Joint Chance-constrained Programming Research Articles

Two-stage chance-constrained programming based on Gaussian mixture model and piecewise linear decision rule for refinery optimization

The two-stage chance-constrained program (CCP) is studied for a refinery optimization problem. In stage-I, the refinery decision-makers determine the type and quantity of crude oil procurement under operational uncertainties to maximize the expected profit under all possibilities. In stage-II, process unit flowrates are adjusted based on the realized uncertainties and available crude oil, while introducing probabilistic constraints to manage the off-spec risk. To solve such a two-stage optimization problem, we propose a novel approach using Gaussian mixture model (GMM) to characterize uncertainties, and piecewise linear decision rule for stage-II operations. Comparing to the conventional scenario-based mixed-integer linear program (MILP), our new approach offers three advantages. First, it leverages a well-developed global optimization scheme for joint CCP to avoid scenario generation and potential bias. Second, the data-driven GMM enables CCP to handle uncertainties with general distributions. Third, the stage-II variables are parameterized via Gaussian component induced piecewise linear decision rule to strike an excellent trade-off between optimality and computational time. A simplified refinery plant, consisting of distillation, cracker, reformer, isomerization, and desulfurization units, is used as a test bed to demonstrate the superiority of the proposed optimization method in solution time, probabilistic feasibility, and optimality over the large-scale scenario-based MILP.

A Fast Polytope-Based Approach for Aggregating Large-Scale Electric Vehicles in the Joint Market Under Uncertainty

The aggregation of electric vehicles (EVs) has been advocated as an effective means to manage and control large-scale plug-in EVs. However, the efficient aggregation is challenging due to the heterogeneity and uncertainty of individual EVs. To cope with this challenge, this paper puts forth a fast polytope-based aggregation approach, which could deal with the time-coupled, uncertain, and heterogeneous individual EV models. First, the feasible region of each EVs power profiles is expressed as a polytope under H-representation considering various operational constraints. To improve the aggregation efficiency, H-representation reconstruction and polytopic projection processes are designed to address the heterogeneity of EVs. The uncertainties of EVs are modeled by the distributionally robust joint chance-constrained programming. The original individual and aggregate feasible region of EVs are subsequently inner approximated by the homothets of a basic prototype. Last, a day-ahead optimal bidding model for charging service provider in a joint market, including energy, peak-shaving and reserve markets, is developed. Numerical simulation results based on real-world dataset demonstrate that the proposed aggregation method can improve both the computation efficiency and accuracy of approximate feasible region.

Joint chance-constrained program based electric vehicles optimal dispatching strategy considering drivers' response uncertainty

Electric vehicles (EVs) bring both opportunities and challenges to the distribution network due to their features of random behavior, storage capacity and charging flexibility. EV aggregator (EVA) can serve to unlock the dispatching potential of large scale of EVs and participate in the joint energy and reserve electricity market. In this paper, the improved EVA model for ancillary services is firstly established. Then, the dispatching potential of EVs considering drivers' response willingness and its uncertainty through EVA is explored. To obtain the optimal incentive price considering the driver response uncertainty, the joint chance-constrained program (JCCP) model is proposed. Due to the nonlinearity and nonconvexity of probability constraints, the developed JCCP model is intractable. A Monte Carlo based sequential convex approximation (SCA) algorithm is further developed to achieve the tractability and solve the JCCP model effectively. Case studies show that the proposed incentive program can reduce 15.1% net cost of EVA compared to uncoordinated charging. Besides, the proposed SCA algorithm can solve the JCCP problem with more accurate and less conservative results compared to the existing method, while achieving a decent computational efficiency for the day-ahead dispatching.

Distributionally robust joint chance-constrained programming: Wasserstein metric and second-order moment constraints

In this paper, we propose a new approximate linear reformulation for distributionally robust joint chance programming with Wasserstein ambiguity sets and an efficient solution approach based on Benders decomposition. To provide a convex approximation to the distributionally robust chance constraint, we use the worst-case conditional value-at-risk constrained program. In addition, we derive an approach for distributionally robust joint chance programming with a hybrid ambiguity set that combines a Wasserstein ball with second-order moment constraints. This approach, which allows injecting domain knowledge into a Wasserstein ambiguity set and thus allows for less conservative solutions, has not been considered before. We propose two formulations of this problem, namely a semidefinite programming and a computationally favorable second-order cone programming formulation. The models and algorithms proposed in this paper are evaluated through computational experiments demonstrating their computational efficiency. In particular, the Benders decomposition algorithm is shown to be more than an order of magnitude faster than a standard solver allowing for the solution of large instances in a relatively short time.

