## 2 Basic Concepts of DQ0-Based Models
### Basic Definitions  
The \(dq0\) transformation and its inverse are defined as in [28]  
\[T_{\theta}=\frac{2}{3}\left[\begin{matrix}\cos\left(\theta\right)&\cos\left( \theta-\frac{2\pi}{3}\right)&\cos\left(\theta+\frac{2\pi}{3}\right)\\ -\sin\left(\theta\right)&-\sin\left(\theta-\frac{2\pi}{3}\right)&-\sin\left( \theta+\frac{2\pi}{3}\right)\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{matrix}\right] \tag{1}\]  
and  
\[T_{\theta}^{-1}=\begin{bmatrix}\cos\left(\theta\right)&-\sin\left(\theta \right)&1\\ \cos\left(\theta-\frac{2\pi}{3}\right)&-\sin\left(\theta-\frac{2\pi}{3}\right) &1\\ \cos\left(\theta+\frac{2\pi}{3}\right)&-\sin\left(\theta+\frac{2\pi}{3}\right) &1\end{bmatrix}. \tag{2}\]  
Note that several variations of (1) are available in the literature [20; 29; 2]. The angle \(\theta\) is called here the _reference angle_ or _reference phase_, and we will discuss different selections of it in the following  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Operating Point** & **Small-Signal** & **High Frequencies** & **Non-Symmetric Networks** \\ \hline time-varying & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ phasors & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \(abc\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \(dq0\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of approaches for dynamic modeling.  
sections. Direct multiplication of these matrices reveals that \(T_{\theta}T_{\theta}^{-1}=T_{\theta}^{-1}T_{\theta}=I_{3\times 3}\). Other useful identities are provided in Appendix A.  
The \(dq0\) transformation maps three-phase signals in the \(abc\) reference frame to new quantities in a rotating \(dq0\) reference frame. Denote \(x_{abc}=[x_{a},x_{b},x_{c}]^{\mathsf{T}}\) and \(x_{dq0}=[x_{d},x_{q},x_{0}]^{\mathsf{T}}\), then \(x_{dq0}=T_{\theta}x_{abc}\) and \(x_{abc}=T_{\theta}^{-1}x_{dq0}\), where the subscripts \(d\), \(q\), and \(0\) represent the direct, quadrature, and zero components. A fundamental property of the \(dq0\) transformation is that it maps symmetric AC signals to constants. For instance, consider a three-phase voltage source modeled as  
\[\begin{split} v_{a}&=A\cos\left(\omega_{s}t\right)+B,\\ v_{b}&=A\cos\left(\omega_{s}t-\frac{2\pi}{3} \right)+B,\\ v_{c}&=A\cos\left(\omega_{s}t+\frac{2\pi}{3} \right)+B.\end{split} \tag{3}\]  
Using the inverse transformation \(T_{\theta}^{-1}\) with \(\theta=\omega_{s}t\), it is easy to see that  
\[\begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix}=\begin{bmatrix}\cos\left(\omega_{s}t\right)&-\sin\left( \omega_{s}t\right)&1\\ \cos\left(\omega_{s}t-\frac{2\pi}{3}\right)&-\sin\left(\omega_{s}t-\frac{2\pi} {3}\right)&1\\ \cos\left(\omega_{s}t+\frac{2\pi}{3}\right)&-\sin\left(\omega_{s}t+\frac{2\pi} {3}\right)&1\end{bmatrix}\begin{bmatrix}A\\ 0\\ B\end{bmatrix}, \tag{4}\]  
and therefore \(v_{d}=A\), \(v_{q}=0\), \(v_{0}=B\). The sinusoidal signals in the \(abc\) reference frame are mapped to constant signals in the \(dq0\) reference frame (see Figure 1).