Successive Linear Programming Method Research Articles

Parameter Estimation Method for Virtual Power Plant Frequency Response Model Based on SLP

In adapting to the double-high development trend of high-voltage direct current (HVDC) receiving-end power systems and solving optimization problems in emergency frequency control (EFC) supporting virtual power plants (VPPs) in large-scale power systems, a parameter estimation method for a VPP frequency response model based on a successive linear programming (SLP) method is proposed. First, a “centralized/decentralized” hierarchical control architecture for VPP participation in EFC is designed. Second, the frequency response characteristics of multiple flexible resources are scientifically analyzed, and the system frequency response (SFR) model and equivalent model of VPP are constructed. Subsequently, parameter estimation of the VPP frequency response model is carried out based on the SLP method, aiming to balance the accuracy and computational efficiency of the model. Finally, the effectiveness of the proposed methodology is verified by using PSD-BPA to simulate and test the three-zone HVDC recipient area grid.

A robust optimization method with successive linear programming for intensity-modulated radiation therapy

Intensity-modulated radiation therapy for cancer is considered to be effective when dealing with complicated tumour shapes because the dose distribution for each irradiation can be modulated. Fluence map optimization is often formulated as an optimization problem with dose volume constraints (DVCs). A linear programming (LP) method that approximated DVCs was proposed, and it was modified to the successive LP method (SLPM) to find a feasible treatment plan in a wider region. In the present paper, we propose a numerical method called SLPM-R (the SLPM with robustness) that enhances the SLPM using a robust optimization approach. We mathematically prove that the proposed method with extended LP problems has the favourable properties of the SLPM, even taking uncertainty in the influence matrix into consideration. In particular, when the optimal value of the LP problem is non-positive, the proposed SLPM-R guarantees that the output solution can satisfy all DVCs. Through numerical experiments, we observed that the proposed method found a feasible plan that the SLPM could not find. In addition, for a test case that even the SLPM-R failed, the largest deviations of 5.65 Gray in the SLPM was reduced to 3.15 Gray by the SLPM-R.

Grid Services Optimization From Multiple Microgrids

We propose a multi-case, multi-period AC model to coordinate the procurement of grid services from an aggregation of connected distributed energy resources and microgrids. The proposed model aggregates the capabilities of all resources in the system to manage the network’s power flow and provide services to the power grid at the point of common coupling. The modeling approach enables incorporating a number of services (e.g., reserve, regulation, and reactive power capacity) with simple addition of service defining equations. We also propose a solution methodology for the resultant mixed-integer quadratically-constrained programming (MICQP) problem based on successive linear programming (SLP). The SLP is augmented with a fix-binary algorithm that promises efficient handling of discrete variables. The paper also provides a decentralized formulation that decomposes the problem into an aggregator problem and multiple small <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> Gs’ problems. We integrate the SLP method with the analytical target cascading (ATC) algorithm to optimize the overall system with minimal shared information between the aggregator and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> Gs. Case studies analyze the convergence of the centralized and the decentralized approaches, present the optimal power and service schedules of the tested system, and study the network conditions under service calls.

Complexity Analysis and Algorithm Design of Pooling Problem

The pooling problem, also called the blending problem, is fundamental in production planning of petroleum. It can be formulated as an optimization problem similar with the minimum-cost flow problem. However, Alfaki and Haugland (J Glob Optim 56:897–916, 2013) proved the strong NP-hardness of the pooling problem in general case. They also pointed out that it was an open problem to determine the computational complexity of the pooling problem with a fixed number of qualities. In this paper, we prove that the pooling problem is still strongly NP-hard even with only one quality. This means the quality is an essential difference between minimum-cost flow problem and the pooling problem. For solving large-scale pooling problems in real applications, we adopt the non-monotone strategy to improve the traditional successive linear programming method. Global convergence of the algorithm is established. The numerical experiments show that the non-monotone strategy is effective to push the algorithm to explore the global minimizer or provide a good local minimizer. Our results for real problems from factories show that the proposed algorithm is competitive to the one embedded in the famous commercial software Aspen PIMS.

A filter successive linear programming method for nonlinear semidefinite programming problems

In this paper we present a successive linear programming method with filter technique for nonlinear semidefinite programming. Such a method is characterized by use of the dominance concept of multiobjective optimization,~instead of a penalty parameter. The Successive Linear Programming with Filter (SLP-Filter) was used to solve the nonlinear programming (see [8]). In this paper, we extend it to deal with nonlinear semidefinite programming, and prove the convergence of the SLP-Filter for nonlinear semidefinite programming. We report numerical experiments to show the validity of the SLP-Filter method for nonlinear semidefinite programming.

Environmental/economic dispatch considering emission benefit factor in the emission trading environment

PurposeThe efficiency of the emission trading system (ETS) may help to control the total emission amount. The purpose of this paper is to investigate the generating cost issue in environmental/economic power dispatch, under the premise that the ETS has already been established.Design/methodology/approachThe emission benefit and price level factors are introduced for transforming the bi‐objective optimization problem with the fuel cost and emission cost minimization into a single objective. In the developed mathematical model, both the total emission amount from all units and the permitted emission amount from each generating unit are taken into account. The successive linear programming method is employed to solve the optimization problem.FindingsSimulation results of the IEEE 30‐bus test system show that a proper trading mechanism of emission permits is very important for generation companies to control the total emission amount and to reduce the overall generation cost.Research limitations/implicationsFurther research is needed to find out the impact on the generating cost caused by trading price fluctuation and the coping strategies.Originality/valueThe results can help to meet the requirements of current generating optimal dispatch.

Investigation on Limit Surfaces in Space Spanned by Generation Parameter

In a space spanned by generation parameter, the different properties of loading margin surface and generation limit surface are investigated at first in this paper. Then, the shortage of taking the generation limit surface as limit boundary for generation rescheduling (GR) is elaborated and a developed GR model is presented. Additionally, in order to take the uncertainty of load pattern into consideration, a bilevel programming model with the opposite structure to each other is proposed. In this model, the upper level is used to optimize generation pattern (GP) for maximizing the loading margin while the lower level is used to determine the worst load pattern for checking the adaptability of the GP to the variation of the load pattern. A Step-by-step Approximation based Primal Dual Interior Point method (SAPDIP method) and Successive Linear Programming method (SLP method) are used to solve the two sub-models, respectively. The GP optimized by the proposed model reflects the balance between the “variation” and “adaption”, and is less sensitive to the changes in load pattern. Finally, the proposed method is tested on multiple standard IEEE test systems as well as a real power system in China.

Optimal Active Power Dispatch with Small-signal Stability Constraints

Conventional optimal active power dispatch without considering small-signal stability constraints may violate the small-signal stability requirement. This article investigates the optimal active power dispatch problem with small-signal stability constraints. A new optimal active power dispatch model with small-signal stability constraints is built in order to ensure the generation optimization result that can satisfy the given small-signal stability requirement. The successive linear programming method is used to solve the new model. The proposed methodology is validated by two test systems.

An algorithm for nonlinear optimization using linear programming and equality constrained subproblems

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the l1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1, 15] test set.

An efficient multi-objective fuzzy logic based successive LP method for optimal reactive power planning

