POWER SYSTEM HARMONICS IN DETAIL: Subject 2: Effect of FFT Window Size on Harmonic Spectrum

     Power system harmonics are integral parts of Power Quality concerns. Harmonics causes lots of harms on fragile loads in the system and they cause overloading of the power system elements such as transformers or cables. Therefore, they should be eliminated, however, in order to eliminate them totally ,at first we should correctly measure their magnitudes by our control system.

     

Our aim : Measuring the whole harmonic spectrum of a signal of any Power System variable.

Tools we have: We have measuring probes, interface electronic card and a microcontroller based control system. 

Methodology: At first we should determine frequency resolution for our harmonic spectrum. Generally 10 Hz resolution is enough for ordinary power system load currents and voltages as stated in the IEC 61000-4-7 recommendation for 50 Hz synchronous frequency. For 10 Hz resolution, the window size will be 1/10 = 0.1 sec. After knowing the window size, we can sample all the data included by the window and calculate the FFT(Fast Fourier Transform) for finding the harmonic spectrum.

     Effect of  FFT Window Size on Harmonic Spectrum

     Up to now it is seem to be well defined, however, this is not the actual case. If we have integer number of complete cycles for a harmonic in a measuring window, the result of FFT gives us the correct magnitudes. However, if the frequency of harmonic component is not the integer multiplication of fundemental frequency, then there won't be complete cycles in an FFT window. Therefore, the magnitude of the actual signal is shared by the nearby frequencies and most importantly high frequency components,which does not exist in reality, will appear at the harmonic spectrum. Let me try to explain what is the reason of this situation. FFT is a linear transform that transforms the time domain signals into the frequency domain. Actually for FFT calculations we never use the whole signal (it can not be practical since the signal is from -infinity to +infinity), instead of that we use a part of it and assume that the signal continues periodically as if it continues from the beginning of the window. Figure 1 illustrates what I try to explain.

Figure 1: Case when window size contains complete cycles of periodic signal

     As can be seen in figure 1 when we have complete cycles of periodic signal, the FFT behaves correctly to the signal. and if we have a 250 Hz signal at window size 100 msec, corresponding harmonic spectrum is as in figure 2.

Figure 2: Harmonic Spectrum of 250 Hz signal in 100 msec window(10 Hz resolution)

     As expected, 250 Hz component is appeared at harmonic order of 25. This is because we have 10 Hz resolution. Please draw your attention to that the signal magnitude is calculated correctly as 1 and there will be no high frequency components, their magnitudes are too small and they can be ignored. As a result, our FFT code calculates the magnitude of our signal which is sampled at 25 kHz correctly as 1, since the window contains 25 complete cycles. 

     Let me pass through the most impressive result. What if our frequency is slightly deviates to 255 Hz. Actually please consider that 255 Hz component is not the harmonic component of an ordinary load, this value can be an individual harmonic in addition to the characteristic harmonics. In other words, 255 Hz can not be generally appear as a characteristic 5th harmonic, this can only be happen if the system frequency  deviates to 51 Hz, which is an unlike situation 50 Hz deviates +-0.2Hz at max. We ,for this article, are dealing with the individual load harmonic of 255 Hz. This kind of harmonic components (inter-harmonics) can be encountered mostly in the loads of metal industry, especially for metal melting Induction Furnaces. If you want to reach this kind of load characteristics please contact me. Since we have 255 Hz signal, our FFT window contains 25 complete cycles and 1 half cycle, which means that our control system calculates FFT of a signal of 255 Hz which is distorted at every 100 msec. I try to explain this case in figure 3.

Figure 3: Case when window size does not contain complete cycles of periodic signal

     Figure 3 shows an exaggerated version of the case I try to mention but this is a good illustration. Actually in our case, the distortion happens once for every 25 cycle. Let us see what happens to frequency spectrum for this case.

Figure 4: Harmonic Spectrum of 255 Hz signal in 100 msec window(10 Hz resolution)

     Figure 4 shows the frequency spectrum when we have 255 Hz signal in 100 msec window. I can hear your criticisms like it is normal that the magnitude of the signal is reduced since our resolution is 10 Hz, thus the magnitude is shared by the nearby points. The criticism is totally correct, because in the IEC recommendation the harmonic components are calculated with geometric sum of nearby components for this kind of cases. However, the point I want to draw attention is that there are higher frequency components appearing in the harmonic spectrum which are actually does not exist. When you thoroughly look at the figure 4, you can see that the magnitude of 400 Hz component is about 3% of the individual harmonic, which is an important result. Unlike the harmonic spectrum in figure 2, this case creates the high frequency harmonic components which do not exist in reality. This situation is totally the result of wrong measurement and may cause limit exceeding for higher order harmonics, whose magnitude in reality do not exceed the limits.
     I tried to explain the effect of FFT window size on Harmonic Spectrum of a signal. Actually, the results I show are inevitable for inter-harmonics, since we can not form a window which contains the complete cycles of all harmonic components. However, if we increase our resolution, meaning size of the FFT window, we can significatly reduce the effect of this measurement defects. This time we should consider the storage and execution time concerns. Our control system can not calculate the FFT if our data is huge ,actually we can not even store them, and our process time increases significantly.

