Chance-constrained Optimization Research Articles

Data-Driven Chance-Constrained Schedule Optimization of Cascaded Hydropower and Photovoltaic Complementary Generation Systems for Shaving Peak Loads

The coordinated scheduling of cascade hydropower with photovoltaic (PV) power stations can significantly improve the utilization rate of delivery transmission lines. However, the inherent uncertainty associated with photovoltaic (PV) forecasts challenges the reliable and economic operation of the complementary energy system. Against this background, in this paper, a day-ahead, chance-constrained scheduling for cascaded hydro–photovoltaic complementary generation systems (CHPSs) considering the transmission capacity is proposed. Firstly, the uncertainty of PV forecast errors is simulated by a probability density function fitted using kernel density estimation with historical sampling data. Then, a chance-constrained optimization model considering peak-shaving demands of the receiving-end power grid is developed to determine the day-ahead optimal schedules of CHPSs. Also, complex hydraulic coupling and unit operation constraints of cascade hydropower are considered in the proposed model. To deal with the nonlinear and stochastic constraints, an efficient linearization method is adopted to transform the proposed model into a mixed-integer linear programming (MILP) problem. Finally, the effectiveness and feasibility are verified by case studies. The results show that the day-ahead schedule optimized by the proposed method can fully balance peak-shaving and photovoltaic accommodation while considering photovoltaic output uncertainty.

Chance-constrained vulnerability assessment of smart water distribution systems against stealthy false data injection attacks

Smart water distribution systems (SWDSs) have not only enhanced the monitoring and control of the entire water infrastructure but have also jeopardized its security and resilience. The state-of-the-art examined the uncertainties in the water supply network to obtain optimal design parameters to ensure hydraulic integrity. However, the investigations of uncertainties in SWDSs from the cybersecurity perspective remain unexplored. To address this limitation, this paper utilizes a robust chance-constrained optimization strategy to scrutinize the vulnerable location of SWDSs against False Data Injection (FDI) attacks. This is achieved by factoring in the probabilistic behavior of water demand at junctions that can potentially facilitate stealthy FDI attacks on the hourly measurements of the pump’s flow rate. The proposed nonlinear model is validated with several case studies reflecting the real-world cyberattack consequences including cutoff water supply of the network, enhanced power consumption of the pump, and the undesirable pressure surge in the system at strategic locations of SWDSs. As a result of the studied cases, the cyberattack consequences on the entire water supply network are obtained while highlighting the threat-prone regions of the network. This paper contributes to embedding additional security layers to the existing water infrastructure at the vulnerable links and junctions.

Simplified Approach to Mixed-Integer Chance-Constrained Optimization with Ensemble Streamflow Forecasts for Risk-Based Dam Operation

Simplified Approach to Mixed-Integer Chance-Constrained Optimization with Ensemble Streamflow Forecasts for Risk-Based Dam Operation

Chance-constrained optimal dispatch of integrated energy systems based on data-driven sparse polynomial chaos expansion

In integrated energy systems, stochastic variations of different energy types of loads and the increasing penetration of renewable energy generation have resulted in considerable uncertainties. These uncertainties pose significant challenges to the economics and safe operation of integrated energy systems. Conventional deterministic methods of optimal dispatch overlook the effects of uncertainties, while stochastic optimization, though accounting for uncertainties, often yields conservative solutions that may adversely affect the economic operations. Chance-constrained optimization can effectively deal with uncertainties and provide more flexibility in balancing operational risks and benefits by expanding the feasible region. Hence, this paper proposes a chance-constrained optimal dispatch method for integrated energy systems and employs data-driven sparse polynomial chaos expansion method to enhance solving efficiency. First, the proposed chance-constrained optimization aims to ensure the optimal economic operation with an affordable security confidence level that balances safety and economics compared with deterministic and stochastic optimizations. The introduced data-driven sparse polynomial chaos expansion method enables the fast computing of the output response using only historical data, i.e., without knowledge pertaining to the distribution functions. Moreover, an improved iterative verification structure is proposed, which further improves the convergence speed and accuracy. Finally, the advantages and feasibility of the proposed method are verified using a test case and compared with those of deterministic and stochastic optimization. Results show that the proposed method successfully reduces the operational cost and controls violation probability.

