Nonlinear Mathematical Optimization Research Articles

Research on Surveying and Mapping Data Processing Based on Nonlinear Mathematical Models and Deep Learning Optimization

In order to deeply understand the mapping data processing of nonlinear mathematical model optimization, this paper uses nonlinear model optimization theory to process mapping data. When the precision of parameter approximation is high, the calculation results of each algorithm are the same, and the iteration times of the fastest descent method and simplex method are significantly increased compared with the other three algorithms. Therefore x01 = 5.42 in the parameter is kept unchanged near the truth value, and the convergence range of Newton method, the fastest descent method, conjugate gradient method and simplex method away from the truth value of parameter x02 = −0.25 is investigated. When the approximate value of undetermined parameters is high, the results of Newton algorithm and simplex algorithm are consistent, but the Newton iterative algorithm has faster convergence speed and higher computational efficiency than the simplex algorithm. Lower value when the undetermined parameter approximation precision, namely the undetermined parameter approximation and its true value is far off, may complete failure type Newton iteration algorithm, the simplex algorithm can be a supplement of the Newton method for the most part, the simplex method significantly reduces the requirements for initial value of parameters, data calculation efficiency has improved significantly.

Lua APIs for mathematical optimization

In this paper we present our Lua programming language libraries that enable modeling and solving mixed integer linear and nonlinear mathematical optimization problems. On one side, these libraries provide a framework for developing algorithms written in Lua that use certain well known solvers within more complex procedures. On the other side, they facilitate the transformation of both input and output data of a mathematical programming problem, in order to compute the standardized coefficients that have to be transmitted to the solvers or prepare solution of solved problems for further data processing. Both of these use cases, together with Lua programming language simplicity, versatility and performance, make Lua a suitable programming language for use in scientific and engineering researches.

Bi-level energy optimization model in smart integrated engineering systems using WSN

Based on fuzzy relational inequality, a bi-level linear programme optimizes the visible light brightness and operating costs of access points in a wireless transmission station system. Consider the first computing problem utilizing a minimum solution matrix. A convex infinite set is generated by a restricted number of closed intervals. Second, computing is an objective-domain nonlinear mathematical optimization problem. A multi-objective optimization problem is used to solve the second programming challenge. The constraint set must be used. Use discrete optimization techniques and branch-and-bound procedures for “digital integer linear programming”. Our technique has been shown to be both practical and successful. The programming complexity increases as the organization expands.

The Least-Cost Optimization of PV-Station DC/AC Equipment Using Battery Energy Storage System

Abstract The auction allocation of state support quotas for renewable generation implementation that will change the existing incentive for their development by providing “green” tariffs to producers is based on competitive selection of the power plant projects with the least cost of electricity generation. The competitive advantage of solar photovoltaic power plant (PV stations) projects can be done by deciding what kind of equipment can provide the minimum cost of electricity generation during the period of operation. In the article, there is improvement of non-linear mathematical least-cost optimization model of the structure of PV-station equipment using DC coupled battery energy storage system (BESS) to store the excess electricity of photovoltaic modules (PV modules), which is lost on inverters when they are overloaded during the hours of the highest intensity of solar radiation. The article presents the modelling results of overall operation of PV station with fixed power of PV modules, as well as determines optimal power of inverter equipment and battery capacity. The model calculations have shown that when the DC equipment costs of the PV station are reduced by almost 62 % and DC equipment costs of BESS are reduced by 86.7 % of the actual value in 2020 at the power of 10 MW PV modules, the optimal power of inverter equipment decreases from 7.08 to 6.29 MW, and the storage capacity increases from 0.22 to 2.51 MWh. The use of BESS allows accumulating the amount of electricity produced by PV modules, which is lost with limitation on inverters, while the cost of their joint electricity production is decreased by 0.42 %.

Optimization of a multi-generation power, desalination, refrigeration and heating system

The optimization of a multi-generation system which represents the integrated dual-purpose desalination plant and a low-scale absorption refrigeration system is addressed. A nonlinear mathematical programming optimization model that integrates a natural gas combined-cycle, a multi-effect distillation desalination plant, a series flow double-effect water-lithium bromide absorption refrigeration system, and a water heater, is developed based on first-principle models. The model is implemented in General Algebraic Modelling System and a generalized gradient-based optimization algorithm is used.Given design specifications for electricity generation (around 37 MW), freshwater production (100 kg/s), refrigeration capacity (2 MW), and thermal load for heating (around 0.7 MW of hot water), the integrated system is optimized by minimizing two objective functions by single-objective optimization: total heat transfer area and total annual cost. As a result, minimum total heat transfer area values of 39148 m2, 36002 m2, and 35161 m2 are obtained when 4, 5, and 6 distillation effects were considered in the multi-effect distillation system, respectively. Also, a minimum annual cost of around 24 MM$/yr. is obtained for 5 distillation effects.The influence of the number of effects in the multi-effect distillation subsystem on the optimal solutions is analyzed. Cost-effective optimal solutions are developed for the studied multi-generation system.

