Network harmonics:
Introduction
Apart from the numerous advantages of variable frequency drives, such as precise speed or torque control, smooth motor starting or energy savings, this technology also comes with specific challenges. One of them are the network harmonics.
The good news is that there is a solution. Actually, there are several solutions to choose from. In this series we will look into them and give some hints how to deal with the topic. However, let’s go step by step and structure the topic in a logical sequence. We will start with little bit of theory. Hopefully not too boring 😉 If you are familiar with this matter you can eventually skip this part. However, it might still be a good refresher.
Origin of harmonics
Where do the harmonics come from? How can we imagine them? Ideal grid is a voltage source of specified magnitude and frequency (e.g. 20 kV/50 Hz or 13.8 kV/60 Hz). The voltage is undistorted, i.e. it is a pure sine wave of 50 Hz or 60 Hz. The short circuit capacity is very high; ideally it is infinitely strong grid. Ideal consumer is linear and passive element. Such consumer draws a sinusoidal current when supplied by sinusoidal voltage source. However, real world is not ideal. Many consumers are non linear devices. We all have them in our households (computers, modern lighting). In the industrial networks VFDs belong to some of the most common non linear devices. In operation they cause the current or voltage or both to be non sinusoidal.
Why do we talk about harmonics? Let’s take a simple example of diode rectifier supplied from the grid (directly or through a transformer – it does not matter at this stage). Well, there is basically one current flowing in each of the phases. This current is not purely sinusoidal, but the pattern is repetitive. As explained in next paragraph, such current can be represented by a series of harmonic (sinusoidal) signals that are summed up together.
Fourier analysis
The French genius mathematician and physicist Jean-Baptiste Joseph Fourier brought the fundamentals of Fourier series and Fourier analysis. According to this theory every non-sinusoidal periodic signal can be decomposed into a combination of sine waves of certain frequencies, magnitudes and angles. The component with the largest magnitude is the fundamental component and its frequency is the fundamental frequency. The sine waves of higher frequencies are known as harmonics. Keep in mind that Fourier theory and Fourier analysis is valid only for periodic signals. With regard to harmonics we often talk about harmonic orders (designated as n or h) which is the ratio of their frequencies to that of the fundamental component. This is very useful as specific devices inject typical (characteristic) harmonics into the grid depending on their topology.
Note that Fourier analysis is an abstraction. However, it is very useful for system analysis and for investigation of effects that non-sinusoidal waveforms have.
Mathematical proof of Fourier theory would probably be too complex for this post.You will find more material in the Internet or in professional literature. A simple example is shown below. By adding harmonics to the fundamental waveform we get a waveform that is close to a rectangular shape. Vice versa, rectangular waveform can be decomposed into a series of sinusoidal harmonics (Fourier series). The fundamental harmonic has the same frequency as the rectangular waveform, the (higher) harmonics have frequencies and amplitudes as explained in next paragraph Characteristic harmonics and amplitude law.
Characteristic harmonics and amplitude law
The characteristic harmonics are the dominant harmonic components apart from the fundamental. The harmonic orders of characteristic harmonics are linked with the pulse number of the rectifier bridge according eq. (1).
h = k⋅p ± 1                                (1)
h … characteristic harmonic order
k …Â integer
p … rectifier pulse number
For 6-pulse rectifier bridge (p = 6) the characteristic harmonics are h = 5, 7, 11, 13, 17, 19, 23, 25, etc.
For 12-pulse rectifier (two 6-pulse full bridges supplied from 30 degree phase shifted sources) the characteristic harmonics are 11, 13, 23, 25, etc.
The equation (1) is the base of harmonic elimination efforts based on multi-pulse rectifiers and corresponding phase shifting transformers. The principle will be shown in next posts in section harmonic mitigation.
The amplitude law sets a relation between harmonic order and corresponding magnitude. The magnitude of frequency component of harmonic order h is inverse proportional to the harmonic order acc. (2).
I(h) = 1/h â‹… I(1)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2)
I(h) … magnitude of component of harmonic order h
I(1) … magnitude of fundamental component (h=1)
Amplitude law in form (2) is valid under ideal conditions (infinite inductance on dc side of rectifier, symmetric ac grid, infinitely strong grid, symmetric control). In reality these conditions cannot be met and the eq. (2) becomes more complex. The non ideal conditions might introduce certain non characteristic components as well, but at the same time the total distortion will be reduced. Moreover, the qualitative character is still valid.
