Steady state modelling of non-linear power plant components

This thesis studies the problem of periodic. waveform distortion in electric power systems. A general framework is formulated in the Hilbert domain to account for any given orthogonal basis such as complex Fourier. real Fourier. Hartley and Walsh.· Particular applications of this generalised framework result in unified frames of reference. These domains are unified frameworks in the sense that they accommodate all the nodes. phases and the full spectrum of coefficients of the orthogonal basis. Linear and linearised, non-linear elements can be combined in the same frame of reference for a unified solution. In rigorous waveform distortion analysis. accurate representation of non-linear characteristics for all power plant components is essential. In this thesis several analytical forms are studied which provide accurate representations of non-linearities and which are suitable for efficient. repetitive waveform distortion studies. Several harmonic domain approaches are also presented. To date most frequency domain techniques in power systems have used the Complex Fourier expansion but more efficient solutions can be obtained when using formulations which do not require complex algebra. With this in mind. two real harmonic domain frames of references are presented: the real Fourier harmonic domain and the Hartley domain. The solutions exhibit quadratic rate of convergence. Also, discrete convolutions are proposed as a means for free-aliasing harmonic domain evaluations; a fact which aids convergence greatly. Two new models in the harmonic domain are presented: the Three Phase Thyristor Controlled Reactor model and the Multi-limb Three Phase Transformer model. The former uses switching functions and discrete convolutions. It yields efficient solutions with strong characteristics of convergence. The latter is based on the principle of duality and takes account of the non-linear electromagnetic effects involving iron core, transformer tank and return air paths. The algorithm exhibits quadratic convergence. Real data is used to validate both models. Harmonic distortion can be evaluated by using true Newton-Raphson techniques which exhibit quadratic convergence. However, these methods can be made to produce faster solutions by using relaxation techniques. Several alternative relaxation techniques are presented. An algorithm which uses diagonal relaxation has shown good characteristics of convergence plus the possibility of parallelisation. The Walsh series are a set of orthogonal functions with rectangular waveforms. They are used in this thesis to study switching circuits which are quite common in modern power systems. They have switching functions which resemble Walsh functions substantially. Accordingly, switching functions may be represented exactly by a finite number of Walsh functions, whilst a large number of Fourier coefficients may be required to achieve the same result. Evaluation of waveform distortion of power networks is a non-linear problem which is solved by linearisation about an operation point. In this thesis the Walsh domain is used to study this phenomenon. It has deep theoretical strengths which helps greatly in understanding waveform distortion and which allows its qualitative assessment. Traditionally, the problem of finding waveform distortion levels in power networks has been solved by the use of repetitive linearisation of the problem about an operation point. In this thesis a step towards a true non-linear solution is made. A new approach, which uses bi-linearisations as opposed to linearisations, is presented. Bi-linear systems are a class of simple, non-linear systems which are amenable to analytical solutions. Also, a new method, based on Taylor series expansions, is used to approximate generic, non-linear systems using a bi-linear system. It is shown that when using repetitive bi-linearisations, as opposed to linearisations, solutions show super-quadratic rate of convergence. Finally, several power system applications using the Walsh approach are presented. A model of a single phase TCR, a model of three phase bank of transformers and a model of frequency dependent transmission lines are developed

Estimation of instantaneous complex dynamics through Lyapunov exponents: a study on heartbeat dynamics

Measures of nonlinearity and complexity, and in particular the study of Lyapunov exponents, have been increasingly used to characterize dynamical properties of a wide range of biological nonlinear systems, including cardiovascular control. In this work, we present a novel methodology able to effectively estimate the Lyapunov spectrum of a series of stochastic events in an instantaneous fashion. The paradigm relies on a novel point-process high-order nonlinear model of the event series dynamics. The long-term information is taken into account by expanding the linear, quadratic, and cubic Wiener-Volterra kernels with the orthonormal Laguerre basis functions. Applications to synthetic data such as the H�non map and R�ssler attractor, as well as two experimental heartbeat interval datasets (i.e., healthy subjects undergoing postural changes and patients with severe cardiac heart failure), focus on estimation and tracking of the Instantaneous Dominant Lyapunov Exponent (IDLE). The novel cardiovascular assessment demonstrates that our method is able to effectively and instantaneously track the nonlinear autonomic control dynamics, allowing for complexity variability estimations

Simulation of Continuous Stirred Tank Reactors (CSTR’s) Using Orthogonal Functions

Over the centuries, several numerical methods have been developed to approximate the solution of mathematical problems that are difficult to be solved by analytical methods. These numerical techniques succeeded in attaining a solution that is close enough to the exact solution with minimum errors and maximum stability. However, there may be the development of several other numerical methods which can be robust and efficient than the existing methods.My proposed research work is about the application of one such method-Orthogonal functions. Orthogonal functions can be broadly classified in to three families; namely, the piecewise constant, polynomial, and sine-cosine family. Walsh function and block pulse function belong to the piecewise constant family. So far orthogonal functions have been used in the optimal control, solving integro-differential equations, trajectory problems and so on. However, orthogonal functions have not been applied to chemical systems and processes. Hence my work is emphasised on simulating reactors using orthogonal functions; mainly block pulse functions and triangular functions. The continuous stirring tank reactors (CSTR’s) are widely used in the chemical industries. Hence the reactions in a CSTR are modelled by a set of differential equations which are discretised to a set of algebraic equations by orthogonal functions. Block-pulse functions have been used to obtain the dynamics of concentration and temperature of the continuous stirring tank reactors (CSTR’s). Further a recurrence relationship developed using block-pulse functions and triangular functions have been used in solving linear and non-linear system of differential equations. The major importance of orthogonal functions lies in its application to optimal control to systems. A recursive algorithm developed using block pulse functions has been applied to a linear control problem to determine the states and optimality criterio

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte