Chance Constraints Research Articles

Optimization of maintenance for complex manufacturing systems using stochastic Remaining Useful Life prognostics

This paper leverages Remaining Useful Life (RUL) prognostic information for preventive maintenance planning in complex manufacturing factories. Such a factory consists of multiple complex systems that use redundant components as backups to ensure system availability. The purpose is to minimize production fluctuations in the factory due to maintenance or system breakdown. To achieve this, we propose a Mixed-Integer Linear Programming (MILP) model that minimizes the overall production loss. Furthermore, we incorporate random RUL decrease rates according to reliability theory and extend the MILP model using chance-constrained programming. A novel approximation method for dealing with chance constraints is proposed to solve the stochastic model and approximation properties are analyzed. Computational results show that (i) our MILP model can provide more reasonable maintenance decisions compared to the literature. (ii) Sensitivity analysis reports that RUL prognostics and thresholds significantly impact maintenance decisions. (iii) The analysis of the impacts of confidence levels in chance constraints provides decision-makers with the flexibility to adjust maintenance planning under different levels of uncertainty.

Rolling dispatch framework of integrated electricity-gas system considering power-to-gas facilities and natural gas network dynamics: Modelling and case studies

Rolling dispatch strategy is widely used in power systems to handle the uncertainty and volatility of renewable energy. Meanwhile, flexible gas-fired units and emerging power-to-gas (P2G) facilities integrate the power system and natural gas system as an integrated electricity-gas system (IEGS). Hence, rolling dispatch should be applied to the whole IEGS. This paper designs the rolling dispatch framework in the IEGS and proposes the partial-differential-equations-involved (PDEs) mathematical models that accurately formulate the natural gas network dynamics and the operation properties of P2G facilities to retain the facticity of the IEGS in the rolling dispatch process (e.g., the linepack characteristics of natural gas networks and the physical limits of the P2G facilities). The impact of uncertain wind power on the IEGS operation is also considered. In order to reduce the computational burdens, this paper employs a combination of implicit difference method and the technique of linearization to transform the original rolling dispatch model with PDEs and chance constraints into mixed-integer linear programming which is solvable by off-the-shelf solvers. The results of the case studies demonstrate that the proposed rolling dispatch strategy promotes renewable energy accommodation, enhances the flexibility, security, and efficiency of the IEGS, and mitigates the impact of renewable energy's uncertainty and volatility on the IEGS.

Optimization Methodologies for Reserve with Large-scale Renewables Integration

Renewable energy has unpredictable intermittency and variability, increasing the complexity of grid dispatch. Fuzziness theory is introduced under the uncertainty of large-scale intermittent power supply. Renewable power is characterized by fuzziness parameters. Meanwhile, the conventional deterministic unit commitment model is transformed into a chance-constraint unit commitment with fuzziness parameters. The explicit equivalence classes of chance constraints with triangular fuzziness parameters are presented to clarify chance constraints. Also, the effectiveness of the proposed model is validated in a standard IEEE test system.

Efficient Safety-Enhanced Velocity Planning for Autonomous Driving With Chance Constraints

Velocity planning is an important module of autonomous driving, which aims to generate the velocity profile given a reference path. However, most existing algorithms fail to adequately address the uncertainty inherent in driving contexts, leading to potentially risky situations. To this end, we propose an efficient safety-enhanced velocity planning algorithm (ESEVP), which uses chance constraints to take uncertainties from trajectory prediction and velocity tracking into account, arising great improvement in driving safety. In addition, ESEVP formulates velocity planning as quadratic programming and explores candidate solutions through a fast planning space construction method, which ensures efficiency and covers all the interaction possibilities. Experimental results obtained from various scenarios demonstrate that ESEVP outperforms recent state-of-the-art methods in terms of safety, comfort, and driving efficiency. Besides, we successfully deploy ESEVP in real traffic, showcasing its competitive capabilities in practice.

A new fuzzy DEA network based on possibility and necessity measures for agile supply chain performance evaluation: A case study

With respect to the necessity (Nec) measure, suggesting by Dubois and Prade, there is a dearth of any procedure or formula to deal with equality chance constraints. Therefore, this study attempted to introduce (Nec) and possibility (Pos) equality constraints in a real-case network non-oriented slack based model problem via agile supply chain (ASC) theory. The efficiency of different levels of agility of a dairy supply chain (SC), using a new fuzzy network Date Envelopment Analysis slack based model (FNDEA-SBM) on the basis of (Pos)/(Nec) measures, has been measured in this paper. The measurement took place in a multi-stage process. Since the agility features were rather qualitative in nature, the measures were likely to be delineated subjectively using linguistic expressions and fuzzy data. The application of (Pos)/(Nec) measures and different confidence levels made decision makers capable of getting information of all efficiency and sub efficiency scores through optimistic-pessimistic lenses and taking appropriate decisions on the basis of different optimistic-pessimistic points of view and confidence levels. The validity of the suggested approach was demonstrated through a real-life case study related to Iranian top twenty dairy firms. The results indicated the success of the suggested approach and computations were straightforward.

Data‐driven distributionally robust supplier selection and order allocation problems considering carbon emissions

AbstractUnder the constraints of uncertainty and carbon emission policies, reasonable and flexible supplier selection and order allocation can minimize the total cost, which is of great significance to the supply chain. We study sustainable supplier selection and order allocation problems with multiple periods, multiple suppliers, and multiple products under a carbon cap‐and‐trade policy. We exploit distributionally robust versions of supplier selection and order allocation problems with an explicit distributional uncertainty set and derive computationally tractable reformulations of the distributionally robust optimization model with chance constraints by cone duality theory and conditional value‐at‐risk measure. Numerical experiments show that the proposed distributionally robust optimization models with nice computational tractability and probability guarantees are superior to the existing distributionally robust chance‐constrained optimization models in terms of cost and risk aversion. Relevant management insights on different carbon emission regulations and the application of distributionally robust optimization models are provided for policymakers and enterprises' decision‐makers based on the derived results.

A statistical moment-based spectral approach to the chance-constrained stochastic optimal control of epidemic models

This paper presents a spectral approach to the uncertainty management in epidemic models through the formulation of chance-constrained stochastic optimal control problems. Specifically, a statistical moment-based polynomial expansion is used to calculate surrogate models of the stochastic state variables of the problem that allow for the efficient computation of their main statistics as well as their marginal and joint probability density functions at each time instant, which enable the uncertainty management in the epidemic model. Moreover, the surrogate models are employed to perform the corresponding sensitivity and risk analyses. The proposed methodology provides the designers of the optimal control policies with the capability to increase the predictability of the outcomes by adding suitable chance constraints to the epidemic model and formulating a proper cost functional. The chance-constrained optimal control of a COVID-19 epidemic model is considered in order to illustrate the practical application of the proposed methodology.

ADMM-based joint rescheduling method for high-speed railway timetabling and platforming in case of uncertain perturbation

The operation of high-speed trains is susceptible to perturbations such as strong winds and heavy snowfall, causing them to deviate from operating schedules. A joint mixed-integer linear programming (MILP) model based on the event-activity network is proposed to reschedule the train timetable and platform assignment jointly. Trains traveling in different directions are simultaneously considered in this model, and they can occupy the available reverse platforms in stations. Furthermore, to strengthen the robustness of the rescheduling scheme in case of uncertain perturbations, such as variable temporary speed restrictions caused by changing wind speed, the uncertainty is modeled using scenario-based chance constraints, which are then converted into deterministic constraints. Model predictive control (MPC) divides the entire time horizon into several stages. In each stage, the proposed model is decomposed according to directions. The constraints for trains traveling in different directions are dualized by the Alternating Direction Method of Multipliers (ADMM), which makes the sub-problems decoupled and solvable in parallel in each iteration of ADMM, promoting real-time performance. The real timetable and line data of Beijing–Shanghai HSR are utilized to investigate the effect of the proposed method. Compared to the commonly used strategies, it can significantly reduce delays and the number of affected events. The proposed method also shows relatively high robustness compared to the method in the case of predefined perturbations. It causes fewer conflicts and fewer needs for rescheduling when facing the changing perturbation and gets rescheduling results with higher quality.

A new uncertain multi-objective programming model with chance-entropy constraint for advertising promotion.

The COVID-19 outbreak has forced people to stay at home to prevent the spread of the virus. In this case, social media platforms have become the main communication venue for people. Online sales platforms have also become the main field for people's daily consumption. So, how to make full use of social media to carry out online advertising promotion, and then achieve better marketing, is one of the core issues that the marketing industry must pay attention to and solve. Therefore, this study takes the advertiser as the decision-maker, maximizes the number of full playing, likes, comments and forwarding, and minimizes the cost of advertising promotion as the decision-making goals, and Key Opinion Leader (KOL) selection as the decision vector. Based on this, a multi-objective uncertain programming model of advertising promotion is constructed. Among them, the chance-entropy constraint is proposed by combining the entropy constraint and the chance constraint. In addition, the multi-objective uncertain programming model is transformed into a clear single-objective model through mathematical derivation and linear weighting of the model. Finally, the practicability and effectiveness of the model are verified by numerical simulation, and decision-making suggestions for advertising promotion are put forward.

