Three-dimensional Space Vector Research Articles

Identities Generalizing the Theorems of Pappus and Desargues

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

Wolfram-технологии в реализации принципа наглядности при обучении математическим дисциплинам в высшей экономической школе

The article focuses on the substantive and methodological features of the implementation of the principle of clarity by means of Wolfram technologies. The need and methodological feasibility of applying Wolfram technologies in the practice of teaching mathematical disciplines at an economic university is justified. Special attention is paid to the management of educational and cognitive activities of students when working with the created base of visualizations of mathematical concepts and objects, as well as the results of economic and mathematical modeling. Recommendations are presented for teachers of mathematical disciplines, compliance with which contributes to a deeper penetration of undergraduate students into the essence of mathematical methods and a meaningful interpretation of the results obtained on their basis. Disassembled and methodically characterized visualizations affect educational situations that arise when studying various mathematical disciplines: Linear Algebra, Higher Mathematics, Probability Theory and Mathematical Statistics, Theory of Optimal Control, Game Theory, etc. Among them is the visualization of the probability of a random variable falling into the interval under the condition of a normal distribution of a random variable, as well as under the condition that a random variable has a Weibul distribution; visualization of probability density of hypergeometric distribution; visualization of a fragment of a function graph on a period length segment; visualization of parallelogram rule for addition of two vectors in three-dimensional space; visualization of the direction field and integral curves when solving the ordinary differential equation of the first order; visualization of the direction field when solving a system of linear differential equations of the first order. The main results of the study of the new instrumental implementation of the classical didactic principle of clarity described in this article can be useful for improving the professional training system of the future bachelor of economics, as well as for designing the content of new professionally significant disciplines in the educational field «Mathematics and mathematical modeling».

Length-Preserving Directions and Some Diophantine Equations

We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail, and also discuss the extension to three-dimensional vector spaces.

Machine Learning-based USD/PKR Exchange Rate Forecasting Using Sentiment Analysis of Twitter Data

This study proposes an approach based on machine learning to forecast currency exchange rates by applying sentiment analysis to messages on Twitter (called tweets). A dataset of the exchange rates between the United States Dollar (USD) and the Pakistani Rupee (PKR) was formed by collecting information from a forex website as well as a collection of tweets from the business community in Pakistan containing finance-related words. The dataset was collected in raw form, and was subjected to natural language processing by way of data preprocessing. Response variable labeling was then applied to the standardized dataset, where the response variables were divided into two classes: “1” indicated an increase in the exchange rate and “ −1” indicated a decrease in it. To better represent the dataset, we used linear discriminant analysis and principal component analysis to visualize the data in three-dimensional vector space. Clusters that were obtained using a sampling approach were then used for data optimization. Five machine learning classifiers—the simple logistic classifier, the random forest, bagging, naïve Bayes, and the support vector machine—were applied to the optimized dataset. The results show that the simple logistic classifier yielded the highest accuracy of 82.14% for the USD and the PKR exchange rates forecasting.

New Simplified and Generalized Three-Dimensional Space Vector Modulation Algorithm for Multilevel Four-Leg Diode Clamped Converter

This article aims to propose a new simplified and generalized three-dimensional space vector modulation (3DSVM) for four-leg multilevel diode clamped converter (FL-DCC). The idea is to use the three-dimensional space vector diagram of the multilevel FL-DCC in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">αβγ</i> space, while all calculation stages of the 3DSVM algorithm will be carried out in a new frame called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ρστ</i> . The proposed 3DSVM algorithm provides, for any inverter level, 1) simple way to calculate the duration time; 2) automatic generation of the adjacent switching states; 3) balancing the dc capacitor voltages, and 4) switching sequence definition without prism and tetrahedron identification stage. All 3DSVM steps are implemented in only one sector, thanks to the proposed new reference voltage vector. Such feature makes possible to build a generalized 3DSVM that can be implemented with minimum computational burden in simple embedded digital platforms. Simulation and experimental results are provided to demonstrate the proposed concept.

