Stochastic Chance-constrained Programming Research Articles

Two-Stage Stochastic Chance-Constrained Fractional Programming Model for Optimal Agricultural Cultivation Scale in an Arid Area

AbstractIn the agricultural irrigation system, agricultural water availability and demand are significantly affected by seasonal variations. Additionally, agricultural land use in arid regions larg...

Optimal production strategy of bimetallic deposits under technical and economic uncertainties using stochastic chance-constrained programming

In order to catch up with reality, all the macro-decisions related to long-term mining production planning must be made simultaneously and under uncertain conditions of determinant parameters. By taking advantage of the chance-constrained programming, this paper presents a stochastic model to create an optimal strategy for producing bimetallic deposit open-pit mines under certain and uncertain conditions. The uncertainties of grade, price per product, and capacities of the various stages in the process of production of the final product were considered. The results of solving the deterministic and stochastic models showed that the stochastic model had a greater compatibility and performance than the other ones.

Public R&D project portfolio selection problem with cancellations

In this study, we address a public R&D project portfolio selection problem with project cancellations. For several reasons, a funded R&D project may be halted before finishing the planned research. When a project is canceled, most of its budget is usually unused and also some of the spendings can return to the funding organization. In the call-based R&D programs, usually project selection decisions are made in one go, and, in the current call, it is not possible to award new projects with the unused budget. Decision-makers (DMs) of funding organizations can benefit from considering possible project cancellation situations to improve the budget utilization. We consider two cases. In the first case, we assume that cancellation probability of a project cannot be assessed but the DM can estimate the number of projects that will be canceled. In the second case, we assume that for each project, a cancellation probability can be assessed. For the first problem, we develop a mixed-integer linear programming formulation and a dynamic programming algorithm. For the second problem, we develop a chance-constrained stochastic programming formulation that can be solved as a mixed-integer second-order cone program. Our computational results show that practical-size problems can be solved by the proposed solution approaches.

A stochastic fuzzy chance-constrained programming model for energy–environment system planning and management in the City of Beijing

ABSTRACTA stochastic and fuzzy chance-constrained programming (SFCCP) model was developed in this study for supporting energy–environment management in the city of Beijing, China. SFCCP was capable of tackling the variables in constraints as fuzzy random variables, which were integration of randomness and vagueness. SFCCP was applied to an energy–environment management system in the city of Beijing. The study results indicated that SFCCP was useful in helping decision makers gain in-depth insights into proposed management system and establish environment-friendly energy allocation alternatives. The application of SFCCP is expected to provide a good demonstration to energy–environment management problems under complex uncertainties.

Resilience-Oriented Critical Load Restoration Using Microgrids in Distribution Systems

A resilience-oriented service restoration method using microgrids to restore critical load after natural disasters is proposed in this paper. Considering the scarcity of power generation resources, the concept of continuous operating time (COT) is introduced to determine the availability of microgrids for critical load restoration and to assess the service time. Uncertainties induced by intermittent energy sources and load are also taken into account. The critical load restoration problem is modeled as a chance-constrained stochastic program. A Markov chain-based operation model is designed to describe the stochastic energy variations within microgrids, based on which the COT is assessed. A two-stage heuristic is developed for the critical load restoration problem. First, a strategy table containing the information of all feasible restoration paths is established. Then the critical load restoration strategy is obtained by solving a linear integer program. Numerical simulations are performed on the IEEE 123-node feeder system under several scenarios to demonstrate the effectiveness of the proposed method. The impacts of fault locations, available generation resources, and load priority on the restoration strategy are discussed.

A Solution to the Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Wind Energy Integration

This paper presents a unit commitment problem with uncertain loads and wind power. A chance-constrained two-stage stochastic program programming formulation is proposed for the stochastic day-ahead scheduling. The loss of load probability (LOLP) and the probability of wind power utilization less than certain level, referred to as loss of wind probability (LOWP), and transmission line overloading probability (TLOP) are modeled as chance constraints, to provide high reliability of power supply and to ensure the high utilization of wind generation. The chance-constrained stochastic programming formulation is converted into an equivalent deterministic formulation by a sequence of approximations and verification. Correlations between loads at different buses and correlations between wind power at different buses are also considered by the proposed approach. Numerical tests are performed on a six-bus system and a modified IEEE 118-bus system. Results validate the viability of the proposed algorithm formulation for the day-ahead stochastic scheduling. The impact of different levels of LOLP and LOWP on the required up/down spinning reserves are also presented in this paper.

Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.

A stochastic equilibrium chance-constrained programming model for municipal solid waste management of the City of Dalian, China

In this paper, a stochastic equilibrium chance-constrained programming (SECCP) model was developed for tackling the municipal waste management issue under uncertainty. The main advantage of this model is that it effectively reflected the dual-random characteristics of uncertain parameters through incorporating the opinions and judgments from various respondents into the parameter identification processes. This will lead to birandom variables, where their mean values and standard deviations are allowed to be the random variables, instead of the fixed values. The generation of birandom variables will enrich the stochastic optimization theory and improve the accuracy and rationality of parameters design and estimation. The equilibrium chance-constrained programming algorithm was used to solve the SECCP model, which is capable of tackling birandom variables and is overcoming limitations of traditional stochastic chance-constrained programming while parameters with normal distribution are required strictly. Currently, the application of SECCP model in the environmental management fields was limited. As the first attempt, the regional waste management of the City of Dalian, China, was used as a study case for demonstration. A variety of solutions are beneficial in providing decision space to the local managers through designing and adjusting the constraints-violation levels. This solution process also reflected trade-off between system economy and reliability. The successful application in regional waste management system is expected to be a good example for tackling other similar problems.

Day-ahead Economic Dispatch of Wind Integrated Power System Considering Optimal Scheduling of Reserve Capacity

This paper proposes a novel day-ahead dynamic economic dispatch model for the wind integrated power system, aiming to minimize the total cost of generation and reserve capacity of all units. The proposed model is formulated as a chance-constrained stochastic nonlinear programming (CSNLP) problem, in which the constraints of regulating characteristics of units are taken into account. A two-stage iterative algorithm, based on sequential quadratic programming (SQP) method and chance-constraint checking, is developed to solve the CSNLP efficiently. Finally, the proposed dispatch strategy is illustrated in an IEEE30 power system with six generators and a wind farm.

A coupled random fuzzy two-stage programming model for crop area optimization—A case study of the middle Heihe River basin, China

Crop area optimization is essential for agricultural water management. However, decision makers are challenged by the complexity of fluctuating stream condition and irrigation quota, varying economic profits and crop yield, as well as grayness and errors in the estimated modeling parameters. A random-fuzzy-variable-based inexact two-stage stochastic chance-constrained programming (RFV-ITSCCP) model is developed for crop area optimization in response to such complexities in more efficient and sustainable ways. The model is capable of tackling parameters’ dual uncertainties of both randomness and fuzziness and meanwhile reflecting uncertainties expressed as intervals and probability distributions. Moreover, the developed model is helpful for managers in gaining insight into the tradeoffs between the system benefit and the constraint-violation risk. The RFV-ITSCCP model is applied in crop area optimization in the middle reaches of Heihe River basin, northwest of China. The results provide crop area allocation under various flow levels with the maximum system benefit under uncertainty, minimizing the penalty due to water deficit especially in arid and semi-arid area. The results show that the system efficiency (benefit per unit area) increases by 13.2% than conventional linear programming (LP) model with the most credible fuzzy number values under the same condition of water supply, irrigation regimes and socio-economic conditions, which can demonstrate the feasibility and applicability of the developed RFV-ITSCCP model. Sensitivity analysis is also conducted to analyze the impacts of key model parameters on the results. The obtained results are valuable for supporting the adjustment of the existing crop area patterns and identifying desired crop area schemes under complex uncertainties.