An Adjustable Scenario Optimisation Approach in Operating PV-Rich Distribution Systems

The embrace of intermittent distributed energy resources (DER) on the demand side increases the concerns around the safe operation of distribution networks. Existing solutions either neglect the available information around the distribution of uncertain parameters or underestimate DER capability in providing real-time recourse actions. In this paper, we propose an adjustable scenario optimisation-based solution approach that maximises the usage of DER generation within the operation horizon, while providing the required level of network security. Our modelling i) guarantees hard constraints satisfaction, ii) allows fast-acting devices, e.g., inverters, to adjust in response to live realisations, and iii) enables the operators to maintain the soft constraints with a required probability using a joint chance-constrained (JCC) program. We reformulate the JCCs using the scenario optimisation approach that directly takes empirical data and provides a pre-specified bound on the risk of constraint violation. To increase the scalability of our proposal, we introduce an approximation of JCCs to be used within the scenario optimisation approach. Through extensive comparisons and simulations, we show that our approach outperforms the conventional scenario optimisation and adjustable robust approaches respectively by 83% and 80% in our experiments.

Distributionally robust joint chance-constrained programming with Wasserstein metric

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations.

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.

Flow shop scheduling with human–robot collaboration: a joint chance-constrained programming approach

ABSTRACT Human–robot collaboration has been incorporated into production and assembly processes to promote system flexibility, changeability and adaptability. However, it poses new challenges to resource allocation and production scheduling due to its intrinsic uncertainty and also the increasing complexity of resources. This paper investigates a stochastic flow shop scheduling problem in the context of human–robot collaboration. The goal is to achieve efficient utilisation of flexible resources including human workers and cobots and take full advantage of human–robot collaboration in production scheduling. A stochastic Cobb–Douglas production function is utilised to evaluate the production efficiency of human–robot collaboration considering instabilities of human performance. A joint chance-constrained programming model is formulated to ensure that the required system performance can be achieved. A CVaR approximation-based approach is proposed to solve the formulated model with mixed-integer variables and a nonconvex constraint. The effectiveness of the formulated model and the efficiency of the proposed solution approach are evaluated via numerical experiments. Computational results show the superiority of our solution approach over three other approaches including Bonferroni approximation, scenario approach and individual chance-constrained programming.

Optimal blending under general uncertainties: A chance-constrained programming approach

An optimization algorithm is proposed for blend planning with linear mixing law and general parameter uncertainties. The objective is to make the final product satisfy all quality specifications with high probability, and maximize the production profit by carefully determining the feedstock ratio. Conventional approaches that rely on deterministic optimization fail to account for parameter uncertainty, and thus may not generate a probabilistic feasible solution. The proposed work formulates the blend planning problem as a joint chance-constrained program (CCP). Using Boole’s inequality to decompose joint constraints and the Gaussian mixture model to characterize uncertainty distributions, a conservative deterministic approximation of CCP can be formulated. Through second-order cone relaxation, branch-and-bound, optimality-based bound tightening, and reformulate-linearization techniques, the global optimum of deterministic approximation can be found. A risk level adjustment procedure is presented to reduce the conservativeness and further improve the objective value of the solution if posterior evaluation is allowed. Two numerical cases, including steel and gasoline productions, are studied to show the solving time, probabilistic feasibility, and solution quality of the proposed optimization method.

Wasserstein distributionally robust chance-constrained program with moment information

This paper studies a distributionally robust joint chance-constrained program with a hybrid ambiguity set including the Wasserstein metric, and moment and bounded support information of uncertain parameters. For the considered mathematical program, the random variables are located in a given support space, so a set of random constraints with a high threshold probability for all the distributions that are within a specified Wasserstein distance from an empirical distribution, and a series of moment constraints have to be simultaneously satisfied. We first demonstrate how to transform the distributionally robust joint chance-constrained program into an equivalent reformulation, and show that such a program with binary variables can be equivalently reformulated as a mixed 0–1 integer conic program. To reduce the computational complexity, we derive a relaxed approximation of the joint DRCCP-H using McCormick envelop relaxation, and introduce linear relaxed and conservative approximations by using norm-based inequalities when the Wasserstein metric uses the lp-norm with p≠1 and p≠∞. Finally, we apply this new scheme to address the multi-dimensional knapsack and surgery block allocation problems. The results show that the model with a hybrid ambiguity set yields less conservative solutions when encountering uncertainty over the model with an ambiguity set involving only the Wasserstein metric or moment information, verifying the merit of considering the hybrid ambiguity set, and that the linear approximations significantly reduce the computational time while maintaining high solution quality.

