Design and analysis of an efficient neural network model for solving nonlinear optimization problems

This paper presents an efficient approach based on a recurrent neural network for solving constrained nonlinear optimization. More specifically, a modified Hopfield network is developed, and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points that represent an optimal feasible solution. The main advantage of the developed network is that it handles optimization and constraint terms in different stages with no interference from each other. Moreover, the proposed approach does not require specification for penalty and weighting parameters for its initialization. A study of the modified Hopfield model is also developed to analyse its stability and convergence. Simulation results are provided to demonstrate the performance of the proposed neural network.

Optimization Strategies Based on Sequential Quadratic Programming Applied for a Fermentation Process for Butanol Production

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On the solution of mathematical programming problems with equilibrium constraints

Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to first-order optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of the sum of squares of the constraints are feasible points of the MPEC. In usual formulations of MPEC all the feasible points are nonregular in the sense that they do not satisfy the Mangasarian-Fromovitz constraint qualification of nonlinear programming. Therefore, all the feasible points satisfy the classical Fritz-John necessary optimality conditions. In principle, this can cause serious difficulties for nonlinear programming algorithms applied to MPEC. However, we show that most feasible points do not satisfy a recently introduced stronger optimality condition for nonlinear programming. This is the reason why, in general, nonlinear programming algorithms are successful when applied to MPEC.

Penalty-based nonlinear solver for optimal reactive power dispatch with discrete controls

The optimal reactive dispatch problem is a nonlinear programming problem containing continuous and discrete control variables. Owing to the difficulty caused by discrete variables, this problem is usually solved assuming all variables as continuous variables, therefore the original discrete variables are rounded off to the closest discrete value. This approach may provide solutions far from optimal or even unfeasible solutions. This paper presents an efficient handling of discrete variables by penalty function so that the problem becomes continuous and differentiable. Simulations with the IEEE test systems were performed showing the efficiency of the proposed approach. © 1969-2012 IEEE.

Saddle Point and Second Order Optimality in Nondifferentiable Nonlinear Abstract Multiobjective Optimization

This article deals with a vector optimization problem with cone constraints in a Banach space setting. By making use of a real-valued Lagrangian and the concept of generalized subconvex-like functions, weakly efficient solutions are characterized through saddle point type conditions. The results, jointly with the notion of generalized Hessian (introduced in [Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28, 789–809 (1990)]), are applied to achieve second order necessary and sufficient optimality conditions (without requiring twice differentiability for the objective and constraining functions) for the particular case when the functionals involved are defined on a general Banach space into finite dimensional ones.

Accuracy of nonlinear feed formulation for broiler chickens: maximising broiler profit

The modeling technique is simple, useful and practical to calculate optimum nutrient density to maximize profit margins, using nonlinear programming by predictive broiler performance. To demonstrate the influence of the broiler price could interact with nutrient density, the experiment aimed to define the quadratic equations for consumption and weight gain, based on modeling, to be applied to nonlinear programming, according to sex (male and female) in the starter (1 to 21 days), grower (22 to 42 days) and finisher phases (43 to 56 days). The experimental design was a randomized, totaling 6 treatments [energy levels of 2800, 2900, 3000, 3100, 3200 and 3300kcal AME/kg with constant nutrient : AME (Apparent Metabolizable Energy)] with 4 replicates and 10 birds per plot, using the program free download PPFR Excel workbook for feed formulation (http://www.foa.unesp.br/downloads/file_detalhes.asp?CatCod=4&SubCatCod=138&FileCod=1677). Data from this trial confirmed that there was a significant relationship between feed intake and total energy consumption of the diet, in which feed intake was increased or decreased simply to keep the amount of energy, with a constant rate of nutrient : AME. Therefore, the data support that if the essential dietary nutrients are kept in proportion to the energy density of the diet, according to the appropriate requirements (male / female) of broilers, the weight and feed conversion are significantly (P<0.05) favored by increasing the energy density of the diet. Thus, it enables the application of models for maximum profit (nonlinear formulation), to estimate the proportion of weight gain most appropriate according to the price paid by the market.