Optimal Modeling of Sustainable Land Use Planning under Uncertain at a Watershed Level: Interval Stochastic Fuzzy Linear Programming with Chance Constraints

In this paper, an uncertain interval stochastic fuzzy chance constraint land use optimal allocation method is proposed and applied to solve the problem of land use planning in river basins. The UISFCL-LUP method is an aggregation of interval parametric programming, fuzzy linear programming and chance constraint programming which can cope with uncertain problems such as interval value, fuzzy set and probability. In this paper, the uncertain mathematical method is explored and studied in the optimal allocation of land use in the next two planning periods of Nansihu Lake Basin in China. Moreover, it was proved that ISFCL-LUP can deal with the uncertainty of interval, membership function and probability representation and can also be used to solve the land use planning and land use strategy analysis under uncertain conditions. On the basis of model calculations, we obtained the optimal allocation results for six types of land use in four regions over two planning periods based on different environmental constraints. The results show that the optimized λ value (that is, the degree of satisfaction with all the model conditions) is in the range of [0.54, 0.79] and the corresponding system benefits are between [18.4, 20.4] × 1012 RMB and [96.7, 109.3] × 1012 RMB. The results indicate that land managers can make judgments based on the different socio-economic development needs of different regions and determine strategic land use allocation plans under uncertain conditions. At the same time, the model obtained interval solutions under different system satisfaction and constraint violation probabilities, which helps land managers to analyze the importance of land system optimization and sustainable development more deeply.

Optimal energy management of integrated energy systems for strategic participation in competitive electricity markets

The uncertainties originating from electricity markets, renewable distributed generations (RDG) and multi-energy loads pose huge challenges to the optimal energy management of the integrated energy system (IES) in competitive electricity markets. This paper develops an energy management framework (EMF) with a two-stage process for IES strategic participation in electricity markets, which can coordinate the operation of IES to comply with the two-settlement mechanism of the day-ahead and real-time markets. The former stage incorporates the conditional value-at-risk to make day-ahead bidding decisions for risk aversion towards uncertainties introduced by the electricity market. The latter coordinates the day-ahead bidding electricity to implement energy dispatch of IES for real-time market participation. Furthermore, supply-demand balances are formulated into chance constraints in the day-ahead bidding while real-time energy dispatch is implemented based on rolling model predictive control in response to uncertainties due to RDG and multi-energy loads. The proposed EMF is extensively assessed through two case studies in the PJM and Guangdong electricity markets. The numerical results demonstrate that the proposed EMF can effectively achieve day-ahead bidding of IES with risk aversion towards unexpected increases in operating costs caused by market uncertainties and determine the optimal operation scheme for its participation in the real-time market.

Two-stage data-driven dispatch for integrated power and natural gas systems by using stochastic model predictive control

The optimal dispatch of the integrated power and natural gas systems can increase the utilization rate of renewable energy and energy efficiency while decreasing operation costs. The common prediction errors of wind power and electric load have the potential to negatively impact the normal operation of the integrated power and natural gas systems. A two-stage data-driven dispatch strategy is proposed to reduce this effect, consisting of the day-ahead dispatch stage and the intraday rolling dispatch stage using stochastic model predictive control (MPC). In the day-ahead dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators are built, and the day-ahead tie-line power is obtained and regarded as input parameters to the intraday dispatch stage. In the intraday dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators with the latest rolling prediction data are built, and the remaining control variables are obtained. The distribution characteristics of the stochastic prediction errors of wind power and electric load are captured and described by the variational Bayesian Gaussian mixture model with massive historical data. Then the original stochastic mixed-integer nonlinear programming problem is converted to a tractable deterministic one by the quantile-based analytical reformulation and convex relaxation technique. Finally, the proposed strategy is verified by the numerical experiments based on a modified IEEE 33-bus system integrated with a 10-node natural gas system and a micro hydrogen system. The numerical results demonstrate that the proposed strategy reduces the actual costs and decreases the violation rate caused by the stochastic prediction errors of wind power and electric load.