A two-stage joint chance-constrained programming considering compound uncertainty of interval, random and fuzzy: a case study for agricultural water planning in an arid area

A two-stage joint chance-constrained programming considering compound uncertainty of interval, random and fuzzy: a case study for agricultural water planning in an arid area

An Analytical Safe Approximation to Joint Chance-Constrained Programming With Additive Gaussian Noises

We propose a safe approximation to joint chance-constrained programming where the constraint functions are additively dependent on a normally-distributed random vector. The approximation is analytical, meaning that it requires neither numerical integrations nor sampling-based probability approximations. Under mild assumptions, the approximation is a standard nonlinear program. We compare this new safe approximation to another analytical safe approximation for joint chance-constrained programming based on Boole's inequality through two examples representing the constrained control of linear Gaussian-Markov models. It is shown that our proposed safe approximation has a lower degree of conservatism compared to the one based on Boole's inequality.

Optimizing grinding operation with correlated uncertain parameters

ABSTRACT Parameters appearing in deterministic optimization (DO) models are kept constant during optimization, though uncertainty in these parameters is an inherent and unavoidable. Investigations about the effect of these uncertain parameters (UP) on the final outcomes are, therefore, necessary. Chance-Constrained Programming (CCP) can handle such situations by introducing the probability of constraint satisfaction and converting optimization under uncertainty problems into an equivalent deterministic optimization problem (EDOP). However, the most popular variant of this method neglects the aspects of parameter dependency for the ease of the analysis. Under the conditions when the correlations exist among UPs, the use of joint CCP with parameter dependency information is necessary to avoid misleading results arising from the assumption of no correlation. To show the impact of dependency of UPs on the outcomes of optimization, joint CCP has been adopted in this study for an industrial grinding process (IGP), which is non-linear in nature in terms of its system states as well as UPs. The effects of both confidence level and correlation coefficient have been shown on the Pareto optimal (PO) front for different sets of distributions among UPs while handling a multi-objective optimization problem (MOOP) of the Grinding Process.

Non-convex chance-constrained optimization for blending recipe design under uncertainties

A global optimization algorithm is proposed to design blending recipes for gasoline production with nonlinear mixing law and parameter uncertainty. Important fuels, such as gasoline, are produced by mixing several intermediate feedstocks in such a way that all quality specifications are met, and total profit is maximized. Conventional blending design approaches that rely on linear models and deterministic optimization may generate a suboptimal or infeasible solution due to model inaccuracy and failure to account for parameter uncertainty. The proposed work designs the blending recipe subject to chance constraints with normally distributed uncertain parameters and nonlinear mixing rule. The resulting non-convex joint chance-constrained program is solved to a near-global optimum through second-order cone relaxation, branch-and-bound, optimality-based bound tightening, and reformulate-linearization techniques. A case study involving nine feedstocks and two grades of gasoline is presented to demonstrate the effectiveness of the proposed method.

Coordinated Chance-constrained Optimization of Multi-energy Microgrid System for Balancing Operation Efficiency and Quality-of-service

To enhance the flexible interactions among multiple energy carriers, i.e., electricity, thermal power and gas, a coordinated programming method for multi-energy microgrid (MEMG) system is proposed. Various energy requirements for both residential and parking loads are managed simultaneously, i.e., electric and thermal loads for residence, and charging power and gas filling requirements for parking vehicles. The proposed model is formulated as a two-stage joint chance-constrained programming, where the first stage is a day-ahead operation problem that provides the hourly generation, conversion, and storage towards the minimization of operation cost considering the forecasting error of photovoltaic output and load demand. Meanwhile, the second stage is an on-line scheduling which adjusts the energy scheme in hourly time-scale for the uncertainty realizations. Simulations have demonstrated the validity of the proposed method, i.e., collecting the flexibilities of thermal system, gas system, and parking vehicles to facilitate the operation of electrical networks. Sensitivity analysis shows that the proposed scheme can achieve a compromise between the operation efficiency and service quality.

Workforce Assignment in Assembly Line Considering Uncertain Demand

In this work, we study a workforce assignment problem in the assembly line with several stations and executing some parts of different types. The objective is to minimize the number of workers over all the production line. Due to the complex business environment, changes in demand have great influence on production scheduling. Therefore, we consider uncertain demand in our investigated problem. To the best of our knowledge, we are the first to consider stochastic demand in assembly line problem with workers assignment. The main contributions include: 1) an ambiguity set is applied to portray the demand uncertainty, 2) a joint chance-constrained programming model is proposed, and 3) we choose two distribution-free approaches, namely Approximations based on Markov Inequality (AMI) and Mixed Integer Second-Order Conic Program (MI-SOCP), to approximate the chance constraints of the model. Finally, computational experiments are conducted to compare performances of the two methods.

