Perform Park transformation from three-phase (abc) reference frame to dq0 reference frame
powerlib_extras/Measurements, powerlib_extras/Discrete Measurements
Note: The Transformations section of the Control and Measurements library contains the abc to dq0 block. This is an improved version of the abc_to_dq0 Transformation block. The new block features a mechanism that eliminates duplicate continuous and discrete versions of the same block by basing the block configuration on the simulation mode. If your legacy models contain the abc_to_dq0 Transformation block, they continue to work. However, for best performance, use the abc to dq0 block in your new models. |
The abc_to_dq0 Transformation block computes the direct axis, quadratic axis, and zero sequence quantities in a two-axis rotating reference frame for a three-phase sinusoidal signal. The following transformation is used:
$$\begin{array}{c}{V}_{d}=\frac{2}{3}\left({V}_{a}\mathrm{sin}(\omega t)+{V}_{b}\mathrm{sin}(\omega t-2\pi /3)+{V}_{c}\mathrm{sin}(\omega t+2\pi /3)\right)\\ {V}_{q}=\frac{2}{3}\left({V}_{a}\mathrm{cos}(\omega t)+{V}_{b}\mathrm{cos}(\omega t-2\pi /3)+{V}_{c}\mathrm{cos}(\omega t+2\pi /3)\right)\\ {V}_{0}=\frac{1}{3}\left({V}_{a}+{V}_{b}+{V}_{c}\right),\end{array}$$
where ω = rotation speed (rad/s) of the rotating frame.
The transformation is the same for the case of a three-phase current; you simply replace the V_{a}, V_{b}, V_{c}, V_{d}, V_{q}, and V_{0} variables with the I_{a}, I_{b}, I_{c}, I_{d}, I_{q}, and I_{0} variables.
This transformation is commonly used in three-phase electric machine models, where it is known as a Park transformation [1]. It allows you to eliminate time-varying inductances by referring the stator and rotor quantities to a fixed or rotating reference frame. In the case of a synchronous machine, the stator quantities are referred to the rotor. I_{d} and I_{q} represent the two DC currents flowing in the two equivalent rotor windings (d winding directly on the same axis as the field winding, and q winding on the quadratic axis), producing the same flux as the stator I_{a}, I_{b}, and I_{c} currents.
You can use this block in a control system to measure the positive-sequence component V_{1} of a set of three-phase voltages or currents. The V_{d} and V_{q} (or I_{d} and I_{q}) then represent the rectangular coordinates of the positive-sequence component.
You can use the Math Function block and the Trigonometric Function block to obtain the modulus and angle of V_{1}:
$$\begin{array}{c}\left|{V}_{1}\right|=\sqrt{{V}_{q}^{2}+{V}_{d}^{2}}\\ \angle {V}_{1}=\mathrm{atan}2\left({V}_{q}/{V}_{d}\right).\end{array}$$
This measurement system does not introduce any delay, but, unlike the Fourier analysis done in the Sequence Analyzer block, it is sensitive to harmonics and imbalances.
abc
Connect to the first input the vectorized sinusoidal phase signal to be converted [phase A phase B phase C].
sin_cos
Connect to the second input a vectorized signal containing the [sin(ωt) cos(ωt)] values, where ω is the rotation speed of the reference frame.
dq0
The output is a vectorized signal containing the three sequence
components [d q o], in the same units as the abc
input
signal.
The power_3phsignaldq
example
uses a Three-Phase Programmable Generator block to generate a 1 pu,
15 degrees positive sequence voltage. At 0.05 second, the positive
sequence voltage is increased to 1.5 pu and at 0.1 second an imbalance
is introduced by the addition of a 0.3 pu negative sequence component
with a phase of −30 degrees. The magnitude and phase of the
positive-sequence component are evaluated in two different ways:
Sequence calculation of phasors using Fourier analysis
abc-to-dq0 transformation using the abc to dq0 block (an improved version of the abc_to_dq0 Transformation block).
Start the simulation and observe the instantaneous signals Vabc (Scope1), the signals returned by the Sequence Analyzer (Scope2), and the abc to dq0 (Scope3) blocks.
The Sequence Analyzer, which uses Fourier analysis, is immune to harmonics and imbalance. However, its response to a step is a one-cycle ramp. The abc-to-dq0 transformation is instantaneous. However, an imbalance produces a ripple at the V1 and Phi1 outputs.
[1] Krause, P. C. Analysis of Electric Machinery. New York: McGraw-Hill, 1994, p.135.