Data‐driven chance‐constrained dispatch for integrated power and natural gas systems considering wind power prediction errors

AbstractStochastic wind power prediction errors hurt the normal operation of integrated power and natural gas systems (IPGS). First, the data‐driven stochastic chance‐constrained programming method is applied to deal with wind power prediction errors, and its probability distribution is accurately fitted by variational Bayesian Gaussian mixture model with massive historical data. In addition, the data‐driven chance constraints of tie‐line power and reserve capacity of gas turbine are built. Next, to utilize wind power more reasonably, the operational characteristics and optimal commitment of power‐to‐hydrogen devices are considered and modelled in proposed strategy to reflect the actual situation of IPGS. Then, the original complicated dispatch problem is converted into a tractable second‐order cone programming problem via convex relaxation and quantile‐based analytical reformulation techniques. Finally, the effectiveness of the proposed strategy is validated by numerical experiments based on a modified IEEE 33‐bus system integrated with a 10‐node natural gas system and a micro hydrogen system.

Wasserstein distance‐based distributionally robust parallel‐machine scheduling

Recent research on distributionally robust (DR) machine scheduling has used a variety of approaches to describe the region of ambiguity of uncertain processing times by imposing constraints on the moments of the probability distributions. One approach that has been employed outside machine scheduling research is the use of statistical metrics to define a distance function between two probability distributions. Adopting such an approach, we study Wasserstein distance-based DR parallel-machine scheduling, where the ambiguity set is defined as a Wasserstein ball around an empirical distribution of uncertain processing times corresponding to finitely many samples. The objective is to minimize a DR objective that concerns the worst-case expected total completion time-related cost over all the distributions arising from the Wasserstein ambiguity set, subject to DR chance constraints on the machine service capacity. We show that the problem can be equivalently re-formulated as a mixed-integer linear program (MILP), which has a more simplified formulation when the bounded support set reduces to a left bounded one. To solve the resulting model, we develop a tailored branch-and-Benders-cut algorithm incorporating some enhancement strategies, including in-out Benders cut generation, aggregated sample group cut generation, and two-stage Benders cut generation, which significantly outperforms the CPLEX solver. Experiment results on comparing our model with the deterministic and stochastic counterparts and the model with first-order moment ambiguity set illustrate the benefits of considering distributional ambiguity and Wasserstein ambiguity set.

Stochastic non‐cooperative game for safety control of shared control vehicle in uncertain environments

AbstractThere are great challenges in the driving of human–machine shared control vehicles (HSCVs) in uncertain environments. Aiming at solving the collision avoidance problem of HSCV considering the uncertainty of surrounding vehicles (SVs), a human–machine cooperative control method for collision avoidance based on stochastic non‐cooperative game (SNCG) is proposed. This method realizes the safe driving of HSCV combined with the prediction of surrounding environment and constructs the interaction model between a driver and an automation system based on the game theory. Firstly, a vehicle–driver coupling model with driver's lateral and longitudinal operation characteristics is established. Next, utilizing the prediction equation, the prediction of the SV motion considering uncertainty is realized. Then, the safety constraints of HSCV are established by considering the chance constraints and the saturation characteristics of the actuators. Finally, based on the distributed stochastic model predictive control (DSMPC) method, the coupled optimization problem with chance constraints is constructed. To solve the coupled optimization problem with chance constraints, the idea of Nash equilibrium solution and the method of chance constraints tightening is adopted. The simulation results illustrate that the proposed method ensures the safe driving of HSCV under uncertain environments and realize the cooperative control between the automation system and the driver.

Computationally Efficient Data-Driven Joint Chance Constraints for Power Systems Scheduling

Although data-driven nonparametric Joint Chance Constraints (JCCs) may lead to more reliable decision-making than individual chance constraints, their computational complexity is a major bottleneck. This paper presents computationally efficient data-driven nonparametric joint chance-constrained programming for multi-interval power systems management. Reserve and transmission line constraints are modeled as data-driven JCCs. Piecewise uniform kernel functions incorporate historical data of uncertain parameters into optimization. Data-driven nonparametric JCCs are modeled as a product of integrated kernel functions. Two approaches are proposed to linearize data-driven nonparametric JCCs. i) The noncontinuous kernel function is linearized with Special Ordered Sets of type 1 (SOS1) variables. ii) A tight convex envelope of multilinear monomial terms, which appear due to the product of kernel functions, is approximated by an optimization subproblem making the scheduling problem bi-level optimization. The continuity and linearity of the lower-level convex envelope approximation subproblem allow replacing it with optimality conditions to form a single-level scheduling problem. Simulation results show the tightness of the proposed linearization approaches and the computational efficiency of data-driven JCC programming.

