Chance-constrained Programming Problem Research Articles

Energy Management Problems Under Uncertainties for Grid-Connected Microgrids: A Chance Constrained Programming Approach

This paper studies two energy management problems under uncertainties for a grid-connected microgrid. The problems are motivated by practical microgrid applications such as peak power shaving and frequency regulation. These applications require constraints on the microgrid energy output, which is uncertain due to the integration with renewable resources and random loads. Both problems are formulated as chance constrained programming problems to systematically incorporate uncertainties. We also show that the resulting chance constrained programming problems can both be solved using linear programming. The proposed formulation and solution are verified by two case studies originated from real world applications.

Chance-Constrained Two-Stage Unit Commitment Under Uncertain Load and Wind Power Output Using Bilinear Benders Decomposition

In this paper, we study unit commitment (UC) problems considering the uncertainty of load and wind power generation. UC problem is formulated as a chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance. In addition to the conventional mixed integer linear programming formulation using Big-M, we present the bilinear mixed integer formulation of chance constraint, and then derive its linear counterpart using McCormick linearization method. Then, we develop a bilinear variant of Benders decomposition method, which is an easy-to-implement algorithm, to solve the resulting large-scale linear counterpart. Our results on typical IEEE systems demonstrate that (i) the bilinear mixed integer programming formula-tion is stronger than the conventional one; (ii) the proposed Benders decomposition algorithm is generally an order of magnitude faster than using a professional solver to directly compute both linear and bilinear chance-constrained UC models.

Stochastic expansion planning of interconnected distribution networks with renewable sources considering uncertainties and power transfer capability

This study proposes a stochastic optimisation model for interconnected distribution network planning (IDNP) in the presence of renewable sources (RSs) considering power transfer capability after an N − 1 contingency. Two types of scenarios, feeder contingency, and substation transformer contingency, are formulated in the IDNP problem. Uncertainties of the RS output and load fluctuation are also integrated. The planning method provides planners with the decisions of feeder reformation, new tie lines, new feeders, substation expansion, new substations and new load allocation. The IDNP model is a chance-constrained mixed integer non-linear programming problem, which is solved by dynamic niche differential evolution algorithm. A case study carried out on a modified 104-bus distribution network demonstrates the effectiveness of those techniques. Compared with the traditional planning approach, the IDNP method could exploit asset utilisation by optimising existing networks efficiently as well as improve system economy and security simultaneously.

Genetic algorithm approach for solving multi-objective fuzzy stochastic programming problem

This paper is concerned with the solution procedure of a multi-objective fuzzy stochastic optimisation problem by simulation-based genetic algorithm. In this article, a multi-objective fuzzy chance constrained programming problem is considered with continuous fuzzy random variables. The uncertain parameters are considered as fuzzy normal and fuzzy log-normal random variables. The feasibilities of the fuzzy chance constraints are checked by the fuzzy stochastic programming with the genetic process without deriving the deterministic equivalents. The proposed procedure is illustrated by a numerical example.

Multi-objective Optimization Operation Considering Environment Benefits and Economy Based on Ant Colony Optimization for Isolated Micro-grids

Abstract A new multi-objective optimization model considering environment benefits and economy is proposed for isolated micro-grids based on ant colony optimization in this paper. First, through fuzzy processing and satisfaction-maximizing scheme, the multi-objective programming problem is reformulated into a nonlinear mono-objective programming problem. Then, to cope with the uncertainty of photovoltaic and wind power generation, the sequence operation theory is introduced to transform such a chance-constrained programming problem into a corresponding deterministic equivalent problem. Next, an improved ant colony algorithm is used to solve the mono-objective mixed-integer linear programming problem. And finally, the test results demonstrate the effectiveness of the proposed model and algorithm.

Sample bound estimate based chance-constrained immune optimization and its applications

This work investigates a simple and practical bio-immune optimization approach to solve a kind of chance-constrained programming problem without known noisy attributes, after probing into a lower bound estimate of sample size for any random variable. Such approach mainly consists of sample allocation, evaluation, proliferation and mutation. The former two, depending on a lower bound estimate acquired, not only decide the sample size of random variable and the importance level of each evolving B cell, but also ensure that such B cell is evaluated with low computational cost; the third makes diverse B cells participate in evolution and suppresses the influence of noise; the last, which associates with the information on population diversity and fitness inheritance, creates diverse and high-affinity B cells. Under such approach, three similar immune algorithms are derived after selecting different mutation rules. The experiments, by comparison against two valuable genetic algorithms, have illustrated that these immune algorithms are competitive optimizers capable of effectively executing noisy compensation and searching for the desired optimal reliable solution.

Energy-efficient resource allocation for multiuser OFDMA system based on hybrid genetic simulated annealing

Orthogonal frequency division multiple access (OFDMA) is adopted in 4G wireless communication standard, where bandwidth resources can be split into smaller granular units. Tailored for slow adaptive OFDMA system, we study the energy-efficient resource allocation problem based on chance-constrained programming. The optimization objective is to minimize the power consumption over subcarrier and power allocation. The constraints include the outage probability constraint and target bit error rate (BER) constraint. Because the optimization problem contains the probabilistic constraint, it is a chance-constrained programming problem. Hence, support vector machine (SVM) is adopted to compute the outage probability constraint. Then, we integrate SVM and genetic simulated annealing (GSA) to develop hybrid genetic simulated annealing (HGSA). Simulation tests verify that HGSA not only has the lower consumption power, but also satisfies the outage probability constraint.

A copula-based chance-constrained waste management planning method: An application to the city of Regina, Saskatchewan, Canada

ABSTRACTThis study proposes a copula-based chance-constrained waste management planning (CCWMP) method. The method can effectively reflect the interactions between random parameters of the waste management planning systems, and thus can help analyze the influences of their interactions on the entire systems. In particular, a joint distribution function is established using preestimated marginal distributions of random variables and an optimal copula selected from widely used Gaussian, Student’s t, Clayton, Frank, Gumbel, and Ali-Mikhail-Haq copulas. Then a set of joint probabilistic constraints in the chance-constrained programming problems is converted into individual probabilistic constraints using the joint distribution function. Further, this method is applied to residential solid waste management in the city of Regina in Canada for demonstrating its applicability. Nine scenarios based on different joint and marginal probability levels are considered within a multiperiod and multizone context to effectively reflect dynamic, uncertain, and interactive characteristics of the solid waste management systems in the city. The results provide many decision alternatives under these scenarios, including cost-effective and environmentally friendly decision schemes. Moreover, the results indicate that even though the effect of the joint probability levels on the system costs is more significant than that of the marginal probability levels, the effect of marginal probability levels is notable, and there exists a trade-off between the total system cost and the constraint-violation risk. Therefore, the results obtained from the present study would be useful to support the city’s long-term solid waste management planning and formulate local policies and regulation concerning the city’s waste generation and management.Implications: The CCWMP method not only can solve chance-constrained problems with unknown probability distributions of random variables in the right-hand sides of constraints, but also can effectively reflect the interactions between the random parameters and thus help analyze the influences of their interactions on the entire systems. The results obtained through applying this method to the city of Regina in Canada can provide many decision alternatives under different joint probability levels and marginal probability levels, and would be useful to support the city’s long-term solid waste management planning.

A Novel Method for Optimal Solution of Fuzzy Chance Constraint Single-Period Inventory Model

A method is proposed for solving single-period inventory fuzzy probabilistic model (SPIFPM) with fuzzy demand and fuzzy storage space under a chance constraint. Our objective is to maximize the total profit for both overstock and understock situations, where the demandD~jfor each productjin the objective function is considered as a fuzzy random variable (FRV) and with the available storage space areaW~, which is also a FRV under normal distribution and exponential distribution. Initially we used the weighted sum method to consider both overstock and understock situations. Then the fuzziness of the model is removed by ranking function method and the randomness of the model is removed by chance constrained programming problem, which is a deterministic nonlinear programming problem (NLPP) model. Finally this NLPP is solved by using LINGO software. To validate and to demonstrate the results of the proposed model, numerical examples are given.

Solving Geometric Programming Problems with Normal, Linear and Zigzag Uncertainty Distributions

Geometric programming is a powerful optimization technique widely used for solving a variety of nonlinear optimization problems and engineering problems. Conventional geometric programming models assume deterministic and precise parameters. However, the values observed for the parameters in real-world geometric programming problems often are imprecise and vague. We use geometric programming within an uncertainty-based framework proposing a chance-constrained geometric programming model whose coefficients are uncertain variables. We assume the uncertain variables to have normal, linear and zigzag uncertainty distributions and show that the corresponding uncertain chance-constrained geometric programming problems can be transformed into conventional geometric programming problems to calculate the objective values. The efficacy of the procedures and algorithms is demonstrated through numerical examples.

Chance-constrained dynamic programming with application to risk-aware robotic space exploration

Existing approaches to constrained dynamic programming are limited to formulations where the constraints share the same additive structure of the objective function (that is, they can be represented as an expectation of the summation of one-stage costs). As such, these formulations cannot handle joint probabilistic (chance) constraints, whose structure is not additive. To bridge this gap, this paper presents a novel algorithmic approach for joint chance-constrained dynamic programming problems, where the probability of failure to satisfy given state constraints is explicitly bounded. Our approach is to (conservatively) reformulate a joint chance constraint as a constraint on the expectation of a summation of indicator random variables, which can be incorporated into the cost function by considering a dual formulation of the optimization problem. As a result, the primal variables can be optimized by standard dynamic programming, while the dual variable is optimized by a root-finding algorithm that converges exponentially. Error bounds on the primal and dual objective values are rigorously derived. We demonstrate algorithm effectiveness on three optimal control problems, namely a path planning problem, a Mars entry, descent and landing problem, and a Lunar landing problem. All Mars simulations are conducted using real terrain data of Mars, with four million discrete states at each time step. The numerical experiments are used to validate our theoretical and heuristic arguments that the proposed algorithm is both (i) computationally efficient, i.e., capable of handling real-world problems, and (ii) near-optimal, i.e., its degree of conservatism is very low.

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.

Satisficing measure approach for vehicle routing problem with time windows under uncertainty

The complexity of evaluating chance constraints makes chance-constrained programming problem difficult to solve. One way to handle this complexity is by devising satisficing measures for the relevant uncertainties. This paper focuses on solving the stochastic vehicle routing problem with time windows (VRPTW) by Satisficing Measure Approach (SMA) that mitigates the dissatisfaction experienced by the customers. Satisficing measures are first proposed for the VRPTW with stochastic demand on various distributions to demonstrate the dependency of customers’ satisfaction towards lack of inventory based on the vehicle's capacity. Similar satisficing measures are extended to VRPTW with stochastic travel times. We integrate the proposed satisficing measures into an existing tabu-search heuristics to solve a set of generalized Solomon instances in a short amount of computation time. Compared with best-known results, the SMA saves the effort to design recourse actions, applicable to many popular probability distributions and produces very competitive results.

A fixed-charge transportation problem in two-stage supply chain network in Gaussian type-2 fuzzy environments

This paper presents two mathematical models representing imprecise capacitated fixed-charge transportation problems for a two-stage supply chain network in Gaussian fuzzy type-2 environment. It is a two-stage transportation process from a manufacturing center to m potential distribution centers (DCs) and then from DCs to business centers of n retailers with particular demands. Retailers are situated at some distances apart. Here unit transportation costs, fixed charges, availabilities, and demands are imprecise and represented by Gaussian type-2 fuzzy numbers. The proposed models are formulated as profit maximization problems in such a way that some DCs are selected in order to satisfy the demands at all retailers. The type-2 fuzziness has been removed by using generalized credibility measure developed with the help of CV-based reduction method and hence the models are reduced to chance constrained programming problems with different credibility labels. The deterministic models are then solved using both genetic algorithm (GA) based on Roulette wheel selection, arithmetic crossover with uniform mutation and modified particle swarm optimization (PSO), where the position of each particle is adjusted according to its own experience and that of its neighbors. Finally models are illustrated with some numerical data. Some sensitivity analyses on the proposed models are presented.

