Chapter 2 – Single Phase Consideration of Series Compensation Issues
Here a single phase model will be considered for the purposes of explanation of the theory and technique that could potentially be used to solve the problem in a single phase model without MOV operation. (i.e. before the MOV or other protective elements operate)
Simple Series Representation of the Fault Circuit
The final result we seek is to know the actual values of series capacitance and series inductance of the line in question if at all possible (and if not, to know if the capacitor was part of the fault loop). Considering two frequencies at which voltage and current measurements are made, there are subsequently two available impedances, one for each frequency.5 This leaves two equations and two unknowns, and as a result, there should be a unique solution to the problem. The model of the line which must be used to solve the problem will neglect series resistance. This is an acceptable assumption since it is possible to separate the real and imaginary parts of the impedances calculated at each frequency and solve using only the imaginary part. One must however ignore the shunt capacitance values, which will likely cause some error. This error is slight and not significant since we only need to have an estimate of the series C value of the line in order to infer the exact value based on the known value(s) of installed C. If the installation has a variable capacitor, then our results should still be accurate enough to make a determination of fault location which is adequate for relaying decisions.
Given this, the resulting single phase model is very simple. It consists only of a series inductor and capacitor which symbolize the lumped impedance and sum of all series capacitance included in the fault loop. A diagram is shown in figure 2.1.1.
Next it would be logical to explore the theory behind the assumption of subsynchronous frequencies in all faults involving a series capacitor
Examination of Presence of Subsynchronous Harmonics in Resulting Waveforms
If we consider the frequency domain representation of our circuit it would appear as shown here with the previous diagram’s inductor’s value replaced by “sL” and the capacitor’s value replaced by “1/sC”.
Figure 2.2.1 – Laplace Domain representation of the single phase test circuit.
Assuming a perfect sinusoidal source of 1 volt, Vsource= s/(s2+w2). Substituting this into equations (1) and (2) above, and performing an inverse Laplace transform on the result yields equations (3) and (4) below.
Thus we can conclude that when conditions are normal, and usual levels of series compensation are applied to a transmission line, the alternate frequency generated will be below 60Hz and thus we will call continue to call it the “subsynchronous frequency”
What is Necessary to Detect the Subsynchronous Component?
As in most computer based relay algorithms, this paper will suggest use of the Fourier Transform in order to measure different frequency components in the voltage and current signals. However, since there is significant interest in frequency components from DC through the ninth harmonic, use of the FFT is recommended instead of the DFT which is normally used in fundamental frequency relaying applications.3 The problem of course lies in detecting the correct magnitudes of these components and the correct frequency of the subsynchronous component given a relatively short data window to work with. Of course if one would like to detect at 30Hz intervals it would be necessary to have two cycles of 60Hz input data. If you would like to examine values at 15Hz intervals you would need 4 cycles of 60Hz data. Let us assume for now that there are six cycles of test data available starting at the instant of fault inception. Therefore, using a standard FFT, a magnitude for each frequency is available at 10 Hz intervals starting at DC or 0Hz. Of course there is no problem to speak of if the subsynchronous frequency in question is either 10, 20, 30, 40, or 50 Hz exactly, but this is highly unlikely! If the frequency falls somewhere in between these values, there is no way to tell from the raw FFT result what the correct magnitude or frequency is. Further, if the value is close to 60Hz, there is a corruption in the 60 Hz component thus destroying the accuracy of its magnitude estimate as well.3
The proposed solution to these problems consists of several steps. First, to accurately detect the magnitude of the 60 Hz component, an estimation of the error coming from the other terms must be made. This estimation is based on a curve fit to the magnitudes present at other frequencies. Once the 60Hz magnitude is properly determined, it should be removed so that the spectrum includes only the subsynchronous frequency component. Finally, the size of the data window is reduced, point by point, in order to “focus” in on the correct value. This is done repeatedly until the stopping point is reached. The stopping point is determined to be reached at that length of data window which produces one single spike of significant magnitude in the spectrum obtained with the reduced window. This process will be more fully explained in chapter three along with the discussion of the algorithm proposed for solving this problem
What is Necessary to Actually Compute L and C Values?