Chance constrained simultaneous optimization of substations, feeders, renewable and non-renewable distributed generations in distribution network

The penetration of distributed generators (DGs) is continually increasing in the power sector due to its ability in enhancing technical specifications as well as providing a promising future for power generation in electric networks. The aforementioned objectives will be realized if DG units are allocated optimally and coordinately simultaneous with distribution network expansion planning. On the other hand, given the stochastic nature of renewable generation and severe fluctuations of load consumption and electricity price, the DGs planning problem should be accomplished under uncertainties. To address these issues, this paper proposes a novel joint chance constrained programming (JCCP) method to fulfill an acceptable level of constraint feasibility for optimal simultaneous expansion planning of HV/MV substations and multiple-DG units along with robust MV feeder routing problem. Our design objective is to determine the optimal site and size of sub-transmission substations and various DG units associated with optimally construction of network by implementing the feeder routing problem with aim to minimize the investment costs, energy not supplied (ENS) cost and energy purchasing cost from upstream network. The diverse objectives are mathematically formulated as an MINLP model and converted into a single-objective function through weighted sum method and subsequently has been minimized by adaptive genetic algorithm. Furthermore, the Taguchi method is utilized in order to furnish an efficient algorithm that can find a satisfactory solution. Finally, the effectiveness of the proposed method is investigated by applying it on the 54-bus distribution network and the obtained results are duly drawn and discussed.

Chance-Constrained Optimization for Refinery Blend Planning under Uncertainty

A near-global optimization approach is proposed to design blending recipes for refinery products under uncertainties. In the refining industry, the most valuable products, such as gasoline and diesel, are produced by blending several intermediate feedstocks to maximize profit and ensure that all qualities are on specification. Because of the presence of property uncertainties, optimal blending recipes under linear mixing laws should be designed by solving a linear program with joint chance constraints. However, joint chance-constrained programming is generally intractable even with Gaussian distributions, and thereby, it is usually converted to an individual chance-constrained (ICC) program to achieve a conservative approximation. To reduce this conservatism, we find a global optimal solution for the ICC program. In case studies, a multiproduct blending problem with 12 chance constraints and a crude oil procurement example with 14 chance constraints are studied to test and compare the proposed scheme with...

A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand

We study the single-item single-resource capacitated lot-sizing problem with stochastic demand. We propose to formulate this stochastic optimization problem as a joint chance-constrained program in which the probability that an inventory shortage occurs during the planning horizon is limited to a maximum acceptable risk level. We investigate the development of a new approximate solution method which can be seen as an extension of the previously published sample approximation approach. The proposed method relies on a Monte Carlo sampling of the random variables representing the demand in all planning periods except the first one. Provided there is no dependence between the demand in the first period and the demand in the later periods, this partial sampling results in the formulation of a chance-constrained program featuring a series of joint chance constraints. Each of these constraints involves a single random variable and defines a feasible set for which a conservative convex approximation can be quite easily built. Contrary to the sample approximation approach, the partial sample approximation leads to the formulation of a deterministic mixed-integer linear problem having the same number of binary variables as the original stochastic problem. Our computational results show that the proposed method is more efficient at finding feasible solutions of the original stochastic problem than the sample approximation method and that these solutions are less costly than the ones provided by the Bonferroni conservative approximation. Moreover, the computation time is significantly shorter than the one needed for the sample approximation method.

A joint chance-constrained programming approach for call center workforce scheduling under uncertain call arrival forecasts

We consider a workforce management problem arising in call centers, namely the shift-scheduling problem. It consists in determining the number of agents to be assigned to a set of predefined shifts so as to optimize the trade-off between manpower cost and customer quality of service. We focus on explicitly taking into account in the shift-scheduling problem the uncertainties in the future call arrival rates forecasts. We model them as independent random variables following a continuous probability distribution. The resulting stochastic optimization problem is handled as a joint chance-constrained program and is reformulated as an equivalent large-size mixed-integer linear program. One key point of the proposed solution approach is that this reformulation is achieved without resorting to a scenario generation procedure to discretize the continuous probability distributions. Our computational results show that the proposed approach can efficiently solve real-size instances of the problem, enabling us to draw some useful managerial insights on the underlying risk-cost trade-off.