On the optimal capacity expansion of electric power systems

The analysis of investment in the electric power has been the subject of intensive research for many years. The efficient generation and distribution of electrical energy is a difficult task involving the operation of a complex network of facilities, often located over very large geographical regions. Electric power utilities have made use of an enormous range of mathematical models. Some models address time spans which last for a fraction of a second, such as those that deal with lightning strikes on transmission lines while at the other end of the scale there are models which address time horizons consisting of ten or twenty years; these usually involve long range planning issues. This thesis addresses the optimal long term capacity expansion of an interconnected power system. The aim of this study has been to derive a new, long term planning model which recognises the regional differences which exist for energy demand and which are present in the construction and operation of power plant and transmission line equipment. Perhaps the most innovative feature of the new model is the direct inclusion of regional energy demand curves in the nonlinear form. This results in a nonlinear capacity expansion model. After review of the relevant literature, the thesis first develops a model for the optimal operation of a power grid. This model directly incorporates regional demand curves. The model is a nonlinear programming problem containing both integer and continuous variables. A solution algorithm is developed which is based upon a resource decomposition scheme that separates the integer variables from the continuous ones. The decompostion of the operating problem leads to an interactive scheme which employs a mixed integer programming problem, known as the master, to generate trial operating configurations. The optimum operating conditions of each trial configuration is found using a smooth nonlinear programming model. The dual vector recovered from this model is subsequently used by the master to generate the next trial configuration. The solution algorithm progresses until lower and upper bounds converge. A range of numerical experiments are conducted and these experiments are included in the discussion. Using the operating model as a basis, a regional capacity expansion model is then developed. It determines the type, location and capacity of additional power plants and transmission lines, which are required to meet predicted electicity demands. A generalised resource decompostion scheme, similar to that used to solve the operating problem, is employed. The solution algorithm is used to solve a range of test problems and the results of these numerical experiments are reported. Finally, the expansion problem is applied to the Queensland electricity grid in Australia.

On the optimal capacity expansion of electric power systems

The analysis of investment in the electric power has been the subject of intensive research for many years. The efficient generation and distribution of electrical energy is a difficult task involving the operation of a complex network of facilities, often located over very large geographical regions. Electric power utilities have made use of an enormous range of mathematical models. Some models address time spans which last for a fraction of a second, such as those that deal with lightning strikes on transmission lines while at the other end of the scale there are models which address time horizons consisting of ten or twenty years; these usually involve long range planning issues. This thesis addresses the optimal long term capacity expansion of an interconnected power system. The aim of this study has been to derive a new, long term planning model which recognises the regional differences which exist for energy demand and which are present in the construction and operation of power plant and transmission line equipment. Perhaps the most innovative feature of the new model is the direct inclusion of regional energy demand curves in the nonlinear form. This results in a nonlinear capacity expansion model. After review of the relevant literature, the thesis first develops a model for the optimal operation of a power grid. This model directly incorporates regional demand curves. The model is a nonlinear programming problem containing both integer and continuous variables. A solution algorithm is developed which is based upon a resource decomposition scheme that separates the integer variables from the continuous ones. The decompostion of the operating problem leads to an interactive scheme which employs a mixed integer programming problem, known as the master, to generate trial operating configurations. The optimum operating conditions of each trial configuration is found using a smooth nonlinear programming model. The dual vector recovered from this model is subsequently used by the master to generate the next trial configuration. The solution algorithm progresses until lower and upper bounds converge. A range of numerical experiments are conducted and these experiments are included in the discussion. Using the operating model as a basis, a regional capacity expansion model is then developed. It determines the type, location and capacity of additional power plants and transmission lines, which are required to meet predicted electicity demands. A generalised resource decompostion scheme, similar to that used to solve the operating problem, is employed. The solution algorithm is used to solve a range of test problems and the results of these numerical experiments are reported. Finally, the expansion problem is applied to the Queensland electricity grid in Australia

