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Introduction to Harmonic Balance Finite Element Method (HBFEM)

1.1 Harmonic Problems in Power Systems

The harmonics problem in power systems is not a new problem. It has existed since the early 1900s - as long as AC power itself has been available. The earliest harmonic distortion issues were associated with third harmonic currents produced by saturated iron in machines and transformers, or so-called ferromagnetic loads. Later, arcing loads, like lighting and electric arc furnaces, were also shown to produce harmonic distortion. The final type, electronic loads, burst onto the power scene in the 1970s and 1980s, and has represented the fastest growing category ever since [1].

Since power system harmonic distortion is mainly caused by non-linear loads and power electronics used in the electrical power system [2, 3], the presence of non-linear loads and the increasing number of distributed generation power systems in electrical grids contributes to changing the characteristics of voltage and current waveforms in power systems (which differ from pure sinusoidal constant amplitude signals). The impact of non-linear loads and power electronics used in electrical power systems has been increasing during the last decade.

Such electrical loads, which introduce non-sinusoidal current consumption patterns (current harmonics), can be found in power electronics [4], such as: DC/AC inverters; switch mode power supplies; rectification front-ends in motor drives; electronic ballasts for discharge lamps; personal computers or electrical appliances; high-voltage DC (HVDC) power systems; impulse transformers; magnetic induction devices; and various electric machines. In addition, the harmonics can be generated in distributed renewable energy systems, geomagnetic disturbances (GMDs) and geomagnetic induced currents (GICs) [5, 6].

Harmonics in power systems means the existence of signals, superimposed on the fundamental signal, whose frequencies are integer numbers of the fundamental frequency. The presence of harmonics in the voltage or current waveform leads to a distorted signal for the voltage or current, and the signal becomes non-sinusoidal. Thus, the study of power system harmonics is an important subject for electrical engineers. Electricity supply authorities normally abrogate responsibility on harmonic matters by introducing standards or recommendations for the limitation of voltage harmonic levels at the points of common coupling between consumers.

1.1.1 Harmonic Phenomena in Power Systems

A better understanding of power system harmonic phenomena can be achieved by consideration of some fundamental concepts, especially the nature of non-linear loads, and the interaction of harmonic currents and voltages within the power system. By definition, harmonic (or non-linear) loads are those devices that naturally produce a non-sinusoidal current when energized by a sinusoidal voltage source. As shown in Figure 1.1, each "waveform" represents the variation in instantaneous current over time for two different loads each energized from a sinusoidal voltage source. This pattern is repeated continuously, as long as the device is energized, creating a set of largely-identical waveforms that adhere to a common time period. Both current waveforms were produced by turning on some type of load device. In the case of the current on the left, this device was probably an electric motor or resistance heater. The current on the right could have been produced by an electronic variable-speed drive, for example. The devices could be single- or three-phase, but only one phase current waveform is shown for illustration. The other phases would be similar.

Figure 1.1 (a) Sine wave. (b) Distorted waveform or non-sinusoidal

A French mathematician, Jean Fourier, discovered a special characteristic of periodic waveforms in the early 19th century. The method describing the non-sinusoidal waveform is called its Fourier Series. The Fourier theorem breaks down a periodic wave into its component frequencies. Periodic waveforms are those waveforms comprised of identical values that repeat in the same time interval, as shown in Figure 1.2. Fourier discovered that periodic waveforms can be represented by a series of sinusoids summed together. The frequency of these sinusoids is an integer multiple of the frequency represented by the fundamental periodic waveform.

Figure 1.2 Distorted waveform and number of harmonics by Fourier series

The distorted (non-linear) waveform, however, deserves further scrutiny. This waveform meets the continuous, periodic requirement established by Fourier. It can be described, therefore, by a series of sinusoids. This example waveform is represented by only three harmonic components, but some real-world waveforms (square wave, for example) require hundreds of sinusoidal components to describe them fully. The magnitude of these sinusoids decreases with increasing frequency, often allowing the power engineer to ignore the effect of components above the 50th harmonic.

1.1.2 Sources and Problems of Harmonics in Power Systems

Harmonic sources generated in power systems can be divided into two categories: established and known; and new and future. Table 1.1 presents sources and problems of harmonics. Harmonic problems in power systems can be traced to a number of factors [3], such as: (a) the substantial increase of non-linear loads resulting from new technologies such as silicon-controlled rectifiers (SCRs), power transistors, and microprocessor controls, which create load-generated harmonics throughout the system; and (b) a change in equipment design philosophy.

Table 1.1 Sources and problems of harmonics

In the past, equipment designs tended to be under-rated or over-designed. Nowadays, in order to be competitive, power devices and equipment are more critically designed and, in the case of iron-core devices, their operating points are more focused on non-linear regions. Operation in these regions results in a sharp rise in harmonics.

1.1.3 Total Harmonic Distortion (THD)

The reduced impedance at the peak voltage results in a large, sudden rise in current flow until the impedance is suddenly increased, resulting in a sudden drop in current. Because the voltage and current waveforms are no longer related, they are said to be "non-linear". These non-sinusoidal current pulses introduce unanticipated reflective currents back into the power distribution system, and the currents operate at frequencies other than the fundamental 50/60 Hz. Ideally, voltage and current waveforms are perfect sinusoids. However, because of the increased non-linear load and power electronic devices based on switch mode power supplies and motor drives, these waveforms quite often become distorted. This deviation from a perfect sine wave can be represented by harmonics - sinusoidal components having a frequency that is an integral multiple of the fundamental frequency, as shown in Figure 1.3. Thus, a non-sinusoidal wave has distortion and harmonics. To quantify the distortion, the term total harmonic distortion (THD) is used, and this expresses the distortion as a percentage of the fundamental voltage and current waveforms.

Figure 1.3 Harmonic distortion of the electrical...