Sensitivity-based chance-constrained Generation Expansion Planning

A Generation Expansion Planning problem with load uncertainty is formulated based on joint chance-constrained programming (CCP) and is solved by incorporating sensitivity into iterative algorithms. These algorithms exploit the characteristics of the system and its response to load variations. Sensitivities help to classify buses according to stress level, and sensitivity-based iterative algorithms distinguish each bus based on its contribution to the overall system reliability. The use of sensitivity overcomes some of the mathematical obstacles to solving joint CCP problems and, in addition, leads to optimal expansion solutions because uncertain loads are correctly estimated. The IEEE 30- and 118-bus test systems are used to demonstrate the proposed algorithms, and the results of these algorithms are compared with those of other algorithms for solving the joint CCP problem.

Chance-Constrained Programming Method of IT Risk Countermeasures for Social Consensus Making

The authors address a social consensus making support in discussing countermeasures for information technology risks (IT risks). For supporting stakeholders’ discussion on which IT risk countermeasures the stakeholders should implement, experts of the risk management estimate parameter values of the countermeasure, define a goal and constraints, and formulate the decision problem of the countermeasures to be implemented as one of 0–1 integer programming problems. Because parameter values and constraint values are uncertain, the decision problem is reformulated as a chance-constrained programming problem. The sample average approximation method is a well-known method for solving the chance-constrained programming problem. However, the computational time is still so long that the opinion leaders cannot use a solution of the chance-constrained programming problem in their discussion. The authors propose a high-speed chance-constrained programming method by aggregating the constraints that are generated by approximation of the problem in the sample average method. By applying the proposed method to real decision problems, the authors confirmed that computational time is decreased to 1 min while obtaining the same error rate and the same rate of the feasible solutions as a conventional method.

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.

Unascertained Support Vector Machine

In this paper, we propose an unascertained support vector machine (USVM) for classification problem with unascertained information (a kind of uncertain information). First, based on unascertained mathematics, classification problems containing unascertained information are converted to solving unascertained chance constrained programming problems. Then, the unascertained chance constrained programming is converted to its equivalent quadratic programming. Based on these theories, the algorithm of a unascertained support vector machine is given. Theoretical analysis and experiments show that the proposed USVM is effective and feasible.

The robust binomial approach to chance-constrained optimization problems with application to stochastic partitioning of large process networks

In this paper, we study an interpretation of the sample-based approach to chance-constrained programming problems grounded in statistical testing theory. On top of being simple and pragmatic, this approach is theoretically well founded, non parametric and leads to a general method for leveraging existing heuristic algorithms for the deterministic case to their chance-constrained counterparts. Throughout this paper, this algorithm design approach is illustrated on a real world graph partitioning problem which crops up in the field of compilation for parallel systems. Extensive computational results illustrate the practical relevance of the approach, as well as the robustness of the obtained solutions.

Joint chance constrained programming for hydro reservoir management

In this paper, we deal with a cascaded reservoir optimization problem with uncertainty on inflows in a joint chance constrained programming setting. In particular, we will consider inflows with a persistency effect, following a causal time series model with Gaussian innovations. We present an iterative algorithm for solving similarly structured joint chance constrained programming problems that requires a Slater point and the computation of gradients. Several alternatives to the joint chance constraint problem are presented. In particular, we present an individual chance constraint problem and a robust model. We illustrate the interest of joint chance constrained programming by comparing results obtained on a realistic hydro valley with those obtained from the alternative models. Despite the fact that the alternative models often require less hypothesis on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models, such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore not useful. We therefore conclude that Joint Chance Constrained programming appears as an approach offering a good trade-off between cost and robustness and can be tractable for complex realistic models.