Model predictive control with constant switching frequency for four‐leg DSTATCOM using three‐dimensional space vector modulation

In this study, model predictive control (MPC) with constant switching frequency is proposed for four-leg distribution static compensator (FL-DSTATCOM) to compensate the current related power quality issues. MPC techniques having many advantages compared to conventional pulse width modulation and hysteresis control techniques. In MPC, the future behaviour of the controlled variable is predicted using the model of the system and a cost function is formed using reference and predicted variable. MPC for DSTATCOM applications consider the difference between the reference and actual DSTATCOM current as a cost function and selects the switching state which minimises the cost function. However, while using MPC, the switches of DSTATCOM operate in variable switching frequency and sometimes the values are also very high. Due to this reason higher switching losses, unequal stress appear across the inverter switches and it also reduces the lifetime of the switch. Therefore, in this study, the concept of three-dimensional space vector modulation is adopted to achieve a constant and user-defined switching frequency MPC for FL-DSTATCOM to conquer the limitations of conventional variable switching frequency MPC. The validation of the proposed work is achieved using simulation and experimental studies.

Minimization of Common-Mode Voltage of Three-Phase Five-Level NPC Inverter Using 3D Space Vector Modulation

In this paper, a three-dimensional Space Vector Modulation (3D SVM) is implemented for minimization of Common-Mode Voltage (CMV) of five-level Neutral Point Clamped (NPC) inverter. The 3D SVM control includes all merits of 2D SVM and provides better control compared to other PWM strategies. The switching state vectors are selected based on the nearest vector Switching State Vector (NSV); it selects the switching vectors which are having the minimum CMV level. It leads to minimization of the bearing voltage and protection of the drive from the damage; also this system reduces the total harmonic distortion. The switching time is calculated by reference vector identification with large and small subcubes tracking and prisms tracking in 3D cubic region. The CMV level with 3D SVM scheme is compared with other PWM methods. The simulation and hardware results are verified using Matlab Simulink and FPGA processor.

Modulated Model Predictive Control for Four-Leg Inverters With Online Duty Ratio Optimization

Conventional finite control set (FCS) predictive load-voltage control for four-leg inverters features simplification in digital implementation compared to its complex counterpart, three-dimensional space vector modulation. However, the FCS approach is typically characterized by a variable switching frequency, low steady-state performance, and a higher average switching frequency in the neutral leg than the other three legs. To overcome these shortcomings, an online duty-cycle-optimization based model predictive control technique is proposed in this article. The performance of the inverter is evaluated experimentally in terms of load voltage tracking in different operating conditions such as nonlinear loads and balanced/unbalanced references. The proposed technique fulfills the requirement of a high-performance converter, constant switching frequency, and enhanced load voltage tracking at a lower sampling rate and, thus, solves the drawbacks of the conventional FCS model predictive control for a four-leg inverter. Furthermore, the proposed approach can facilitate the overmodulation operation, which is desirable for this type of power converters under nonlinear and unbalanced loading conditions.

Single‐stage high‐frequency‐isolated three‐phase four‐leg buck–boost inverter with unbalanced load

A three-phase four-leg inverter shows its preponderance on providing energy to unbalanced load and high DC-link utilisation. To increase the power density of the traditional three-phase four-leg inverter with power frequency isolation, this study proposes a single-stage isolated three-phase four-leg inverter. To ensure the symmetry of three-phase output voltages, the proposed converter provides the path for neutral current resulted by the unbalanced load. It achieves buck-boost DC/AC conversion by introducing the transformer whose turns ratio provides additional voltage gain. The working frequency of the transformer is in accord with the switching frequency, so high-frequency electrical isolation is realised with a single-stage structure as well. The specific mode analysis and the relationship between the switching states and the voltage vectors are presented. A modified three-dimensional space vector pulse width modulation algorithm with a simplified section partition method is proposed, which guarantees the voltage-second balance of the transformer within one switching cycle as well. In order to explore the feasibility of the topology and control strategy, the simulation is done by MATLAB/Simulink. Then a prototype was built in the lab and the experimental results verify that the inverter can produce balanced voltages under both balanced and unbalanced load.