Distributionally-robust chance-constrained optimization of selective maintenance under uncertain repair duration

The joint selective maintenance and repairperson assignment problem with uncertain duration of maintenance actions is studied in this paper. This problem arises in mission-oriented assets that perform sequences of missions interspersed with scheduled pauses to allow maintenance activities. The goal is to find the optimal subset of components to maintain and the maintenance level to be carried out by each repairperson that maximizes system reliability for the following mission. We first reformulate the complex objective function via single-variable concave functions before applying piecewise-linear approximation. This tight approximation enables large instances of the problem to be solved efficiently. We also consider the commonly encountered case of probabilistic maintenance duration with their distributions estimated from limited historical data samples. Distributionally-robust chance constraints (DRCCs) are employed to ensure that the maintenance tasks assigned to each repairperson can be completed within the break for a set of distributions contained in a Wasserstein-1 ball around the empirical distribution of the field data. A CVaR-based tractable approximation is applied to the DRCCs such that the problem can be reformulated into a mixed integer linear program readily handled by standard solvers. Results from the numerical experiments conducted on large benchmark instances demonstrated the computational efficiency of the proposed models and the added value of considering distributional ambiguity when dealing with uncertain maintenance duration.

Chance-Constrained Optimization Formulation for Ship Conceptual Design: A Comparison of Metaheuristic Algorithms

This paper presents a new chance-constrained optimization (CCO) formulation for the bulk carrier conceptual design. The CCO problem is modeled through the scenario design approach. We conducted extensive numerical experiments comparing the convergence of both canonical and state-of-the-art metaheuristic algorithms on the original and CCO formulations and showed that the CCO formulation is substantially more difficult to solve. The two best-performing methods were both found to be differential evolution-based algorithms. We then provide an analysis of the resulting solutions in terms of the dependence of the distribution functions of the unit transportation costs and annual cargo capacity of the ship design on the probability of violating the chance constraints.

Characterizing Rational Transplant Program Response to Outcome-Based Regulation

U.S. health agencies periodically evaluate transplant programs based on their patients’ posttransplant survival outcomes, and a program is flagged for review if the number of unsuccessful transplants far exceeds what would be expected based on national averages. Some researchers have expressed concerns that these regulations might cause programs to reject high-risk patients, whereas others have questioned if such a response would be rational. In “Characterizing Rational Transplant Program Response to Outcome-Based Regulation,” Mildebrath et al. use chance-constrained optimization to demonstrate that it may, in fact, be rational for a transplant program to become more selective when evaluating transplant candidates for admittance to the waitlist. They also demonstrate that the regulations may unfairly penalize medium-sized programs. Moreover, their model quantifies which patients may be most at risk for adverse selection by programs. Their results provide insights to policymakers by quantitatively characterizing the response of rational programs to outcome-based regulations.

Chance-Constrained Optimization of Photovoltaic System Allocation considering Power Loss, Voltage Level, and Line Current

Changes are emerging that will significantly alter structure and operation of this century’s distribution networks, and photovoltaic (PV) systems will play greater role in energy sector, with implications for power system reliability. Considering uncertainties in solar irradiance and electrical loads and incorporating them into the optimization problem within an appropriate methodology is becoming increasingly important in reshaping distribution networks. In this paper, uncertainty scenarios are handled with Monte Carlo Simulation (MCS) under genetic algorithm (GA) and differential evolution- (DE-) based optimization, and probability distribution functions (pdf) of bus voltages and line current are obtained to be used in chance-constrained stochastic programming. This present study focuses on investigating impact of uncertainties in PV system operating under different irradiance scenarios on power loss with probabilistic constraints in distribution networks instead of precise deterministic limits to contribute more efficient and reliable use of energy. By combining meta-heuristic optimization and MCS technique under one framework, this paper contributes to knowledge base of how to allocate PV plants within distribution networks under chance-constrained strategy. In order to show the effectiveness of the proposed methodology, obtained optimization results are tested using MCS under set of uncertainty conditions and network constraints are evaluated for limit violation probabilities. The effectiveness of this method is investigated based on comparative results of two different optimization methods through probabilistic analysis and simulation.

A joint machine learning and optimization approach for incremental expansion of electric vehicle charging infrastructure

As Electric vehicle (EV) adoption increases worldwide, public charging infrastructure must be expanded to meet the growing charging demand. Furthermore, insufficient and improperly deployed public charging infrastructure poses a real risk of slowing EV adoption. The infrastructure thus needs to be expanded intelligently and flexibly while accounting for uncertain dynamics in future charging demand. Nevertheless, current methods for demand-based expansion often rely on rigid and error-prone tools, such as travel surveys and simple rules of thumb. The former is more appropriate for long-term, equilibrium scenarios, where we consider the charging network as a whole rather than incrementally expanding it. At the same time, the latter relies on business experience in a rapidly changing field. We propose a predictive optimization approach for intelligent incremental expansion of charging infrastructure. At each time step, we estimate the future charging demand through a Gaussian Process, which is subsequently used in a linear chance-constrained optimization method to expand the charging infrastructure incrementally. To develop and validate this framework, we account for environmental feedback by simulating user behavior changes based on historical charging records and considering an optimized charging network at every iteration. We apply this approach to a case study of EV charging in Dundee, Scotland. We compare different strategies and reasons for their pros and cons for monthly incremental expansion of the charging network. In particular, combining machine learning and optimization results in the cheapest expansion and one that serves the most demand.

Chance-Constrained Generic Energy Storage Operations Under Decision-Dependent Uncertainty

Compared with large-scale physical batteries, aggregated and coordinated generic energy storage (GES) resources provide low-cost, but uncertain, flexibility for power grid operations. While GES can be characterized by different types of uncertainty, the literature mostly focuses on decision-independent uncertainties (DIUs), such as exogenous stochastic disturbances caused by weather conditions. Instead, this manuscript focuses on newly-introduced decision-dependent uncertainties (DDUs) and considers an optimal GES dispatch that accounts for uncertain available state-of-charge (SoC) bounds that are affected by incentive signals and discomfort levels. To incorporate DDUs, we present a novel chance-constrained optimization (CCO) approach for the day-ahead economic dispatch of GES units. Two tractable methods are presented to solve the proposed CCO problem with DDUs: (i) a robust reformulation for general but incomplete distributions of DDUs, and (ii) an iterative algorithm for specific and known distributions of DDUs. Furthermore, reliability indices are introduced to verify the applicability of the proposed approach with respect to the reliability of the response of GES units. Simulation-based analysis shows that the proposed methods yield conservative, but credible, GES dispatch strategies and reduced penalty cost by incorporating DDUs in the constraints and leveraging data-driven parameter identification. This results in improved availability and performance of coordinated GES units.

Learning and optimization under epistemic uncertainty with Bayesian hybrid models

Hybrid (i.e., grey-box) models are a powerful and flexible paradigm for predictive science and engineering. Grey-box models use data-driven constructs to incorporate unknown or computationally intractable phenomena into glass-box mechanistic models. The pioneering work of statisticians Kennedy and O’Hagan introduced a new paradigm to quantify epistemic (i.e., model-form) uncertainty. While popular in several engineering disciplines, prior work using Kennedy–O’Hagan hybrid models focuses on prediction with accurate uncertainty estimates. This work demonstrates computational strategies to deploy Bayesian hybrid models for optimization under uncertainty. Specifically, the posterior distributions of Bayesian hybrid models provide a principled uncertainty set for stochastic programming, chance-constrained optimization, or robust optimization. Through two illustrative case studies, we demonstrate the efficacy of hybrid models, composed of a structurally inadequate glass-box model and Gaussian process bias correction term, for decision-making using limited training data. From these case studies, we develop recommended best practices and explore the trade-offs between different hybrid model architectures.

Data-driven distributionally robust chance-constrained optimization with large data set and outliers: Sequential sample removal algorithm for solution improvement

Data-driven distributionally robust chance-constrained optimization (DRCCP) is a powerful technique to handle optimization problems involving uncertainty in constraint functions. However, the outliers and extreme samples in the data set may deteriorate the decision quality of DRCCP. Although there are numerous outlier detection techniques, they are either unable to pinpoint the samples causing overly conservative solutions, or incompatible with DRCCP models. This work proposes a novel and widely compatible algorithm that generates a representative subset of the original data set and removes samples causing overly conservative solutions for the DRCCP problem. With the proposed approach, the DRCCP solution quality can be enhanced while simultaneously ensuring the solution feasibility. To illustrate its effectiveness, we examine two numerical examples and a nonlinear process optimization problem in our study.

Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.

Efficient Scenario Generation for Heavy-Tailed Chance Constrained Optimization

We consider a generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. As the most popular generic method for solving chance constrained optimization, the scenario approach generates sampled optimization problem as a precise approximation with provable reliability, but the computational complexity becomes intractable when the risk tolerance parameter is small. To reduce the complexity, we sample the risk factors from a conditional distribution given that the risk factors are in an analytically tractable event that encompasses all the plausible events of constraints violation. Our approximation is proven to have optimal value within a constant factor to the optimal value of the original chance constraint problem with high probability, uniformly in the risk tolerance parameter. To the best of our knowledge, our result is the first uniform performance guarantee of this type. We additionally demonstrate the efficiency of our algorithm in the context of solvency in portfolio optimization and insurance networks. Funding: The research of B. Zwart is supported by the NWO (Dutch Research Council) [Grant 639.033.413]. The research of J. Blanchet is supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397], the National Science Foundation [Grants 1820942, 1838576, 1915967, and 2118199], Defense Advanced Research Projects Agency [Award N660011824028], and China Merchants Bank.

Chance-constrained optimization for active distribution networks with virtual power lines

Active distribution networks (ADNs) experience uncertainties because of the high integration of distributed renewable energy resources (RESs). Stochastic analysis approaches are attracting increasing attention owing to the uncertain nature of RESs and load demands. This study proposes a chance-constrained optimal operation framework for ADNs considering virtual power lines (VPLs) solved via the second- order cone programming (SOCP) optimal power flow (OPF) problem. VPLs comprise virtual lines connected to each network’s energy storage systems (ESSs). If one ESS is charged, another ESS is discharged by the same amount. We reformulate the constraints into tractable forms using a simplified Z-bus sensitivity matrix and first-order Taylor expansion to solve via SOCP OPF because the chance constraints on voltage magnitudes are generally non-convex and intractable. The efficacy of the proposed approach was verified through its application to two modified IEEE 33-bus systems, each featuring different consumer units, including commercial, residential, and industrial units. The effectiveness of the proposed methods was demonstrated by comparing three different types of constraints: conventional hard constraints, soft constraints with penalties, and chance constraints. The results showed that although the chance constraints case had a slightly higher voltage violation rate, the network power loss was reduced by 3.4% compared to the conventional hard constraints case. Furthermore, modified 85-bus and 65-bus systems are carried out to verify the algorithm’s scalability. The proposed approach achieves a trade-off between voltage regulation and power loss minimization, with a voltage violation rate of 0.88% and a power loss reduction to 1.02%.

Robust Power Allocation for Integrated Visible Light Positioning and Communication Networks

Integrated visible light positioning and communication (VLPC), capable of combining advantages of visible light communications (VLC) and visible light positioning (VLP), is a promising key technology for the future Internet of Things. In VLPC networks, positioning and communications are inherently coupled, which has not been sufficiently explored in the literature. We propose a robust power allocation scheme for integrated VLPC Networks by exploiting the intrinsic relationship between positioning and communications. Specifically, we derive explicit relationships between random positioning errors, following both a Gaussian distribution and an arbitrary distribution, and channel state information errors. Then, we minimize the Cramer-Rao lower bound (CRLB) of positioning errors, subject to the rate outage constraint and the power constraints, which is a chance-constrained optimization problem and generally computationally intractable. To circumvent the nonconvex challenge, we conservatively transform the chance constraints to deterministic forms by using the Bernstein-type inequality and the conditional value-at-risk for the Gaussian and arbitrary distributed positioning errors, respectively, and then approximate them as convex semidefinite programs. Finally, simulation results verify the robustness and effectiveness of our proposed integrated VLPC design schemes.

