Interval Linear Programming Method Research Articles

Day-ahead interval scheduling strategy of power systems based on improved adaptive diffusion kernel density estimation

The increasing penetration of renewable energy causes high uncertainty, which further complicates the optimal scheduling operation of power systems. The uncertainty of renewable energy output is represented by empirical prediction intervals in the traditional interval optimization scheduling model, but the interval range is not precise enough. To address this problem, a novel improved interval optimization method is proposed in this paper. First, the improved adaptive diffusion kernel density estimation (IADKDE) is used to obtain more accurate intervals for the renewable energy output. Furthermore, a data-driven adaptive optimal bandwidth selection is adopted to select the optimal bandwidth instead of normal reference rules in IADKDE. In addition, the day-ahead scheduling optimization model is developed by IADKDE considering the driving requirements of electric vehicles (EVs) owners. The proposed model is described in details and solved by interval linear programming method. Finally, the effectiveness and accuracy of the proposed method are validated, and the comparative analysis with interval optimization by empirical prediction intervals, extreme learning machine(ELM) and stochastic optimization is given. It is demonstrated that the proposed method can obtain more accurate interval ranges of uncertain variables and has strong applicability.

Interval optimization for integrated electrical and natural-gas systems with power to gas considering uncertainties

This paper is aimed at the optimal operation of the integrated electrical and natural-gas systems (IENGS) with power to gas (P2G) considering uncertainties of the wind turbine (WT) and photovoltaic (PV) via interval mathematics. The optimization model of the IENGS is established under deterministic conditions. The model takes the operation and maintenance cost, environmental cost, purchasing electricity/gas costs and selling electricity/gas incomes into account. Based on the interval arithmetic, the outputs of WTs and PVs are represented in interval forms and the interval optimization model is established considering the uncertainties of WTs and PVs. The model is converted into a mixed-integer linear programming (MILP) model and solved by the interval linear programming method. In the numerical test, the simulation results under deterministic conditions and the influence of the P2G device are analyzed, then the interval outputs of components and the operation cost of the IENGS are discussed when the uncertainties of the WT and PV are considered. The simulation results show that a higher conversion rate of P2G will lower the operation cost of the IENGS and how the outputs of components change under different uncertain levels of the WT and PV.

A modified two-step method for solving interval linear programming problems

In this paper, we propose a new method for solving interval linear programming (ILP) problems. For solving the ILP problems, two important items should be considered: feasibility (i.e., solutions satisfy all constraints) and optimality (i.e., solutions are optimal for at least a characteristic model). In some methods, a part of the solution space is infeasible (i.e., it violates any constraints) such as the best and worst cases method (BWC) proposed by Tong in 1994 and two-step method (TSM) proposed by Huang et al. in 1995. In some methods, the solution space is completely feasible, but is not completely optimal (i.e., some points of the solution space are not optimal) such as modified ILP method (MILP) proposed by Zhou et al. in 2009 and improved TSM (ITSM) proposed by Wang and Huang in 2014. Firstly, basis stability for the ILP problems is reviewed. Secondly, the solving methods are analysed from the point of view of the feasibility and optimality conditions. Later, a new method which modifies the TSM by using the basis stability approach is presented. This method gives a solution space that is not only completely feasible, but also completely optimal.

A modified two-step method for solving interval linear programming problems

In this paper, we propose a new method for solving interval linear programming (ILP) problems. For solving the ILP problems, two important items should be considered: feasibility (i.e., solutions satisfy all constraints) and optimality (i.e., solutions are optimal for at least a characteristic model). In some methods, a part of the solution space is infeasible (i.e., it violates any constraints) such as the best and worst cases method (BWC) proposed by Tong in 1994 and two-step method (TSM) proposed by Huang et al. in 1995. In some methods, the solution space is completely feasible, but is not completely optimal (i.e., some points of the solution space are not optimal) such as modified ILP method (MILP) proposed by Zhou et al. in 2009 and improved TSM (ITSM) proposed by Wang and Huang in 2014. Firstly, basis stability for the ILP problems is reviewed. Secondly, the solving methods are analysed from the point of view of the feasibility and optimality conditions. Later, a new method which modifies the TSM by using the basis stability approach is presented. This method gives a solution space that is not only completely feasible, but also completely optimal.