Fatigue life prediction method for notched 3D woven composites based on progressive damage and field intensity

To solve the problem of fatigue life prediction of notched 3D woven composites, a novel fatigue life prediction method based on progressive damage and field intensity theory is proposed. By analyzing the initial loading process of 3D woven composite structure based on progressive damage method, the stress field when the fatigue damage develops gently and the dangerous area where the fatigue damage occurs first are obtained. Then combining with the concept of notch field intensity function, the failure criteria of composites and the three-dimensional space vector stress field intensity method, the fatigue damage degree field intensity of the fiber yarns in the dangerous area is analyzed and the most damaged fiber yarn is found. The fatigue life of the structure is calculated with the field intensity equivalent stress and the S-N curve of this fiber yarn. In order to verify the effectiveness of this method, the fatigue life of notched 3D woven composites under several stress levels is predicted and compared with the existing experimental data. The compared results show that the fatigue life prediction values are all within the two-fold error bounds of the experimental data, which are in good agreement with experimental results. Compared with the progressive damage fatigue life prediction method of woven composites, the prediction accuracy of the method proposed in this paper is improved by 31.5% on average, and the calculation efficiency is also greatly improved.

Examples of Gibbs States of Mechanical Systems with Symmetries

In a previous paper, the notion of Gibbs state for the Hamiltonian action of a Lie group on a symplectic manifold was given, together with its applications in Statistical Mechanics, and the works in this field of the French mathematician and physicist Jean-Marie Souriau were presented. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar{\'e} disk and the Poincar{\'e} half-plane.

Provision of Balanced Voltages Under Severe Unbalanced Load Currents in a Microgrid Using a 3D-SVM Technique

The current source inverter (CSI) is more suitable for use in photovoltaic applications than the voltage source inverter (VSI), but this structure has two main constrains: inability to feed unbalanced load and high common mode voltage (CMV). Despite the merits, use of CSI is limited due to these constrains. A three-phase CSI with a neutral-leg is presented in this paper, compared to the conventional three-phase CSI to overcome mentioned limitations, in which load unbalance under a PV system is dealt in a microgrid. An innovative three-dimensional space vector modulation (3D-SVM) is suggested for the four-leg CSI inverter. This novel 3D-SVM technique is very simple compared to the existing methods and computational complexity is too low. Operating principles, equivalent circuit and SVM is discussed for the four-leg inverter. Then, a balancing technique is applied to the four-leg inverter using the developed 3D-SVM, where simulations validate the capability of the inverter in supplying balanced load voltages; 4-leg inverter generates balanced sinusoidal voltages even under severe unbalanced load current.

Classic Discrete Control Technique and 3D-SVPWM Applied to a Dual Unified Power Quality Conditioner

The unified power quality conditioners (UPQCs) are useful to correct distortions in voltage and current waveforms in case of problems related to harmonics, flicker, and power factor. The dual topology of the UPQC allows a loop control in the inverters less demanding in terms of switching functions; however, the control techniques proposed in the literature have some disadvantages as large computation time and some calculation delays. Therefore, this paper presents the application of a classic discrete-time control model for a three-dimensional space vector pulse width modulation (3D-SVPWM) to be used with the dual UPQC, obtaining a fixed commutation frequency and a low computation cost. The results show that the applied method helps to reduce the harmonics in the source, voltage sags and load current and improve the power factor of the electrical circuit tested. Furthermore, the high frequency created by the switching of elements in the UPQC is filtered by the impedance of the power source and it does not represent a problem for the circuit. Due to the simplicity of the model, simulations demonstrate that the application of this classical control technique is enough to achieve good results to compensate harmonics and power factor with the dual UPQC. The development of the controllers is carried out by discretization of the transfer functions of the control in continuous time and their application with the three-dimensional space vector pulse width modulation (3D-SVPWM) technique.

An Improved Reduced Complexity Model Predictive Current Controller for Grid-Connected Four-Leg Multilevel Inverter

Model predictive current controller offers numerous benefits over the traditional controllers in grid-tied inverters. However, the real-time execution duration reduction of model predictive controller in multilevel inverters is a big challenge, which has been investigated by several researchers. The work presented in this article exploits the three-dimensional space vector representation of a four-leg neutral point clamped three-level inverter in a-b-c coordinates to lower the real-time execution duration of model predictive controller. The objective function determination counts of model predictive controller in the process of controlling neutral point voltage and source current of four-leg neutral point clamped inverter are significantly reduced in the proposed scheme. Therefore, real-time execution duration is reduced as well. In the proposed scheme, three virtual mutually orthogonal sliding planes decide the objective functions to be determined. This method evades the use of any weighing factor in the objective function, which completely obviates the parameter tuning requirement for the control of the four-leg inverter. The extension procedure for higher level four-leg inverter is also explained. Experimental evidence is presented to prove the superiority of the core idea with respect to a recent work.

MODELING THE MOTION OF A SOLID BODY UNDER THE ACTION OF THE MOMENT OF LIGHT PRESSURE IN THE MEDIUM WITH RESISTANCE

The paper describes the simulation of fast rotational motionof a dynamically asymmetric satellite relative to the centre of mass under the influence of the joint effect of the moment of forces of light pressure and resistance. It is assumed that the surface of the spacecraft is a surface of revolution. The medium creates a weak resistance proportional to the angular velocity of the rigid body's own rotation relative to the centre of mass. Orbital motions with an arbitrary eccentricity are considered given. The mathematical model of satellite motion in this formulation is described by a rigid system of differential equations: the fast variables are Euler angles, and the slow variables are the modulus of the angular momentum vector, the kinetic energy, and the angles of ori-entation of the angular momentum vector in space. Averaging is performed over the Euler-Poinsot motion. The averaged system of equations of body motion allows numerical simulation of the satellite's motion relative to the centre of mass. The study is carried out in a dimensionless form for a multiparameter system of equations. For numerical calculation, an implicit third-order Adams method is used to integrate systems of differential equations. A personal computational package was developed for the constructed mathe-matical model of the satellite, as well as a library for calculating the complete elliptic integrals of the first and second kinds. Numer-ical calculation allows one to obtain the functions of modulating the modulus of the satellite kinetic moment vector, its orientation angles to the orbit, as well as the satellite kinetic energy values. The analysis of the influence of the parameters of the problem on the nature of the motion of the satellite relative to the centre of mass is carried out. A qualitative picture was obtained of the influence of the initial values of the angles of orientation of the kinetic moment vector, the geometry of the masses, the eccentricity of the orbit, the characteristic numbers of disturbing moments on the hodograph character of the kinetic moment vector. The hodograph of the kinetic moment vector in three-dimensional space is simulated for various values of the system parameters. To construct three-dimensional objects on the scene, according to the carried out numerical calculations, we developed our own software using DirectX technology in C# language, simulating a virtual laboratory of a numerical experiment.

Attitude Estimation of Underwater Vehicles Using Field Measurements and Bias Compensation.

This paper proposes a method of estimating the attitude of an underwater vehicle. The proposed method uses two field measurements, namely, a gravitational field and a magnetic field represented in terms of vectors in three-dimensional space. In many existing methods that convert the measured field vectors into Euler angles, the yaw accuracy is affected by the uncertainty of the gravitational measurement and by the uncertainty of the magnetic field measurement. Additionally, previous methods have used the magnetic field measurement under the assumption that the magnetic field has only a horizontal component. The proposed method utilizes all field measurement components as they are, without converting them into Euler angles. The bias in the measured magnetic field vector is estimated and compensated to take full advantage of all measured field vector components. Because the proposed method deals with the measured field independently, uncertainties in the measured vectors affect the attitude estimation separately without adding up. The proposed method was tested by conducting navigation experiments with an unmanned underwater vehicle inside test tanks. The results were compared with those obtained by other methods, wherein the Euler angles converted from the measured field vectors were used as measurements.

