I have been working with multi-machine power systems, and I have been struggling with the transformation between the *dq0* and *DQ0* frames. I will first introduce my understanding of these frames, and then ask my question about it.
In power systems, the *dq0* transformation, also called Park's transformation, consists in converting from the static *abc* frame to a rotating *dq0* frame. This transformation slightly differs from author to author depending on the choice of the leading and lagging components.
When considering multi-machine networks, usually all the devices are converted to a same rotating frame with rotation speed equal to the synchronous speed \$\omega_s\$. For this reason, this rotating frame is called synchronously rotating frame, and it is represented by *DQ0* coordinates (capital letters).
However, for transforming from *dq0* to *DQ0* and vice-versa, the transformation matrix also differ from author to author depending on the choice of the leading and lagging components. For instance, when the *q* axis and *D* axis are considered initially aligned, the following transformation matrix is used for converting from *dq0* to *DQ0*, where \$\delta\$ is the angle between the *D* and *q* axes.
$$
\begin{equation*}
T_{dqos}T_{dqoi}^{-1} =
\begin{bmatrix}
\sin{\delta} & \cos{\delta}\\ -\cos{\delta} & \sin{\delta}
\end{bmatrix}
\end{equation*}
$$
On the other hand, when the *d* axis is considered initially aligned with the *D* axis, the following transformation matrix is used (here \$\delta^{\star}\$ is the angle between *d* and *D* axes).
$$
\begin{equation}
T_{dqos}^{\star}T_{dqoi}^{\star-1} =
\begin{bmatrix}
\cos{\delta^{\star}} & -\sin{\delta^{\star}}\\ \sin{\delta^{\star}} & \cos{\delta^{\star}}
\end{bmatrix}
\end{equation}
$$
Therefore, my question is: how should I define the leading and lagging components? In the case of a multi-machine power system, should the *dq0* frames of all machines be considered either lagging or leading the *DQ0* frame?
I really appreciate any kind of help!
You should analyse all the conditions of the authors and see which ones are common. If the choice is rigid (no intermediate cases) you will be left with a conditional or switch with the number of cases.
Hello,
First of all, don't try to imitate other authors. instead, try to understand how they got their results. Then try to reproduce them using tools like matlab scripts to make it easier. If you understand the procedure and the logic behind it, you can manipulate it as you want according to your data and situations, without the need to closely follow the steps of others.
I invite you to read these documents to understand how these transformations work
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