How do you model harmonic injection with Thevenin or Norton equivalents?
The Thevenin and Norton equivalents are characterized by their open-circuit voltage and short circuit current. The inner voltage source of the Thevenin equivalent equals the open circuit voltage, while the inner current source of the Norton equivalent equals the short-circuit current.
Thevenin and Norton equivalents are equivalent to each other. Thus, the admittance G(f) + j B(f) of the Norton equivalent (G and B in parallel) can be replaced by the impedance R(f) + j X(f) taken from the Thevenin equivalent (R and X in series).
In the resulting Norton equivalent, the impedance (series connection of R and X) is in parallel to the inner current source of the Norton equivalent. This is the modelling approach for the Norton equivalents in PowerFactory. The relations between the quantities of the admittance and the impedance are as follows:
R(f) = G(f) / [ G(f)2 + B(f)2 ]
X(f) = -B(f) / [ G(f)2 + B(f)2 ]
The procedure to model harmonic injections with the Norton equivalent in PowerFactory is as follows:
1. Enter the frequency characteristics for real and imaginary components of the equivalent’s impedance in the positive and negative sequence (r1h, x1h, r2h and x2h) using vector characteristic *.ChaVec.
- In order to set up the vector characteristic, first the frequency range has to be defined (TriFreq). Usually it is configured to range from second order to 40th or 50th order harmonic.
- The R(f) and X(f) characteristic are entered in per unit. The per unit values are derived by dividing the values R(f) and X(f) by the base impedance of the element:
r(f)[p.u.] = R(f)[Ohm] / Zbase[Ohm]
x(f)[p.u.] = X(f)[Ohm] / Zbase[Ohm]
- The base impedance is derived from the nominal voltage of the terminal and the rated power of the element: Zbase = Un2/Sr.
- It is recommended to use the characteristics in an absolute manner, in order to enter the characteristics independently from equivalent’s impedance at fundamental frequency.
2. For the Harmonic Source, the currents of the Norton equivalent are to be defined in a Harmonic Current Source type (TypHmccur) as percentage values.