Abstract This paper presents a new and efficient method to solve the Optimal Reactive Power Planning (ORPP) problem, simultaneously minimising transmission losses and improving voltage profile using Multi-objective Fuzzy logic based LP (MFLP) model in Successive Linear Programming (SLP) framework. The ORPP problem is concerned with optimally siting and sizing new capacitors at prospective sites in a planned grid with projected load demands such that, minimum number and quantum of new capacitors are allocated, the transmission losses are minimised and a satisfactory voltage profile is obtained. In this paper, all the prospective capacitor locations are ranked using Voltage Performance Index (VPI) based on sensitivity factors. From the top of the ranked list, a minimum number of locations are chosen for siting new capacitors. Reactive powers at these selected capacitor sites along with generator voltage magnitudes, reactive powers of existing switchable VAR sources and on-load tap changer (OLTC) settings of transformers are used as control variables in this method. In the MFLP framework, each of the objectives and each of the constraints are assigned a satisfaction parameter. The satisfaction parameter corresponding to any objective quantifies the degree of closeness of the current state of the objective to the optimum. The satisfaction parameter corresponding to a constraint describes the degree of enforcement of that constraint. By maximising the minimum of these satisfaction parameters, the objectives are optimised and the constraint enforcements are maximised. The MFLP based SLP method is found to be very effective while planning for systems which have severe under-voltage violations due to insufficient reactive powers, as the method is capable of selectively relaxing certain load voltage constraints, while simultaneously enforcing all other constraints strictly. The MFLP technique optimises all the objectives, i.e. minimises number and quantum of new capacitors, minimises transmission losses and maximises constraint enforcement of all violated constraints, while simultaneously imposing all other problem constraints strictly. This method uses compactly stored, factorised constant matrices in all MFLP steps, both for construction of the MFLP model as well as for the power flow solution. The method was tested on IEEE test systems and on a practical 191 bus electric utility system. The merits of the method, compared with SLP method using non-fuzzy approach, are brought out.

Fuzzy Logic Based Successive LP Method for Reactive Power Optimization

This paper presents a new Successive Fuzzy Linear Programming (SFLP) method for Reactive Power Optimization (RPO) to minimize the transmission loss and improve the voltage profile. The SFLP method uses Fuzzy LP (FLP) model in the Successive Linear Programming (SLP) framework to solve the RPO problem. In the FLP model, the objective and each of the constraints are assigned a satisfaction parameter. The satisfaction parameter corresponding to the objective quantifies the degree of closeness of the objective in the current state to the optimum. The satisfaction parameter corresponding to a constraint describes the degree of enforcement of that constraint. By maximizing the minimum of these satisfaction parameters, the objective as well as the constraint enforcements are maximized. The SFLP method of RPO is efficient and reliable in scheduling systems having severe under-voltages and insufficient reactive power, as the method is capable of maximizing the enforcement of the violated load bus voltage constraints, minimizing the transmission loss while simultaneously enforcing all the other constraints strictly. This method uses compactly stored, factorized constant matrices in all the LP steps, both for the construction of the FLP model as well as for the power flow solution. The control variables used in this method are generator bus voltage magnitudes, reactive powers of switchable VAR sources, and on-load tap changer (OLTC) settings of transformers. The method was tested on IEEE test systems and on a practical electric utility system. The merits of the proposed method compared to the SLP method using non-fuzzy approach are brought out.

A method for multiphase equilibrium calculations

Methods for multiphase equilibrium calculations are generally based on Gibbs free energy minimization or equation-solving. Equation-solving methods, although superior to free energy minimization methods for phase equilibrium calculations without chemical reactions, often involve sequential procedures for a priori phase identification. A simultaneous equation-solving method ( τ-method), based on modifying mole fraction summations, is proposed for these calculations. It requires the solution of a minimization problem only once, and provides phases actually present at equilibrium, their quantities and compositions simultaneously; and phase identification in advance is not required. Phase characteristic variable and pseudo phase are introduced, and their significance discussed. τ-method is shown by analysis and numerical results, to be consistent with Nelson’s (1987, Comput. Chem. Engng 11, 581–591) criteria for phase existence. The method is tested on typical examples for two-phase (vapor–liquid and liquid–liquid) and three-phase (vapor–liquid–liquid) equilibrium calculations. For each example, several conditions and/or different initializations are used, and the minimization problem is solved by a version of successive linear programming method. The results show that τ-method is successful and reliable for multiphase equilibrium calculations.

A method for calculation of vapor-liquid and liquid-liquid equilibria

A new parameter, pseudo temperature ( T p) is introduced to solve two-phase equilibrium calculations. The resulting T p-method has a single set of equations to describe different phase regions encountered in vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE). It does not require a priori the number of phases existing at equilibrium, and involves a minimization problem having a linear objective function subject to nonlinear constraints, which can be solved by a version of successive linear programming method. T p-method is tested on typical examples for VLE and LLE, and compared with pseudo pressure ( P p) method of Bullard and Biegler [ Computers and Chemical Engineering 17, 95–109 (1993)] for VLE. In each example, several operating conditions are considered to cover both single- and two-phase situations. The results show that T p-method is successful for both VLE and LLE, and that T p- and P p-methods are comparable for VLE.

