This thesis describes a novel harmonic domain approach for assessing the steady state performance of Flexible AC Transmission System (FACTS) devices. Existing harmonic analysis techniques are reviewed and used as the basis for a novel iterative harmonic domain model for PWM FACTS devices. The unified Newton formulation adopted uses a combination of positive frequency real valued harmonic and three-phase fundamental frequency power-flow mismatches to characterise a PWM converter system. A dc side mismatch formulation is employed in order to reduce the solution size, something only possible because of the hard switched nature of PWM converters. This computationally efficient formulation permits the study of generalised systems containing multiple FACTS devices. This modular PWM converter block is applied to series, shunt and multi-converter FACTS topologies, with a variety of basic control schemes. Using a three-phase power-flow initialisation and a fixed harmonic Jacobian provides robust convergence to a solution consistent with time domain simulation. By including the power-flow variables in the full harmonic solution the model avoids unnecessary assumptions regarding a fixed (or linearised) operating point, fully modelling system imbalance and the associated non-characteristic harmonics. The capability of the proposed technique is illustrated by considering a range of harmonic interaction mechanisms, both within and between FACTS devices. In particular, the impact of transmission network modelling and operating point variation is investigated with reference to ac and dc side harmonic interaction. The minor role harmonic distortion and over-modulation play in the PWM switching process is finally considered with reference to the associated reduction in system linearity.

The thesis describes the development of a model of the three-phase Voltage Source Converter (VSC) in the Harmonic State Space (HSS) domain, a Linear Time Periodic (LTP) extension to the Linear Time Invariant (LTI) state space. The HSS model of the VSC directly captures harmonic coupling effects using harmonic domain modelling concepts, generalised to dynami- cally varying signals. Constructing the model using a reduced-order three-phase harmonic signal representation achieves conceptual simplification, reduced computational load, and direct inte- gration with a synchronous frame vector control scheme. The numerical switching model of the VSC is linearised to gain a small-signal controlled model, which is validated against time-domain PSCAD/EMTDC simulations. The controlled model is evaluated as a STATCOM-type system, exercising closed-loop control over the reactive power flow and dc-side capacitor voltage using a simple linear control scheme. The resulting state- space model is analysed using conventional LTI techniques, giving pole-zero and root-locus analyses which predict the dynamic behaviour of the converter system. Through the ability to independently vary the highest simulated harmonic order, the dependence on the closed-loop response to dynamic harmonic coupling is demonstrated, distinguishing the HSS model from fundamental-only Dynamic Phasor models by its ability to accurately model these dynamics.

This thesis describes the development of several small signal analysis methods for the modelling of power electronic converters. The methods are written generally and are intended to be able to be applied to all converter classes. In the penultimate chapter these general models are used to model the capacitor commutated converter. All the contained methods are based around a time domain small signal model. This time domain small model is a linearization of a power electronic system of passive components and ideal switches described as a hybrid system. The key problem in the derivation of the small signal model is the correct determination and description of the linearized effect of switching instant variation. Three analysis methods based upon the small signal model are advanced in this thesis, these are; time domain sensitivity matrices for use in a Newton determination of the cyclic steady state of a power electronic converter, partial waveform construction of harmonic sensitivity matrices for studying sensitivity of converters to harmonic disturbances, and harmonic state space models also for the construction of harmonic sensitivity matrices and for study of dynamic systems. Each modelling technique is applied to the more common converter topologies of the Buck-Boost converter and the Graetz Bridge before being finally applied to the capacitor commutated converter. Each technique is compared to PSCAD-EMTDC simulations for verification.

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Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3D. / The scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations.