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Let's I have three phase voltage grid which is unbalanced. The phase voltages are sampled by the adc and I would like to determine in the C language software the positive sequence, negative sequence and zero sequence (i.e. symmetrical components). My first idea how to do that was to use the transformation invented by the C.L. Fortescue i.e.

\begin{eqnarray} \left[ \begin{array}{c} \widehat{U}_{A1} \\ \widehat{U}_{A2} \\ \widehat{U}_{A0} \end{array} \right] = \frac{1}{3} \left[ \begin{array}{ccc} 1 & \widehat{a} & \widehat{a}^2 \\ 1 &\widehat{a}^2 & \widehat{a} \\ 1 & 1 & 1 \\ \end{array} \right] \cdot \left[ \begin{array}{c} \widehat{U}_{A} \\ \widehat{U}_{B} \\ \widehat{U}_{C} \end{array} \right] \end{eqnarray}

where the \$\widehat{U}_{A}\$, \$\widehat{U}_{B}\$ and \$\widehat{U}_{C}\$ are the phasors of the individual phase voltages of the unbalanced three phase voltage grid, \$\widehat{U}_{A1}\$, \$\widehat{U}_{A2}\$, \$\widehat{U}_{A0}\$ are the phasors of the voltage in phase A for the positive, negative and zero sequence and \$\hat{a} = e^{j\cdot\frac{2\pi}{3}}\$.

The obstacle which I see in this method is that the transformation is defined for the phasors and not for the time domain signals. Due to that I have been looking for another method. Does anybody know another method for decomposition of the three phase unbalanced grid into the symmetrical components which is applicable directly in the time domain?

Steve
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    How to implement it in C is another story - first come up with an algorithm. An algorithm not necessarily being the same thing as a mathematical formula. Algorithms describe how things work, math formulas describe relations. – Lundin Apr 26 '22 at 08:32

2 Answers2

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Yes, you can go straight from waveform sampled values to symmetrical component waveforms. Below are the formulas given in one of the old ABB REL356 instruction manual - it is valid when data samples are collected at 12 samples/cycle. All you need are the present 3 samples (\$k\$) and the 3 samples taken at previous sample interval (\$k-1\$) of each of the three phase-quantities (currents or voltages).

$$i_{a1}(k)=\frac{1}{3}[i_a(k)+i_b(k)-\sqrt{3}i_b(k-1)-2i_c(k)+\sqrt{3}i_c(k-1)]$$

$$i_{a2}(k)=\frac{1}{3}[i_a(k)-2i_b(k)+\sqrt{3}i_b(k-1)+i_c(k)-\sqrt{3}i_c(k-1)]$$ and of course, the easy one $$i_{a0}=\frac{1}{3}[i_a(k)+i_b(k)+i_c(k)]$$

I worked on this for a long while years ago after seeing these formulas in the REL356 manual and I derived formulas to find the constants to scale the sampled values and calculate the symmetrical component waveforms at sample frequencies other than 12/cycle. If 12 samples/cycle is too fast for your processor let me know.

Note, this approach assumes fundamental frequency sinusoids. If you need to worry about harmonic energy then you need to take the DFT approach and also use frequency tracking depending on how accurate you want to be.

relayman357
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If there are no harmonics, I would first look for each group of samples (voltages UA, UB, UC) for the values of the RMS voltages and the associated phases...
And then, I would apply Fortescue.

This can be done (as in single-phase) by DFT calculation at the mains frequency (assumed to be correct: 50 or 60 Hz, if not "correction" must be done ...).
Just take the right number of samples (example: 21, @50 Hz ... each 1 ms) corresponding to a period and do the calculation with reference waves (0°, 120° and 240°).
Just also remember that samples (all voltages) must be taken "simultaneously".
If not, apply a "phase" correction for the "delay" of sampling ...

For info, see my answer at this post:
Is there a way to measure the Phase Shift between the input and the output voltages related by the equation: \$V_o = V_i(a+jb).\$?

Antonio51
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