Robust nonlinear mathematical transmission expansion planning based on German electricity market simulation

Requirements on electrical energy transmission and the work of system operators are changing due to the shift from centralized towards decentralized generation. This leads to an increased need of transport capacities in grids. In this paper, a robust method is presented to solve the resulting transmission expansion planning (TEP) problem using a nonlinear mathematical optimization. In terms of TEP optimization successive measures are considered in the used mathematical formulation such as grid optimization, grid enhancement and grid extension. The mathematical optimization is based on a DC power flow formulation using unreduced grid models. The concept of the TEP model design and the implementation of different measures are given in detail in the paper. The applicability of the developed approach is shown by the application to a model of the German transmission grid based on a German electricity market simulation for a yearly simulation calculating an expansion demand of 3047 GWkm.

Advanced Non-linear Mathematical Model for the Prediction of the Activity of a Putative Anticancer Agent in Human-to-mouse Cancer Xenografts.

Mathematical models have long been considered as important tools in cancer biology and therapy. Herein, we present an advanced non-linear mathematical model that can predict accurately the effect of an anticancer agent on the growth of a solid tumor. Advanced non-linear mathematical optimization techniques and human-to-mouse experimental data were used to develop a tumor growth inhibition (TGI) estimation model. Using this mathematical model, we could accurately predict the tumor mass in a human-to-mouse pancreatic ductal adenocarcinoma (PDAC) xenograft under gemcitabine treatment up to five time periods (points) ahead of the last treatment. The ability of the identified TGI dynamic model to perform satisfactory short-term predictions of the tumor growth for up to five time periods ahead was investigated, evaluated and validated for the first time. Such a prediction model could not only assist the pre-clinical testing of putative anticancer agents, but also the early modification of a chemotherapy schedule towards increased efficacy.

A Universal Trajectory Planning Method for Automated Lane-Changing and Overtaking Maneuvers

Lane-changing and overtaking are conventional maneuvers on roads, and the reference trajectory is one of the prerequisites to execute these maneuvers. This study proposes a universal trajectory planning method for automated lane-changing and overtaking maneuvers, in which the trajectory is regarded as the combination of a path and its traffic state profiles. The two-dimensional path is represented by a suitable curve to connect the initial position with final position of the ego vehicle. Based on the planned path, its traffic state profiles are generated by solving a nonlinear mathematical optimization model. Moreover, the study discretizes the time horizon into several time intervals and determines the parameters to obtain the continuous and smooth profiles, which guarantees the safety and comfort of the ego vehicle. Finally, a series of simulation experiments are performed in the MATLAB platform and the results show the feasibility and effectiveness of the proposed universal trajectory planning method.

FEM Based Preliminary Design Optimization in Case of Large Power Transformers

Since large power transformers are custom-made, and their design process is a labor-intensive task, their design process is split into different parts. In tendering, the price calculation is based on the preliminary design of the transformer. Due to the complexity of this task, it belongs to the most general branch of discrete, non-linear mathematical optimization problems. Most of the published algorithms are using a copper filling factor based winding model to calculate the main dimensions of the transformer during this first, preliminary design step. Therefore, these cost optimization methods are not considering the detailed winding layout and the conductor dimensions. However, the knowledge of the exact conductor dimensions is essential to calculate the thermal behaviour of the windings and make a more accurate stray loss calculation. The paper presents a novel, evolutionary algorithm-based transformer optimization method which can determine the optimal conductor shape for the windings during this examined preliminary design stage. The accuracy of the presented FEM method was tested on an existing transformer design. Then the results of the proposed optimization method have been compared with a validated transformer design optimization algorithm.

Parameters identification of photovoltaic models using niche-based particle swarm optimization in parallel computing architecture

Determined parameters for photovoltaic (PV) model is of great practical significance in prediction of output power in PV array and tracing its maximum power point. An optimization algorithm based on niche particle swarm optimization in parallel computing (NPSOPC) is proposed to identify the parameters of PV model. A diode equivalent circuit model is applied to simulate the output characteristics of PV model. On the support of the output current-voltage data of PV model, the parameters identification is transformed to be a multivariate, nonlinear mathematical optimization problem. A mathematical model with objective optimization function is established to quantify the discrepancy between the current experimental data and the simulation data. A particle swarm optimization algorithm based parameters extraction model is established with niches in parallel architecture to improve the extraction performance. Contrast experiments of these three models are carried out in the different condition with different light intensity and temperatures to verify the good performance of the proposed approach. The results indicate that the proposed algorithm can be utilized as an accurate, reliable and promising alternative approach for parameters acquisition in single diode model, double diode model and PV module model.

Nonlinear chance-constrained problems with applications to hydro scheduling

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

Evolution and Modern Approaches of the Power Transformer Cost Optimization Methods

Transformer design is a challenging engineering task, where the different physical fields have to be harmonized together to fulfill the implied specifications. Due to the difficulty of this task, it can be separated into several subproblems. The first subproblem, in the pre-concept phase, during the transformer design is the calculation of the cost optimal key-design parameters, where not only the technical but also the economical parameters have to be considered, as well. This subproblem belongs to the most general branch of the non-linear mathematical optimization problems. This paper presents the main directions of the evolution and trends in the power transformer design. Main directions of the considered research and the future trends in the field of preliminary design transformer optimization methods are summarized.

Solving a Classical Optimization ProblemUsing GAMS Optimizer Package: Economic Dispatch ProblemImplementation

This paper presents an effectiveness strategy to model and solve nonlinearmathematical optimization problems in electrical engineering using Gene-ral Algebraic Modeling System (GAMS) for undergraduate students. Aclassical problem known as economic dispatch has been selected to showthe need of using mathematical tools to solve a large scale optimizationproblem related with engineering. The economic dispatch is a classical op-timization problem in operation of thermal electric systems, being the mainidea to find an economical operation for thermal generators. This opera-tion is based on a quadratic cost curve with some operating constraints,i.e., power balance and generation capacities. A numerical simulation isimplemented using GAMS optimization package in demo version. Thisresearch has been developed with the support of 36 students of the courseof Regulation and Operation of Electrical Systems (ROES) in the programof Electrical Engineering at Universidad Tecnológica de Pereira (UTP).

Impact of short-circuit impedance and tap changing method selection on the key-design parameters of core-form power transformers

Making a competitive transformer design is a nonlinear mathematical optimization task which should be solved in the tendering design stage. Due to the complexity of the power transformer design, a wide range of optimization algorithms are been proposed in the literature. Despite the large number of design optimization methods, most of them neglect the effect of tapping arrangement from their transformer model in the tendering design stage. In this paper, the effect of these different regulation types is examined by a power transformer design methodology which applies geometric programming-based meta-heuristic algorithm. The impact of the selection between a two-winding built-in, a reversed and a coarse-fine regulated tapping arrangement is examined focusing on the total ownership cost and the key-design parameters. A design example on a 63-MVA, three-phase core-type power transformer with the three types of tapping arrangements is presented, and their effect on the parameters is analyzed.

Economic Analysis of Biomass Gasification-Solid Oxide Fuel Cell-Gas Turbine Hybrid Cycle

In this study, an economic analysis of a hybrid system, a pressurized solid oxide fuel cell-gas turbine (SOFC-GT) with a capacity of 1.7 MW, which includes several sub-units, is conducted. Using the formulation, a cost-oriented economic survey including the equipment and the system costs, is also analyzed based on a sensitivity analysis. In compared with the previous studies, the presented model consists of some distinctions and the following considerations: First, considering the field equipment, the formulations is updated; Second, considering the variable cost, a range of variations for parameters’ prices is selected; Third, the objective function in non-linear mathematical optimization format is prioritized. The economic performances for the average price of energy in three different locations (Europe, US and Iran) during the cycle life time, is evaluated in these regions. The results indicate that based on current energy prices and economic conditions, Europe is the most economically justifiable by having internal rate of return (IRR) and a payback period equal to 15.5% and the 6.7 years respectively. It has also been compared with the results of the previous economic studies.

A novel MILP approach for simultaneous optimization of water and energy: Application to a Canadian softwood Kraft pulping mill

An optimization methodology based on Mixed Integer Linear Programming (MILP) has been developed for simultaneous optimization of water and energy (SOWE) in industrial processes. The superstructure integrates non-water process thermal streams and optimizes the consumption of water, while maximizing internal heat recovery to reduce thermal utility consumption. To address the complexity of water and energy stream distribution in pulp and paper processes, three features have been incorporated in the proposed SOWE method: (a) Non-Isothermal Mixing (NIM) has been considered through different locations to reduce the number of thermal streams and decrease the investment cost by avoiding unnecessary investment on heat exchangers; (b) the concept of restricted matches combined with water tanks has been added to the superstructure; and (c) the Integer-Cut Constraint technique has been combined with the MILP model to systematically generate a set of optimal solutions to support the decision-making for cost-effective configurations. The performance of the proposed improved MILP approach has been evaluated using several examples from the literature and applied to a Canadian softwood Kraft pulping mill as an industrial case study. The results indicate that this approach provides enhanced key performance indicators as compared to conceptual and non-linear complex mathematical optimization approaches.

A Mathematical Approach to Neuromarketing: A Weapon – Target Assignment Model

This study developed a nonlinear mathematical optimization model. The aim of the study is to look neuromarketing from a mathematical perspective. The model originates from the wellknown weapon target assignment problem in military operations research. This study proposes that the model can be properly adapted to neuroscience and business applications. The objective of the model is maximizing the total expected satisfaction ratio of people by understanding the functioning of their brain activities under neuroscientific and budget constraints. The recipe is applied to neuromarketing by a hypothetical example which is solved by MS Excel’s Solver.

Adaptive Digital Predistortion for Wideband High Crest Factor Applications Based on the WACP Optimization Objective: A Conceptual Overview

This paper proposes a method of digital predistortion suitable for wideband high crest factor applications such as those encountered in DAB, DVB-T and WCDMA transmitters. The proposed method is advantageous for four main reasons. Firstly, it utilizes a reliable frequency domain measure of transmitter output nonlinearity, specifically the Weighted Adjacent Channel Power (WACP), as the objective for predistortion filter parameter estimation. This is in direct contrast to traditional approaches which utilize a time domain measure obtained via a full feedback path and potentially corrupted by gain and phase compensation error as well as ADC distortion. Secondly, the method models predistortion filter parameter estimation as a generic nonlinear mathematical optimization problem. This model assumes a nonconvex objective function and therefore utilizes both global and local optimization algorithms to achieve true global convergence. This is once again in direct contrast to traditional approaches which model predistortion filter parameter estimation as a linear regression problem. Such a model incorrectly assumes a convex error surface and therefore restricts itself to inadequate local optimization algorithms which unfortunately cannot guarantee true global convergence. Thirdly, the method's predistortion filter is a pruned Volterra Series with memory which utilizes a hybrid pruning strategy in order to keep high order kernels to a practically manageable size, suitable for optimization parameter estimation. Predistortion filter memory ultimately makes the method highly suited to wideband applications. Finally, predistortion filter parameter estimation does not require known test signals to be injected into the transmitter and therefore the technique is on-air adaptive. This means any transmitter using this method of digital predistortion will be both on-air and optimally linearized for its entire operational life. Preliminary results obtained from actual hardware are presented.

Comparative Study of Firefly Algorithm and Particle Swarm Optimization for Noisy Non-Linear Optimization Problems

There are various noisy non-linear mathematical optimization problems that can be effectively solved by Metaheuristic Algorithms. These are iterative search processes that efficiently perform the exploration and exploitation in the solution space, aiming to efficiently find near optimal solutions. Considering the solution space in a specified region, some models contain global optimum and multiple local optima. In this context, two types of meta-heuristics called Particle Swarm Optimization (PSO) and Firefly algorithms were devised to find optimal solutions of noisy non-linear continuous mathematical models. Firefly Algorithm is one of the recent evolutionary computing models which is inspired by fireflies behavior in nature. PSO is population based optimization technique inspired by social behavior of bird flocking or fish schooling. A series of computational experiments using each algorithm were conducted. The results of this experiment were analyzed and compared to the best solutions found so far on the basis of mean of execution time to converge to the optimum. The Firefly algorithm seems to perform better for higher levels of noise.

Fast Layout Generation of RF Embedded Passive Circuits Using Mathematical Programming

In this paper, we present a methodology for the automatic generation of layout for radio frequency (RF) designs using embedded passives. Our methodology is divided into three steps: 1) pre-layout optimization; 2) placement and routing; and 3) post-layout optimization. We show that our methodology generates layouts with small area and good performance response within a fraction of design time compared with fully manual design effort. We make use of circuit models to represent and optimize the physical layout of the resistance-inductance-capacitance (RLC) components as well as the entire RF circuit that uses them. We perform non-linear mathematical programming-based optimization at various stages of the methodology to achieve high quality layouts. Full wave electromagnetic simulations are kept completely out of the design loop, so our methodology significantly reduces design time. We have used our methodology to successfully generate layouts for large-scale filter circuits.