An improved three-step method for solving the interval linear programming problems

Feasibility condition, which ensures that the solution space does not violate any constraints, and optimality condition, which guarantees that all points of the solution space are optimal, are very significant conditions for the solution space of interval linear programming (ILP) problems. Among the existing methods for ILP problems, the best-worst cases (BWC) method and two-step method (TSM) do not ensure feasibility condition, while the modified ILP (MILP), robust TSM (RTSM), improved TSM (ITSM), and three-step method (ThSM) guarantee feasibility condition, whose solution spaces may not be completely optimal. We propose an improved ThSM (IThSM) for ILP problems, which ensures both feasibility and optimality conditions, i.e., we introduce an extra step to optimality.

Improving the modified interval linear programming method by new techniques

In this study, we consider interval linear programming (ILP) problems, which are used to deal with uncertainties resulting from the range of admissible values in problem coefficients. In most existing methods for solving ILP problems, a part of the solution region is not feasible. The solution set obtained through the modified ILP (MILP) method is completely feasible (i.e., it does not violate any constraints), but is not completely optimal (i.e., some points of the region are not optimal). In this paper, two new ILP methods and their sub-models are presented. These techniques improve the MILP method, giving a solution region that is not only completely feasible, but also completely optimal.

Convex contractive interval linear programming for resources and environmental systems management

It is likely that the most reliable estimation of system uncertainty in resources and environmental systems management (RESM) is a value range with an unknown distribution. Stochastic programming would be challenged by distortion of the original uncertain information through fabricating an inexistent probabilistic distribution function. Instead, interval linear programming (ILP), i.e. a synthesis of interval-set coefficients and the conventional linear programming, has been employed to identify the desired schemes for a number of RESM problems under interval uncertainty. However, its effectiveness is disabled by constraint violation which may lead to severe penalties on socio-economic or eco-environmental development. To mitigate such a challenge, a convex contractive interval linear programming (CCILP) approach is proposed in this study. It mainly consists of six modules: parameterizing an RESM problem as an ILP model, initializing a hyperrectangle decision space by two linear programming sub-models, revealing causes of constraint violation given a criterion, inferring feasibilities of potential solutions, finalizing a feasible hyperrectangle decision space by another linear programming sub-model, and supporting RESM of various complexities through alternative variants. A simple ILP model for RESM is introduced to demonstrate the procedures of CCILP and verify its advantages over existing ILP methods. The result indicates that CCILP is capable of robustly incorporating interval uncertainties into the optimization process, avoiding heavy computation burdens on complicated sub-models, eliminating occurrence of constraint violation, enabling provision of a hyperrectangle decision space, adapting to diverse system requirements, and increasing reliability of decision support for interval linear RESM problems.

Interval Recourse Linear Programming for Resources and Environmental Systems Management under Uncertainty

An interval recourse linear programming (IRLP) approach is proposed in this study for mitigating constraint violation problems in resources and environmental systems management (REM) under interval uncertainties. Based on a review of interval linear programming (ILP) and its significances to REM, two linear programming sub-models are employed to initialize a decision space for IRLP. Causes of constraint violation are examined based on identification of a violation criterion. Contraction ratios are defined after revelation of violation ranges of constraints. As a recourse measure to constraint violation problems, another two linear programming sub-models are constructed given a series of contraction ratios. A hypercube decision space where infeasible solutions are excluded is obtained. Post-optimality analysis is conducted to deal with barriers for applying the IRLP approach to real-world ILP models for REM. An REM problem is introduced to demonstrate procedures and effectiveness of the IRLP approach. Comparisons with existing ILP methods reveal that the IRLP approach is effective at resolving the constraint-violation problem, reproducing the largest decision space which does not include infeasible solutions, and enhancing reliability of decision support for REM.

Solid waste management under uncertainty: a generalized fuzzy linear programming approach

In this study, a generalized fuzzy linear programming (GFLP) method is developed to identify optimal waste-flow-allocation schemes under uncertainty. A stepwise interactive algorithm (SIA) is advanced to solve the GFLP model and generate solutions expressed as fuzzy sets. This solution method can handle fuzzy sets with known membership functions, regardless of the shapes of these functions. Moreover, solutions expressed as fuzzy sets can also be obtained through SIA. The developed method is applied to a case study of waste allocation planning problem under uncertainty. The results indicate that reasonable solutions can be obtained for planning waste allocation practices. Compared with interval solutions derived from interval linear programming method, the fuzzy solutions obtained through GFLP can provide more information. Therefore, the decision-makers can make tradeoffs between system stability and plausibility and thus identify desired policies for solid waste planning under uncertainty.