POWER SYSTEM HARMONICS IN DETAIL: Subject 1: Effect of Two Closely Located Harmonics

     Harmonics in power system can be basicly introduced as the unintended frequency components, generally higher than system frequency, of current or voltage in the power system. The sources of current harmonics and the harm of them to the power system is stated in an article about Active Power Filters in the blog, thus; I will skip that part. Instead of the drowning in details, I will mention the case of two very closely located harmonics and their effect on harmonic spectrum. Almost everyone who is dealing with power system signal processing can know that the close harmonics tend to increase the magnitude of frequencies between them in FFT spectrum. However, the point that I will draw attention will be a different side of this situation, actually, the behaviors while they are approaching each other to almost 2 Hz. 

     Some loads in Metal Industry behaves very different than other widely known industrial load. For example, arc furnaces, laddle furnaces or induction furnaces for metal melting purposes have very distructive behaviours to the power system. For the introduction of this article, I will briefly mention some behaviours of induction furnaces. For one of the induction furnaces that I take measurements with my collegues in the field, We recognised that the load creates harmonic spectrum other than the characteristics of any other load in industry. The induction furnace creates variety of interharmonics whose frequency value is changing dynamically between 280 Hz and 550 Hz. In other words, there are not fixed frequency harmonics, there can be for example 312 Hz harmonic currents, while the harmonic was in 290 Hz 10 minutes before 312 Hz. The harmonic frequencies are changing during the melting process. I and my collegues are recognised that there can be two harmonics ,which are 2 Hz apart each other, at the same time. Let me know explain what happens at this case.

     

     As you may know the whole current or voltage signal can be expressed as the sum of fundemental and harmonics frequency sine waves. Therefore, for our example ,for the sake of simplicity, we can consider the sum of 282 Hz and 280 Hz harmonic components whose magnitudes are A and B, respectively. Considering the fundemental Cosine Sum formulation in equation (1), we can apply the rule to our power system signal to get equation (2).

     For the sake of simplicity, we can consider that the phases of our harmonics are equal to zero in order to reach equation (3). If we consider that the magnitudes A and B of the harmonic components are equal to each other as A, we can reach to equation (4).

     When we apply our example to the equation (3) , by putting the frequency values 282 and 280 Hz, we can reach to the result of equation (5). Similarly if we consider that the magnitudes of the harmonics are the same at A, the equation (6) is reached. What is the interpretation of the equation (5) ? Equation (5) means a 282 Hz signal whose amplitude is modulated with a sine wave of 1 Hz. The physical meaning of this situation is that the 282 Hz harmonic component has high magnitude during 0.5 seconds and low magnitude in the following 0.5 seconds. Actually, if you have an active or passive filter for filtering this harmonic, you can hear an audible voice whose intensity is increasing and decreasing at every 0.5 seconds. What if the magnitudes are the same for two harmonics ? This causes the most severe result. The magnitude of the harmonic is increased up to double and decreased down to zero at a frequency of 1 Hz. Similarly, the physical meaning of this situation in active or passive filter case gives that the audible voice is extremely increased and totally ceased with a period of 0.5 seconds.

     Now let me verify this situation using MATLAB.

MATLAB can be get from Amazon at link,

http://www.amazon.com/MATLAB-SIMULINK-Student-Version-R2007a/dp/0979223903/ref=sr_1_1?s=software&ie=UTF8&qid=1364394510&sr=1-1&keywords=matlab+software

Used MATLAB code for different magnitudes;

t = 0:1.5625e-4:1; % 6400 samples/second sampling frequency

A = 1; % magnitude of 282 Hz component

B = 0.3; % magnitude of 280 Hz component

X = A*cos(2*pi*282*t) + B*cos(2*pi*280*t); % Signal to be analysed

plot(X); % plotting the resultant waveform

Figure 1: Resultant Waveform with A = 1 and B = 0.3

Figure 2: Resultant Waveform(zoomed) with A = 1 and B = 0.3

Used MATLAB code for same magnitudes;

t = 0:1.5625e-4:1; % 6400 samples/second sampling frequency

A = 1; % magnitude of 282 Hz component

B = 1; % magnitude of 280 Hz component

X = A*cos(2*pi*282*t) + B*cos(2*pi*280*t); % Signal to be analysed

plot(X); % plotting the resultant waveform

Figure 3: Resultant Waveform with A = 1 and B = 1

Figure 4: Resultant Waveform with(zoomed) A = 1 and B = 1