#### Year

2022

#### Degree Name

Doctor of Philosophy

#### Department

School of Electrical, Computer and Telecommunications Engineering

#### Abstract

There has already been significant work done to aid in the understanding and quantification of the effects of harmonic emissions from controlled and uncontrolled rectifiers, as well as pulse width modulated (PWM) rectifiers, in electrical networks. Models of each type of rectifier have been developed specifically for power system harmonic studies in order to accurately evaluate voltage and current distortion levels and assess potential side effects in electricity supply networks. Mitigation equipment has been extensively developed to counteract the negative effects of harmonics. Effective mitigation is heavily reliant on the accuracy of harmonic studies and the same of existing rectifier models that are incorporated in the harmonic studies. Though the work introduced in rectifier modelling for the purpose of harmonic analysis may appear to be sufficient, the majority of modelling effort for each type of rectifier has been expended while the rectifier of concern is isolated assuming no background supply distortion, abandoning the ability to quantify the influence of already existing supply distortion levels on equipment harmonic emissions.

The accuracy of modelling harmonic producing equipment, specifically uncontrolled and controlled rectifiers, and PWM rectifiers, under the influence of distorted supply is investigated in this research. Given the inaccuracy of existing models of uncontrolled and controlled rectifiers, as well as PWM rectifiers, the main goal of the research is to develop a mathematical relationship between supply voltage distortion and the change in the harmonic current spectrum, specifically when background distortion levels are significant, i.e., near or above the limits set by standards, so that harmonic mitigation can be achieved in a more robust manner compared to what is possible presently.

The harmonic spectrum of uncontrolled and controlled rectifiers, as well as PWM rectifiers under the influence of supply distortion will be derived employing the harmonic spectrum of the same devices connected to an undistorted supply via a function reflecting the effect of background distortion for each individual harmonic order. A harmonic generic model of an uncontrolled and controlled rectifier is proposed based on a PWM rectifier model to represent the effect of supply voltage distortion on the harmonic current which a PWM rectifier will draw. A mathematical model for the resulting harmonic current is derived using the Double Fourier Integral method. A new method for controlling the PWM rectifier is presented, which replicates the harmonic current spectrum of a typical uncontrolled rectifier with greater accuracy, particularly when operating under the influence of supply voltage distortion. The Double Fourier Integral offers a quick and efficient solution for evaluating the change in harmonic current in uncontrolled rectifiers as voltage distortion levels change, which is then used to refresh the harmonic spectrum of these rectifiers when performing harmonic analysis. In doing so, an evaluation benchmark utilizing the PWM rectifier is put in place to initially test the methodology. Therefore, an uncontrolled rectifier can be treated as a special case of a PWM rectifier operating under the influence of supply voltage distortion. Similarly, the model can be extended to describe the operation of controlled rectifiers. PWM rectifiers, are also analysed with their harmonic emissions tested under the influence of significant supply voltage distortion levels.

To verify the developed mathematical model a representative electrical system, made up of an uncontrolled rectifier and a PWM rectifier, is simulated whilst operating under the influence of varying levels of supply voltage distortion levels. Likewise, a simulation model, is established using ATPDraw, to emulate changes in the current harmonic spectrum at the terminals of PWM rectifier and the uncontrolled rectifier under varying voltage distortion levels. Simulation outcomes show that the harmonic current drawn by the PWM rectifier is affected by the severity of voltage distortion for each harmonic order as expected. The simulations also demonstrate that the harmonic current drawn by the PWM rectifier can be replicated by superimposing the voltage distortion on the carrier frequency of the PWM control circuit as opposed to voltage distortion present at its terminals. A benchmark mathematical model of the power and control circuits for the PWM rectifier is derived using Double Fourier Integral method. The superimposed voltage distortion is mathematically derived by incorporating the voltage distortion in the Double Fourier Integral limits therefore, offering a novel approach to mathematically determine the change in PWM rectifier harmonic current distortion as supply voltage distortion levels are varied. The derived mathematical model is then re-applied to evaluate harmonic currents for uncontrolled rectifier under the influence of supply distortion in a similar manner.

A hardware-based scaled model of a PWM rectifier was built using dSPACE, MATLAB and SEMIKRON laboratory kit. Correspondingly, harmonic current spectrum of a variable speed drive comprising from a six-pulse uncontrolled rectifier was measured under the influence of varying supply distortion. As supply voltage distortion was introduced to the input of the PWM rectifier test circuit, the harmonic current spectrum for each individual order changed while the rectifier was operating under the same load conditions. The result of the measurements is in agreement with the mathematical model and simulation results. A novel mathematical expression is presented that can be used to evaluate the change in harmonic current of the converter when voltage distortion is present in the supply. The mathematical expressions offer a simple modification to the widely used current source-based model of controlled, uncontrolled and PWM rectifiers used to replicate the harmonic spectrum with varying harmonic levels present in the supply. The existing current source model is replaced by a voltage controlled current source performing harmonic analysis. The method will enhance the evaluation of harmonic emissions from uncontrolled rectifiers and state of art renewable energy based PWM rectifiers implemented in distorted electrical networks.

#### Recommended Citation

Abdelqader, Raed, Double Fourier Integral for Modelling Harmonic Distortion of Rectifiers to Improve Harmonic Emission Estimation in Industrial Installations, Doctor of Philosophy thesis, School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, 2022. https://ro.uow.edu.au/theses1/1691

#### FoR codes (2020)

400802 Electrical circuits and systems, 400806 Electrical machines and drives, 400911 Power electronics

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**