Classification of harmonics
Harmonics can be classified according several criteria. Specifically for variable frequency drive systems we can distinguish between:
– Line side harmonics / Motor side harmonics
– Characteristic and non-characteristic harmonics
– Integer and non-integer harmonics
In this series we focus on network harmonics, i.e. on the line side. Motor side harmonics (motor current harmonics, torque pulsations) are discussed in their dedicated series.
Sources of harmonics
Every non-linear device, either power consumer or power source, is a source of harmonics. It means that the current or voltage waveform does not constitute of fundamental component only, but contains components of other frequencies as well (‘harmonics’).
Among non-linear loads belong e.g. LED lights, computer loads, uninterrupted power supplies (UPS), welding supplies, rectifiers, motor starters, variable frequency drives etc.
Effects of harmonics
In some specific applications harmonics are wanted. The working principle might even be based on them. However, in this blog series we will discuss the unwanted harmonics as by-products of frequency conversion. What are the negative effects of harmonics?
– Harmonic heating
The harmonic currents of frequencies that are multiples of fundamental frequency cause additional losses. The resistance of a conductor increases with frequency which means more losses. This effect is generally known as skin effect [3]. You can easily see it when looking e.g. into a catalog or datasheet of a cable. There you will find the resistance for dc current as well as resistance for 50 Hz AC. In table 1 below you can see that DC resistance of a 240 mm2 cable is 0.0754 Ω while the AC resistance at 50 Hz is 0.098 Ω (1.3 times higher). No need to say that for harmonic component of several hundred Hz the resistance gets even higher.
Table 1: DC and AC resistance of a medium voltage cable [4]
Note: This is the reason why High Voltage Direct Current, HVDC, is used for high capacity transmission – to minimize the transmission losses and to increase the line capacity and stability.
The network harmonics cause loss heating in power cables, input transformer and basically all components that they flow through. The losses are not limited to conductors (winding) only. The eddy currents flowing in ferromagnetic materials are responsible for the resistive heating of the core material.
– Audible noise
The harmonics also caused increased noise. Human ear is more sensitive to particular frequencies, especially in the range 1-2 kHz, while for other frequencies the sensitivity is lower. In many countries the limits of permissible noise level became very strict. If for instance the transformer is located in or close to a residential area, low noise level might be required. It is a challenge to reach the noise level for sinusoidal supply and then every extra decibel (dB) caused by harmonics is very critical.
– Vibration / forces
Harmonic components may also cause vibration that usually goes hand in hand with audible noise described above. Vibration is mainly a topic for the motor side (torque pulsations). It can also be seen on the grid side, but usually does not have any significant impact apart from possible unpleasant noise.
– Distortion of grid voltage
This is one of the main concerns regarding network harmonics on the system level. The harmonics generated by the non-linear devices cause distortion of grid voltage. Weak networks (i.e. networks with low short circuit capacity) tend to have higher voltage distortion. This rule does not have a general validity, but holds for most cases. If the frequencies of the injected harmonics are close to a resonance frequency of the grid a significant distortion might be expected. This distortion then:
– disturbs other consumers connected to the same grid (point of common coupling)
– might cause a malfunction of some of the consumers
– might cause interferences with radio signals
– Resonance effects
In a resonance condition the system oscillates with larger magnitude than at other frequencies. The resonance in the grid is not dangerous unless it is excited by a source of harmonics where one of these harmonics coincides with the resonance (the frequency of harmonic component matches the resonance frequency of the grid). In such condition, larger harmonic distortion is expected. The effect depends on the magnitude of harmonic component and inherent system damping. The lower is the damping the more dangerous is the resonance condition.
How does the series continue?
The introduction provided some theoretical background on network harmonics. Where does the journey go?
Next posts will be more specific. We will touch following topics:
– Harmonic spectrum of VFD (depending on topology)
– Harmonic mitigation methods
– Harmonic calculations and network studies
– Standards and grid codes
– Measurements
If harmonics is a topic you want to learn more about, simply follow this series. Keep in harmony.
References
[1] A. Kloss, A basic guide to power electronics, John Wiley & Sons, 1984
[2] J. Arrillaga, N.R. Watson, Power System Harmonics, John Wiley & Sons, 2003
[3] Skin effect – Wikipedia, https://en.wikipedia.org/wiki/Skin_effect (accessed on 2019-10-18)
[4] 6 – 36 kV Medium Voltage Underground Power Cables, Nexans catalog, available online https://www.nexans.co.uk/UK/files/Underground%20Power%20Cables%20Catalogue%2003-2010.pdf (accessed on 2019-10-18)