Detecting flow regime changes from the spatial RGB distribution of turbid water images using a three-dimensional vector space model

Detecting flow regime changes from the spatial RGB distribution of turbid water images using a three-dimensional vector space model

A three-dimensional active zero state PWM for common-mode voltage reduction of a three-phase four-leg voltage-source inverter

PurposeThis paper aims to investigate the common-mode voltage (CMV) issue of a three-phase four-leg voltage-source inverter. A new space vector modulation method, named as three-dimensional active zero state Pulse-width modulation (PWM) (3-D AZSPWM), is proposed to reduce the CMV level.Design/methodology/approachPWM is a general method to generate the switching signals of the power converters in order to obtain high-quality output voltages. However, the CMV produced by PWM methods has become a serious problem. 3-D AZSPWM is proposed to solve this issue. In 3-D AZSPWM, instead of using zero voltage vectors with high CMV level, appropriate complementary non-zero vectors are introduced to synthesize reference vector. The proposed method is classified into four types of AZSPWM1(a), AZSPWM1(b), AZSPWM2(a) and AZSPWM2(b) based on different complementary vectors chosen for each type. An extend software simulation using MATLAB/Simulink is performed to verify the superior performance of the proposed methods.FindingsCompared to other reduced CMV methods, the proposed method not only reduces the CMV but also retains the positive characteristics of the three-dimensional classical space vector PWM (3-D CSVPWM).Originality/valueThe proposed method does not suffer from linear modulation region limitation and also does not impose additional switching loss. Furthermore, calculated output voltage harmonic distortion factor illuminates acceptable quality of output voltage produced by the proposed method.

Three-Dimensional Space Vector Modulation Strategy for Capacitor Balancing in Split Inductor Neutral-Point Clamped Multilevel Inverters

Capacitor imbalance is one of the major drawbacks of a neutral-point clamped multilevel inverter (NPC-MLI). The capacitor imbalance rises due to the nonuniform switching, nonideal DC link capacitors, improper commutation, and various asymmetrical phase currents in switching states. The imbalance can be minimized by using proper switching using a redundancy switching method and avoiding the usage of medium vectors. The computational cost of this system is to be decided by the level of the inverter system. This paper presents a comprehensive three-dimensional space vector modulation (3D-SVM) strategy to eliminate the imbalance in the DC link capacitor voltage, which is across the applied input DC source. The technique is easy to implement without using any trigonometric functions, lookup tables, or angle determinations. This proposed scheme has been verified using MATLAB Simulink and authenticated using field-programmable gate array (FPGA) controller.

The uncertainty mapping of ontologies based on three-dimensional combination weight vector space model

PurposeThis paper aims to apply vector space model (VSM)-PCR model to compute the similarity of Fault zone ontology semantics, which verified the feasibility and effectiveness of the application of VSM-PCR method in uncertainty mapping of ontologies.Design/methodology/approachThe authors first define the concept of uncertainty ontology and then propose the method of ontology mapping. The proposed method fully considers the properties of ontology in measuring the similarity of concept. It expands the single VSM of concept meaning or instance set to the “meaning, properties, instance” three-dimensional VSM and uses membership degree or correlation to express the level of uncertainty.FindingsIt provides a relatively better accuracy which verified the feasibility and effectiveness of VSM-PCR method in treating the uncertainty mapping of ontology.Research limitations/implicationsThe future work will focus on exploring the similarity measure and combinational methods in every dimension.Originality/valueThis paper presents an uncertain mapping method of ontology concept based on three-dimensional combination weighted VSM, namely, VSM-PCR. It expands the single VSM of concept meaning or instance set to the “meaning, properties, instance” three-dimensional VSM. The model uses membership degree or correlation which is used to express the degree of uncertainty; as a result, a three-dimensional VSM is obtained. The authors finally provide an example to verify the feasibility and effectiveness of VSM-PCR method in treating the uncertainty mapping of ontology.