A unified approach to global convergence of trust region methods for nonsmooth optimization

This paper investigates the global convergence of trust region (TR) methods for solving nonsmooth minimization problems. For a class of nonsmooth objective functions called regular functions, conditions are found on the TR local models that imply three fundamental convergence properties. These conditions are shown to be satisfied by appropriate forms of Fletcher's TR method for solving constrained optimization problems, Powell and Yuan's TR method for solving nonlinear fitting problems, Zhang, Kim and Lasdon's successive linear programming method for solving constrained problems, Duff, Nocedal and Reid's TR method for solving systems of nonlinear equations, and El Hallabi and Tapia's TR method for solving systems of nonlinear equations. Thus our results can be viewed as a unified convergence theory for TR methods for nonsmooth problems.

Optimization of Axial Enrichment and Gadolinia Distributions for BWR Fuel under Control Rod Programming, (II)

A method has been developed for optimizing the axial enrichment and gadolinia distributions for the reload BWR fuel under control rod programming. The problem was to minimize the enrichment requirement subject to the criticality and axial power peaking constraints. The optimization technique was based on the successive linear programming method, each linear programming problem being solved by a goal programming algorithm. A rapid and practically accurate core neutronics model, named the modified one-dimensional core model, was developed to describe the batch-averaged burnup behavior of the reload fuel. A core burnup simulation algorithm, employing a burnup-power-void iteration, was also developed to calculate the rigorous equilibrium cycle performance. This method was applied to the optimization of axial two- and 24-region fuels for demonstrative purposes. The optimal solutions for both fuels have proved the optimality of what is called burnup shape optimization spectral shift. For the two-region fuel with a practical power peaking of 1.4, the enrichment distribution was nearly uniform, because a bottom-peaked burnup shape flattens the axial power shape. Optimization of the 24-region fuel has shown a potential improvement in BWR fuel cycle economics, which will guide future advancement in BWR fuel designs.

Algorithms for solving some problems of the nuclear reactor optimization

Optimization algorithms using backward diffusion calculation are developed. The fuel distribution optimization problem is reduced to a linear programming problem using power densities as control variables. In order to determine optimal placement of the assemblies two algorithms are developed. The successive linear programming method and mixed integer programming are used in the first algorithm. In the second one, the problem is formulated with the addition of some nonlinear restrictions and then the sequential unconstrained minimization technique is applied.

Superlinear convergence of a trust region-type successive linear programming method

The convergence rate of the SLP method suggested in Ref. 1 is discussed for composite nondifferentiable optimization problems. A superlinear rate is assured under a growth condition, and it is further strengthened to a quadratic rate if the inside function is twice differentiable. Several sufficient conditions are given which make the growth condition true. These conditions can be relaxed considerably in practical use.

Knowledge-based successive linear programming for linearly constrained optimization

Combining knowledge engineering technology with some operations research algorithms will get novel efficient optimization methods. As an example, this paper discusses a knowledge-based successive linear programming method for linearly constrained optimization problems. In this new method we use both traditional successive linear programming algorithm in operations research and the knowledge base which is constructed with the expertise of an optimization expert and valuable experience data, so that this knowledge-based program can solve optimization problems somewhat like a human expert who is great at operations research and has a lot of practical experience of problem-solving. The improvement of efficiency in problem-solving depends mainly on the skilful use of plausible reasoning based on incomplete experience knowledge. In addition, man-machine interaction during the computation procedures is also used. Finally, two numerical examples illustrate that the proposed method is much more flexible and efficient than the traditional operations research algorithms concerned.

Nuclear power plant optimal control by successive linear programming

An attempt to find optimal controls for an extremely load-following nuclear power plant during large load pick-ups is reported in this paper. The choice of the numerical method to solve this highly constrained dynamic optimization problem is discussed. The results reported demonstrate the efficacy of the successive linear programming method in tackling this problem without recourse to model linearization.