Once the values of voltages and currents have been correctly determined at the fundamental frequency (60 Hz) and at some subsynchronous frequency, then it is a simple matter of solving the equations which describe the model. Recall that the model of the line being used is a simple series L and C. If we consider looking into the system from the terminals of the relay.
Therefore, once values have been computed for the voltages and currents, we take their ratios in order to yield a 60Hz impedance, and a subsynchronous impedance. The imaginary part (since we are neglecting resistance) of these impedances is then put into equations (10) and (11) along with the appropriate frequency values and the series L and C of the line are calculated
Discussion of the Challenges faced in Finding a Solution
Again the limitations of the Fast Fourier Transform must be considered as well as how these limitations will effect the solution. Further, the available means to get around these problems must also be examined. As mentioned previously, when there is frequency content in the signal which does not fall directly on the fundamental or one of its submultiples or harmonics, there is a corruption of all neighboring values in the spectrum obtained using an FFT. This is of particular concern in this application as the solution deals with a signal containing very closely spaced frequency components. For instance, typical values could be a 60Hz fundamental frequency with a 45Hz-55Hz subsynchronous value. In this case, if both signals have a true value of 100, we can see that a 45Hz signal adds or takes away an error of greater than 10% from the magnitude of the 60Hz component. When this is added to error in other steps of the solution, it is not surprising that the results would be very inaccurate.
Even after this problem is solved, there is still the question of how to deal with the problem of determining the correct frequency and magnitude of the subsynchronous component of the signals. Here we must try to detect a component whose frequency may be located at a point of high attenuation on the frequency response plot of the particular FFT we are calculating. Quite a few frequency components that are not present in the actual signal will show up as a result of corruption, and the raw FFT result will be practically worthless for determining the magnitude or frequency of this component
Curve Fitting to Determine Corruption in 60 Hz Component
In order to remove corruption of the fundamental frequency component (60Hz) caused by the subsynchronous component, it is necessary to make an estimation of what the corruption in the fundamental is so that it can be removed. Let’s consider again the spectrum of the 45Hz and 60Hz combined signal shown in figure 3.2.1
Introduction – The Relaying Problems Associated With Series Compensation
Chapter 1 – Overview of Principles of Distance Protection for Transmission Lines
1.1 Commonly Used Methods Available for Distance Protection of Transmission Lines
1.2 Non-Pilot Distance Protection of Transmission Lines
1.3 Pilot Distance Protection of Transmission Lines
1.4 Limits to Fault Location Accuracy
Chapter 2 – Single Phase Consideration of Series Compensation Issues
2.1 Simple Series Representation of the Fault Circuit
2.2 Examination of Presence of Subsynchronous Harmonics in Resulting Waveforms
2.3 What is Necessary to Detect the Subsynchronous Component?
2.4 What is Necessary to Actually Compute L and C Values?
Chapter 3 – Designing an Algorithm – Improved Accuracy for Phasor Computations in the Dual Frequency Post Fault Environment
3.1 Discussion of the Challenges faced in Finding a Solution
3.2 Curve Fitting to Determine Corruption in 60 Hz Component
3.3 Point by Point Reduction of the Data Window
3.4 Flowcharting the possible approaches
3.5 Design of the Most Efficient Algorithm for Relaying Purposes
3.6 Test of the Single Phase Algorithm
Chapter 4 – Application to a Three Phase System
4.1 Differences Between Single Phase and Three Phase Systems
4.2 Three Phase System with Series Capacitance
4.3 Three Phase Algorithm – Flowchart
Chapter 5 – Additional Considerations of Three Phase Systems
5.1 Addition of Mutual Inductance and Shunt Capacitance
5.2 Addition of Metal Oxide Varistors in Shunt with the Series C Element
Chapter 6 – Conclusions
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