Optimal allocation and adaptive VAR control of PV-DG in distribution networks

The development of distributed generation (DG) has brought new challenges to power networks. One of them that catches extensive attention is the voltage regulation problem of distribution networks caused by DG. Optimal allocation of DG in distribution networks is another well-known problem being widely investigated. This paper proposes a new method for the optimal allocation of photovoltaic distributed generation (PV-DG) considering the non-dispatchable characteristics of PV units. An adaptive reactive power control model is introduced in PV-DG allocation as to balance the trade-off between the improvement of voltage quality and the minimization of power loss in a distribution network integrated with PV-DG units. The optimal allocation problem is formulated as a chance-constrained stochastic programming (CCSP) model for dealing with the randomness of solar power energy. A novel algorithm combining the multi-objective particle swarm optimization (MOPSO) with support vector machines (SVM) is proposed to find the Pareto front consisting of a set of possible solutions. The Pareto solutions are further evaluated using the weighted rank sum ratio (WRSR) method to help the decision-maker obtain the desired solution. Simulation results on a 33-bus radial distribution system show that the optimal allocation method can fully take into account the time-variant characteristics and probability distribution of PV-DG, and obtain the best allocation scheme.

Chance-Constrained Day-Ahead Scheduling in Stochastic Power System Operation

This paper proposes a day-ahead stochastic scheduling model in electricity markets. The model considers hourly forecast errors of system loads and variable renewable sources as well as random outages of power system components. A chance-constrained stochastic programming formulation with economic and reliability metrics is presented for the day-ahead scheduling. Reserve requirements and line flow limits are formulated as chance constraints in which power system reliability requirements are to be satisfied with a presumed level of high probability. The chance-constrained stochastic programming formulation is converted into a linear deterministic problem and a decomposition-based method is utilized to solve the day-ahead scheduling problem. Numerical tests are performed and the results are analyzed for a modified 31-bus system and an IEEE 118-bus system. The results show the viability of the proposed formulation for the day-ahead stochastic scheduling. Comparative evaluations of the proposed chance-constrained method and the Monte Carlo simulation (MCS) method are presented in the paper.

Multi‐objective stochastic optimal planning method for stand‐alone microgrid system

To achieve economic and environmental benefit for the stand-alone microgrid consisting of diesel generators, wind turbine generators, photovoltaic generation system and lead-acid batteries, a multi-objective stochastic optimal planning method and a stochastic chance-constrained programming model are presented. In the model, the optimal objective is to simultaneously minimise the total net present cost and carbon dioxide emission in life cycle; the type and capacity of distributed generation units have been selected as the optimal variables; the loss of capacity is adopted as probability index constraint; the coordinated operation strategies between diesel generators and battery, the multi-unit operation constraints of diesel generators and the reserve capacity have been considered in the hard-circle operation strategy. Considering the uncertainties of wind speed, clearness index and load demand, Markov process transition probability matrix is adopted to synthesise those time series data. Optimal planning for an island microgrid system has been carried out by the planning system for microgrid (PSMG), a self-developed optimal planning software based on the multi-objective stochastic optimal planning method for stand-alone microgrid system.

Supply Chain Risk Management in Chinese Chemical Industry based on Stochastic Chance-Constrained Programming Model

It is necessary to research on how to manage risks in Chinese chemical industry supply chain effectively. After analyzing supply chain risks and mitigation strategies in Chinese chemical industry supply chain, firstly this paper defines the supply chain risk management problem in Chinese chemical industry. Then this paper develops a stochastic chance-constrained programming model to select optimal mitigation strategies effectively to achieve risk reduction goa ls without exceeding the expenses budget. Finally, this paper gives a numerical example and discusses how the change of model parameter affects the optimal solution.

An Algorithm for Solving Stochastic Chance-Constrained Programming Problem

Stochastic chance-constrained programming which is one of important stochastic programming widely exits in different fields. For searching an algorithm that can more effectively solve this problem,a new algorithm for its combined stochastic particle swarm optimization with stochastic simulation for approximation of the fitness function and checking feasibility of solution is presented. It overcomes the defaults such as needing a long time, complex calculation,resapsing into local optimum in the hybrid intelligence algorithm based on GA. After testing its performance and comparing with GA, the results show that the algorithm is more preferable.

A robust single row facility layout problem with fuzzy random variables

The facility layout problem is the optimal arrangement of facilities in the plant area. Based on layout configurations, there are several types of the facility layout problem. The single row facility layout problem (SRFLP) is one them in which facilities have to be placed along a line. Although there are exact approaches, based on the integer programming model for solving SRFLP, a huge number of variables and constraints should be used in this model. This paper presents a new exact method to SRFLP based on a new class of variables and an extended branch and bound method (B&B). First, the SRFLP is formulated by new decision variables. To solve it, a new branching scheme for B&B algorithm is presented. Subsequently, we introduce a fuzzy robust single row facility layout problem (FRSRFLP) and solve it by a real expected value method and a fuzzy stochastic chance-constrained programming based on possibility and necessity measures together with an extended B&B method. Furthermore, the proposed methods are applied for solving some benchmark problems to show their efficiency.

An inexact double-sided chance-constrained model for air quality management in Nanshan District, Shengzhen, China

In this study, an inexact double-sided fuzzy-random-chance-constrained programming (IDSFRCCP) model was developed for supporting air quality management of the Nanshan District of Shenzhen, China, under uncertainty. IDSFRCCP is an integrated model incorporating interval linear programming and double-sided fuzzy-random-chance-constrained programming models. It can express uncertain information as both fuzzy random variables and discrete intervals. The proposed model was solved based on the stochastic and fuzzy chance-constrained programming techniques and an interactive two-step algorithm. The air quality management system of Nanshan District, including one pollutant, six emission sources, six treatment technologies and four receptor sites, was used to demonstrate the applicability of the proposed method. The results indicated that the IDSFRCCP was capable of helping decision makers to analyse trade-offs between system cost and risk of constraint violation. The mid-range solutions tending to lower bounds with moderate αh and qi values were recommended as decision alternatives owing to their robust characteristics.

Sample average approximation method for the chance-constrained stochastic programming in the transportation model of emergency management

This study proposes a stochastic programming model for the transportation of emergency resource during the emergency response. Since it is difficult to predict the timing and magnitude of any disaster and its impact on the urban system, resource mobilisation is treated in a random manner, and the resource requirements are represented as random variables. Randomness is represented by the chance constraints in this paper. To deal with the difficulty in calculating the chance constraint function, we use conditional value at risk (CVaR) to approximate the chance constraint, and solve the approximation problem of the chance–constrained stochastic programming by using the sample average approximation (SAA) method. For a given sample, the SAA problem is a deterministic nonlinear programming (NLP) and any appropriate NLP code can be applied to solve the problem. The model and method provide a new way for the emergency logistics management engineering.

A solid waste management model with fuzzy random parameters

A novel solid waste management model with fuzzy random parameters (SWMMFRP) was proposed. The model improves the traditional stochastic mathematical programming (SMP) and fuzzy mathematical programming (FMP) methods through tackling dual uncertain parameters in the right-hand side of model constraints. The dual uncertainty is associated with both fuzzy imprecision and randomness. Based on stochastic and fuzzy chance-constrained programming (FCCP) techniques, SWMMFRP was capable of offering more-optimal solution scenarios under multiple uncertainties through incorporating various confidence information into a model framework, and evaluating tradeoffs between system economy and reliability. A long-term municipal waste management case was used to demonstrate the applicability of the proposed method. The results indicated that the model could help generate a spectrum of waste management patterns under different constraint-satisfaction and violation levels and provide support for decision makers to identify desired waste-management strategies. The study case also demonstrated that SWMMFRP is suitable in handling a municipal waste management system that allows somewhat compromise of environmental quality for achieving high economic benefits.

Hybrid intelligent algorithm for solving stochastic chance-constrained programming and its application

In order to find an algorithm which can solve the Stochastic Chance-Constrained Programming(SCCP) problem more effectively,a hybrid intelligence algorithm based on Clonal Selection Algorithm(CSA),random simulation technology and neural network was proposed.Random simulation was used to produce random variables sample matrix for training Back Propagation(BP) neural network to approximate the stochastic function.Fitness value was calculated and feasible solution was checked by the trained neural network in CSA until it could get the solution to the optimization problems.In order to make the searching rapid and effective,double cloning operators and double mutation operators were adopted in CSA.The simulation results show that satisfactory result has been achieved before 500 generation;moreover,the precision in the single objective optimization problem is improved by 2.2% and the precision in multi-objective optimization problems is increased by 65% compared with other existing algorithms.In addition,the algorithm was applied to solve the problem of optimal reservoir scheduling.The simulation results also show the correctness and effectiveness of the model and the algorithm.