relies on the accuracy of the system model. Nevertheless, process models naturally contain random parameters. To derive a reliable solution, it is necessary to design a stochastic MPC. This work studies the chance constrained MPC of systems described by parabolic partial differential equations (PDEs) with random parameters. Inequality constraints on time- and space-dependent state variables are defined in terms of chance constraints. Using a discretization scheme, the resulting high-dimensional chance constrained optimization problem is solved by our recently developed inner–outer approximation which renders the problem computationally amenable. The proposed MPC scheme automatically generates probability tubes significantly simplifying the derivation of feasible solutions. We demonstrate the viability and versatility of the approach through a case study of tumor hyperthermia cancer treatment control, where the randomness arises from the thermal conductivity coefficient characterizing heat flux in human tissue.

A Data-Driven Joint Chance-Constrained Game for Renewable Energy Aggregators in the Local Market

The local market can promote more self-generated electricity consumed locally. However, the unstable and intermittent generation of renewable energy sources brings uncertainty and overbidding risk to the local market. We proposed a competitive local energy market (LEM) where renewable energy aggregators (REAs) can submit bids instead of acting as price takers. A data-driven joint chance-constrained game of REAs is proposed for the optimal bidding problem under the network constraints and the risk-averse chance constraint. We consider the case that the maximum outputs of REAs are random variables whose probability distributions are unknown, but the decision-maker has access to finite samples. Each player’s strategy set is defined in a data-driven distributionally robust framework with the Wasserstein ambiguity set. The exact reformulation of the distributionally robust chance constraint and the corresponding inner approximation version are derived. We prove the existence of the Nash equilibrium in the proposed distributionally robust chance-constrained game. Finally, an efficient best response seeking algorithm is developed to find the normalized Nash equilibrium, and the convergence proof of the proposed algorithm is also provided. Numerical simulations demonstrate that the proposed game model can effectively increase the revenue of reliable players and reduce the overbidding rate of REAs.

Peer-to-Peer Energy Trading Using Prediction Intervals of Renewable Energy Generation

The rapid development of renewable energy generation and demand side flexible resource makes the operation of distribution network and the organisation of power market facing greater uncertainty challenges. This paper proposes a novel receding horizon peer-to-peer energy transaction model based on the prediction intervals of renewable energy generation to manage the volatility in the range of a distribution network. A peer-to-peer energy interval matching algorithm is proposed to fully explore the flexibility in demand side for mitigating the output fluctuation of renewable energy generation locally. Then the responsibilities of undertaking the uncertainty risk from renewable generations are assigned to the counter-part consumers who have been matched with the renewable energy generations in a peer-to-peer market. The autonomy energy management problem under distribution network of each consumer is formulated as a cooperative gaming problem using the Nash bargaining theory. The uncertainty risk is considered into the Nash bargaining problem by utilizing voltage chance constraints and conditional value at risk based return-risk utility, of which the quantile connotations are consistent with the quantile results of the probability prediction of renewable energy generations. Moreover, an alternating direction method of multipliers algorithm based distributed methodology is developed to solve the Nash bargaining problem in a distributed manner. Numerical results demonstrate the effectiveness of the presented peer-to-peer energy trading model.

On approximations of data-driven chance constrained programs over Wasserstein balls

Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem parameters affecting the safety condition(s) is only known to belong to some ambiguity set. We study three popular approximation schemes for distributionally robust chance constrained programs over Wasserstein balls, where the ambiguity set contains all probability distributions within a certain Wasserstein distance to a reference distribution. The first approximation replaces the chance constraint with a bound on the conditional value-at-risk, the second approximation decouples different safety conditions via Bonferroni's inequality, and the third approximation restricts the expected violation of the safety condition(s) so that the chance constraint is satisfied. We show that the conditional value-at-risk approximation can be characterized as a tight convex approximation, which complements earlier findings on classical (non-robust) chance constraints, and we offer a novel interpretation in terms of transportation savings. We also show that the three approximations can perform arbitrarily poorly in data-driven settings, and that they are generally incomparable with each other.

Capturing delays in response of emergency services in Delhi

Response time of an ambulance plays a significant role in pre-hospital care. The absence of response standards in India has made it challenging for emergency services to provide efficient and timely pre-hospital services. Also, traffic congestion in a city like Delhi may prove detrimental to a patient who needs urgent transport. Since travel times fluctuate throughout the day, solving an ambulance location problem with average travel times would not suffice. Therefore, the Gaussian Mixture Model (GMM) has been used to capture the variability in travel time between each origin-destination pair for Delhi's vast transportation network. This variation in travel times and delays in the dispatch of an ambulance (pre-trip delay) has been incorporated into the traditional double standard model as a chance constraint. The study thus builds three different variations of model, one being deterministic and the other two stochastic with probabilistic response times. These models are referred to as the Chance Constrained Double Standard Model (cc-DSM), Chance Constrained Double Standard Stochastic Model (cc-DSSM) and Double Standard Stochastic Model (DSSM). This study shows the similarity of previously used relocation approaches with the current approach and highlights the difference between vehicle busyness concept from the concept of multiple coverage.

An integrated chance constraints approach for optimal vaccination strategies under uncertainty for COVID-19

Despite concerted efforts by health authorities worldwide to contain COVID-19, the SARS-CoV-2 virus has continued to spread and mutate into new variants with uncertain transmission characteristics. Therefore, there is a need for new data-driven models for determining optimal vaccination strategies that adapt to the new variants with their uncertain transmission characteristics. Motivated by this challenge, we derive an integrated chance constraints stochastic programming (ICC-SP) approach for finding vaccination strategies for epidemics that incorporates population demographics for any region of the world, uncertain disease transmission and vaccine efficacy. An optimal vaccination strategy specifies the proportion of individuals in a given household-type to vaccinate to bring the reproduction number to below one. The ICC-SP approach provides a quantitative method that allows to bound the expected excess of the reproduction number above one by an acceptable amount according to the decision-maker’s level of risk. This new methodology involves a multi-community household based epidemiology model that uses census demographics data, vaccination status, age-related heterogeneity in disease susceptibility and infectivity, virus variants, and vaccine efficacy. The new methodology was tested on real data for seven neighboring counties in the United States state of Texas. The results are promising and show, among other findings, that vaccination strategies for controlling an outbreak should prioritize vaccinating certain household sizes as well as age groups with relatively high combined susceptibility and infectivity.

Two‐stage distributionally robust optimization model for a pharmaceutical cold supply chain network design problem

AbstractPharmaceutical safety has received increasing attention from governments and corporations, and building a safe and effective pharmaceutical cold supply chain network has become an important issue. This paper proposes a two‐stage pharmaceutical cold supply chain network design problem considering drug safety, in which the drug demand, transportation costs, and drug safety risk costs are assumed to be random variables. Due to the presence of uncertainty and the fact that information about the distribution of uncertain parameters is often only partially known, a distributionally robust optimization method is used to handle the uncertainty. A two‐stage distributionally robust optimization model is constructed, in which the reliability of the estimation of the demand necessary to satisfy the entire cold chain network is ensured to be greater than a certain predetermined level by introducing an ambiguous joint chance constraint. The decision process for the problem can be divided into strategic and operational decisions with the goal of minimizing the total costs related to facility construction, the purchase of raw materials, drug production, transportation, and safety risks. By introducing an ambiguity set with mean and covariance information to describe the uncertain parameters, the two‐stage model is eventually reformulated as a standard second‐order cone program, thus making it computationally tractable. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed models and optimization methods.