Hierarchical Central-Local Inverter-Based Voltage Control in Distribution Networks Considering Stochastic PV Power Admissible Range

This paper proposes a hierarchical distribution network voltage control method considering active and reactive power coordination of PV units in both central and local control stages. In contrast to the traditional local control methods, the proposed method defines the admissible range (AR) of PV power and determines it via centralized optimization to realize active PV power curtailments in the local control stage. The affine decision rule (ADR) is adopted to control the reactive power with respect to active power within AR. A distributionally robust chance constraint is designed, based on statistical indices of active PV power, to assess the probability that active PV power generation would fall inside AR. A two-stage optimization problem that can simultaneously provide central and local control strategies is proposed, which is transformed into tractable formulas based on the Gauss Inequality as well as ADR and binary expansion techniques. The proposed method solutions are compared with those of the theoretically optimal method, a robust central-local control method, and a local control method to show the value of hierarchical inverter control with AR in reducing PV power curtailment and ensuring nodal voltages within limits.

Locational Marginal Pricing for Flexibility and Uncertainty With Moment Information

The continuous growth of variable renewable integration has greatly increased uncertainty and variability in the power system, resulting in unprecedentedly high demand for flexibility. Recently, there has been much attention in the power industry to manage the flexibility and uncertainty via market mechanisms. This paper presents a novel approach to jointly clearing energy, reserve, and uncertainty with moment information based on distributionally robust chance constraints (DR-CC). Locational marginal prices (LMP) are derived for both energy and reserve. Uncertainty is modeled with statistical moments of historical forecast errors, and uncertainty marginal prices (UMP) are defined to reflect the marginal cost of uncertainty following the cost-causation principle. The congestion cost of reserve trading is explicitly modelled, and is shown to fully cover financial transmission right (FTR) underfunding. We show that the proposed mechanism guarantees revenue adequacy and provides appropriate incentives for efficient operation and investment. Case studies are carried out to illustrate the impact of the proposed market scheme, and its feasibility is demonstrated on a testbed system based on a real-world 1934-bus grid.

A General Framework for Learning-Based Distributionally Robust MPC of Markov Jump Systems

We present a learning model predictive control (MPC) scheme for chance-constrained Markov jump systems with unknown switching probabilities. Using samples of the underlying Markov chain, ambiguity sets of transition probabilities are estimated which include the true conditional probability distributions with high probability. These sets are updated online and used to formulate a time-varying, risk-averse optimal control problem. We prove recursive feasibility of the resulting MPC scheme and show that the original chance constraints remain satisfied at every time step. Furthermore, we show that under sufficient decrease of the confidence levels, the resulting MPC scheme renders the closed-loop system mean-square stable with respect to the true-but-unknown distributions, while remaining less conservative than a fully robust approach. Finally, we show that the value function of the learning MPC converges from above to its nominal counterpart as the sample size grows to infinity. We illustrate our approach on a numerical example.

Simulated Annealing-Based Heuristic for Multiple Agile Satellites Scheduling Under Cloud Coverage Uncertainty

Agile satellites are the new generation of Earth observation satellites (EOSs) with stronger attitude maneuvering capability. Since optical remote sensing instruments cannot see through the cloud, the cloud coverage brings a significant influence on the satellite observation missions. The scheduling of multiple agile EOSs (AEOSs) is already complicated due to strong satellite maneuverability and onboard satellite energy constraints. Moreover, introducing cloud coverage uncertainty further increases scheduling complexity. Motivated by these challenges, we address the multiple AEOSs scheduling problem under cloud coverage uncertainty where the objective aims to maximize the entire observation profit. A chance constraint programming model is adopted to describe the uncertainty initially, and the observation profit under cloud coverage uncertainty is then calculated via a sample approximation method. To overcome the solving difficulty, an improved simulated annealing (ISA)-based heuristic including a fast insertion strategy is proposed for large-scale observation missions. Finally, extensive experiments are conducted to verify the effectiveness and efficiency of the proposed method. Experimental results show that the ISA-based heuristic outperforms other algorithms for the multiple AEOSs scheduling problem under cloud coverage uncertainty, which validates the proposed algorithm.