Energy management for regional microgrids considering energy transmission of electric vehicles between microgrids

As the proliferation of electric vehicles (EVs) continues to accelerate, the inherent attributes of EVs warrant meticulous consideration in the realm of energy dispatch. In order to evaluate the ability of EVs as mobile energy storage, this paper presents an energy management framework for the microgrids' online dispatch, which accounts for the spatio-temporal energy transmission of EVs between microgrids. The energy management framework contains two iterative processes: optimizing charging price and guiding charging dispatch. To sufficiently capture the uncertainties of the renewable energy and load demand, chance-constrained optimization is utilized to determine the charging price by reasonable power allocation. To achieve a continuous and efficient control policy, a normalized advantage function-deep Q learning network (NAF-DQN) is developed for EV dispatch under V2G technology. The above two processes as a coupled optimization problem are solved alternately until convergence. Numerical cases considering energy transmission of EVs between microgrids are studied to demonstrate the superiority of the proposed dispatch framework. The simulation results indicate improved computational efficiency and higher-quality solution.

A distributionally robust production planning model for maximizing customer satisfaction with budget and carbon emissions constraints

This paper investigates a new stochastic green production planning problem without full knowledge about the probability distribution on customer demands. The only information regarding the demand is the means and covariance matrix. The goal of the study is to provide an eco-conscious manufacturer with solution methods capable of yielding, within a reasonable amount of computation time, a robust solution that maximizes customer satisfaction while taking into account the constraints of budgets and periodic carbon caps spontaneously set out of environmental awareness. We formulate the problem into a novel distributionally robust chance-constrained optimization model under demand uncertainty set. To effectively solve the model, two heuristics are proposed. The widely used sample average approximation (SAA) scheme is first deployed as a benchmark method. However, the computation time increases rapidly with the problem size. In some instances, even feasible solutions cannot be found within a reasonable amount of computation time. We then develop an approximated mixed-integer reformulation on the basis of second order cone program (SOCP). A real-world case is solved and extensive numerical experiments show that the SOCP heuristic is superior over the SAA heuristic in terms of computation time while the solution quality is only slightly lower in most of instances. Finally, sensitivity analysis on budgets, periodic carbon caps, the maximum budget overrun risk level, initial inventory levels, and the magnitude of demand uncertainty are conducted and useful managerial insights are derived.

Proactive Scheduling for Zero-Energy Device Networks With Fast Uplink Grant

To address the inefficiency in terms of resource utilization of grant-free (GF) protocols, the fast uplink (FU) grant medium-access protocol has been proposed. In this work, we design a proactive wireless power transfer (WPT) framework for FU grant zero-energy massive machine-type communication. Specifically, zero-energy devices (ZEDs) first harvest energy during the WPT phase and then transmit data based on the FU grant protocol. Moreover, a multi-arm bandit traffic prediction scheme is adopted. Then, considering that due to the FU grant protocol ZEDs may harvest energy for several WPT rounds prior to data transmission, we extract the ZEDs’ outage probability during uplink. Based on the derived expressions, a chance-constrained optimization problem is formulated, which minimizes the energy consumption for WPT, while guaranteeing an acceptable outage probability for the ZEDs during data transmission. Simulations validate the theoretical analysis and illustrate that the proposed scheme outperforms GF access.

Unlock the Thermal Flexibility in Integrated Energy Systems: A Robust Nodal Pricing Approach for Thermal Loads

The thermal flexibility of buildings in integrated energy systems (IES) has great potential to improve the operational economy and wind power consumption. To incentivize the thermal flexibility of buildings under uncertainties, we propose a robust nodal pricing (RNP) model for thermal loads based on the Stackelberg game approach. The RNP model adopts a bilevel framework in which the IES operator plays the leader at the upper level while the thermal load aggregators (TLA) play the followers at the lower level. The IES operator problem optimizes the heat prices and dispatch plan, which is modeled as a two-stage robust optimization problem to address the uncertainties in the renewables and the loads. The TLA problem optimizes the thermal loads of buildings to minimize the energy cost and thermal comfort loss, which is modeled as a distributionally robust chance-constrained optimization problem to address the uncertainty in outdoor temperature. Then, we convert the TLA model into deterministic quadratic programming and prove that Slaters condition holds under a mild assumption. Using it, the TLA model is equivalently converted into Karush-Kuhn-Tucker conditions, leading to a reformulation of a classical two-stage robust optimization model for the RNP model. Case studies compare different pricing methods and verify the superiority of the proposed method.