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.