Planning of distribution networks for medium voltage and low voltage

Determination of the placement and rating of transformers and feeders are the main objective of the basic distribution network planning. The bus voltage and the feeder current are two constraints which should be maintained within their standard range. The distribution network planning is hardened when the planning area is located far from the sources of power generation and the infrastructure. This is mainly as a consequence of the voltage drop, line loss and system reliability. Long distance to supply loads causes a significant amount of voltage drop across the distribution lines. Capacitors and Voltage Regulators (VRs) can be installed to decrease the voltage drop. This long distance also increases the probability of occurrence of a failure. This high probability leads the network reliability to be low. Cross-Connections (CC) and Distributed Generators (DGs) are devices which can be employed for improving system reliability. Another main factor which should be considered in planning of distribution networks (in both rural and urban areas) is load growth. For supporting this factor, transformers and feeders are conventionally upgraded which applies a large cost. Installation of DGs and capacitors in a distribution network can alleviate this issue while the other benefits are gained. In this research, a comprehensive planning is presented for the distribution networks. Since the distribution network is composed of low and medium voltage networks, both are included in this procedure. However, the main focus of this research is on the medium voltage network planning. The main objective is to minimize the investment cost, the line loss, and the reliability indices for a study timeframe and to support load growth. The investment cost is related to the distribution network elements such as the transformers, feeders, capacitors, VRs, CCs, and DGs. The voltage drop and the feeder current as the constraints are maintained within their standard range. In addition to minimizing the reliability and line loss costs, the planned network should support a continual growth of loads, which is an essential concern in planning distribution networks. In this thesis, a novel segmentation-based strategy is proposed for including this factor. Using this strategy, the computation time is significantly reduced compared with the exhaustive search method as the accuracy is still acceptable. In addition to being applicable for considering the load growth, this strategy is appropriate for inclusion of practical load characteristic (dynamic), as demonstrated in this thesis. The allocation and sizing problem has a discrete nature with several local minima. This highlights the importance of selecting a proper optimization method. Modified discrete particle swarm optimization as a heuristic method is introduced in this research to solve this complex planning problem. Discrete nonlinear programming and genetic algorithm as an analytical and a heuristic method respectively are also applied to this problem to evaluate the proposed optimization method.

Optimal coordination of overcurrent relays in distribution systems with distributed generators based on differential evolution algorithm

Distributed generators (DGs) are defined as generators that are connected to a distribution network. The direction of the power flow and short-circuit current in a network could be changed compared with one without DGs. The conventional protective relay scheme does not meet the requirement in this emerging situation. As the number and capacity of DGs in the distribution network increase, the problem of coordinating protective relays becomes more challenging. Given this background, the protective relay coordination problem in distribution systems is investigated, with directional overcurrent relays taken as an example, and formulated as a mixed integer nonlinear programming problem. A mathematical model describing this problem is first developed, and the well-developed differential evolution algorithm is then used to solve it. Finally, a sample system is used to demonstrate the feasiblity and efficiency of the developed method.

Reliability Optimization using SLUMT

This paper is concerned with the reliability optimization of a spatially redundant system, subject to various constraints, by using nonlinear programming. The constrained optimization problem is converted into a sequence of unconstrained optimization problems by using a penalty function. The new problem is then solved by the conjugate gradient method. The advantages of this method are highlighted.

A four-timescale algorithm for constrained stochastic optimization of RED

The overall performance of random early detection (RED) routers in the Internet is determined by the settings of their associated parameters. The non-availability of a functional relationship between the RED performance and its parameters makes it difficult to implement optimization techniques directly in order to optimize the RED parameters. In this paper, we formulate a generic optimization framework using a stochastically bounded delay metric to dynamically adapt the RED parameters. The constrained optimization problem thus formulated is solved using traditional nonlinear programming techniques. Here, we implement the barrier and penalty function approaches, respectively. We adopt a second-order nonlinear optimization framework and propose a novel four-timescale stochastic approximation algorithm to estimate the gradient and Hessian of the barrier and penalty objectives and update the RED parameters. A convergence analysis of the proposed algorithm is briefly sketched. We perform simulations to evaluate the performance of our algorithm with both barrier and penalty objectives and compare these with RED and a variant of it in the literature. We observe an improvement in performance using our proposed algorithm over RED, and the above variant of it.

Improved Inverted V-Notch or Chimney Weir

This paper is devoted to the improvement of the measuring range of inverted V-notch (IVN) weir, a practical linear sharp-crested weir, designed earlier by the writers. The range of linearity of IVN can be considerably enhanced (by more than 200%) by the addition of a retangular weir of width 0.265W (W = half crest width) at a depth of 0.735d (d = altitude of IVN), above the crest of the weir, which is equivalent to providing at this depth two vertical straight lines to the IVN, resulting in a chimney-shaped profile; hence, the modified weir is named chimney weir. The design parameters of the weir, that is, the linearity range, base flow depth, and datum constant, which fixes the reference plane of the weir, are estimated by solving the nonlinear programming problem using a numerical optimization procedure. For flows through this weir above a depth of 0.22d, the discharges are proportional to the depth of flow measured above a reference plane situated at 0.08d above the weir crest for all heads in the range 0.22d <= h <= 2.43d, within a maximum percentage deviation of ±1.5 from the theoretical discharge. A significant result of the analysis is that the same linear head-discharge relationship governing the flow through the IVN is also valid for the extended chimney weir. Experiments with three different chimney weirs show excellent agreement with the theory by giving a constant average coefficient of discharge for each weir.

Metaheurística para o Problema de Planejamento de Redes de Transmissão de Energia Elétrica com Redimensionamento

Com o passar do tempo, a demanda elétrica de diversas áreas varia tornando necessária a construção de novos geradores elétricos e a expansão da rede de transmissão de energia elétrica. Nesta dissertação, focamos no problema de expansão da rede de transmissão, assumindo que novos geradores estão construídos para suprir as novas demandas. Essa expansão exige altos investimentos que precisam ser cuidadosamente planejados. O problema pode ser modelado como um problema de otimização não linear inteira mista e pertence à classe dos problemas NP-difíceis. Desta forma, uma abordagem heurística pode ser adequada para a sua solução pois pode vir a fornecer boas soluções em tempo computacional aceitável. Esta dissertação se propõe a apresentar um estudo do problema de planejamento da expansão de redes de transmissão de energia elétrica estático e multiestágio. Mostramos o que já existe na literatura para o que é chamado de problema sem redimensionamento e as inovações feitas por nós para o problema com redimensionamento. Quanto aos métodos de solução, utilizamos a metaheurística GRASP para o problema estático e combinamos o GRASP com o procedimento Backward-Forward quando falamos em problema multiestágio. Nesta dissertação comparamos os resultados computacionais obtidos com resultados encontrados na literatura.

Minimum maneuver time calculation using convex optimization

The problem of calculating the minimum lap or maneuver time of a nonlinear vehicle, which is linearized at each time step, is formulated as a convex optimization problem. The formulation provides an alternative to previously used quasi-steady-state analysis or nonlinear optimization. Key steps are: the use of model predictive control; expressing the minimum time problem as one of maximizing distance traveled along the track centerline; and linearizing the track and vehicle trajectories by expressing them as small displacements from a fixed reference. A consequence of linearizing the vehicle dynamics is that nonoptimal steering control action can be generated, but attention to the constraints and the cost function minimizes the effect. Optimal control actions and vehicle responses for a 90 deg bend are presented and compared to the nonconvex nonlinear programming solution. Copyright © 2013 by ASME.