## 2 Basic Concepts of DQ0-Based Models
### Comparison of Time-Varying Phasors and DQ0 Models  
This section shows that \(dq0\) models are a natural extension of time-varying phasor models, and explains the underlying link between these two approaches [11]. To present the quasi-static model and \(dq0\)-based model using similar terminology, we first need to establish the relationships between \(dq0\) signals and time-varying phasors. Assume a unit with balanced three-phase AC voltages given by  
Figure 6: Example—linear network with mutual ground resistances. \[v_{a}(t) =A(t)\cos\left(\omega_{s}t+\psi(t)\right), \tag{44}\] \[v_{b}(t) =A(t)\cos\left(\omega_{s}t+\psi(t)-\frac{2\pi}{3}\right),\] \[v_{c}(t) =A(t)\cos\left(\omega_{s}t+\psi(t)+\frac{2\pi}{3}\right).\]  
Assuming that variations in the magnitude \(A(t)\) and phase \(\psi(t)\) are slow in comparison to the nominal system frequency \(\omega_{s}\), these voltages may be represented by the time-varying phasor  
\[V(t)=\frac{A(t)}{\sqrt{2}}e^{i\psi(t)}. \tag{45}\]  
In addition, based on (2) with \(\theta(t)=\omega_{s}t\), and using trigonometric identity \(\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)\),  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix}  = \begin{bmatrix}A(t)\cos\left(\omega_{s}t+\psi(t)\right)\\ A(t)\cos\left(\omega_{s}t+\psi(t)-\frac{2\pi}{3}\right)\\ A(t)\cos\left(\omega_{s}t+\psi(t)+\frac{2\pi}{3}\right)\end{bmatrix}  \tag{46}\] \[= \begin{bmatrix}\cos\left(\omega_{s}t\right)&-\sin\left(\omega_{ s}t\right)&1\\ \cos\left(\omega_{s}t-\frac{2\pi}{3}\right)&-\sin\left(\omega_{s}t-\frac{2\pi}{3 }\right)&1\\ \cos\left(\omega_{s}t+\frac{2\pi}{3}\right)&-\sin\left(\omega_{s}t+\frac{2\pi} {3}\right)&1\end{bmatrix}\begin{bmatrix}A(t)\cos\left(\psi(t)\right)\\ A(t)\sin\left(\psi(t)\right)\\ 0\end{bmatrix} =T_{\omega_{s}t}^{-1} \begin{bmatrix}v_{d}\\ v_{q}\\ v_{0}\end{bmatrix} ,\]  
and therefore  
\[v_{d}(t) =A(t)\cos(\psi(t)), \tag{47}\] \[v_{q}(t) =A(t)\sin(\psi(t)),\] \[v_{0}(t) =0.\]  
Note that in this paper quasi-static systems are assumed to be balanced, and therefore \(v_{0}(t)=0\). Following (45) and (47), if the system is quasi-static, time-varying phasors relate to \(dq0\) quantities by  
\[V(t)=\frac{A(t)}{\sqrt{2}}e^{i\psi(t)}=\frac{A(t)}{\sqrt{2}}\left(\cos(\psi(t ))+j\sin(\psi(t))\right)=\frac{1}{\sqrt{2}}\left(v_{d}(t)+jv_{q}(t)\right), \tag{48}\]  
or alternatively,  
\[v_{d}(t)=\sqrt{2}\operatorname{Re}\{V(t)\},\quad v_{q}(t)=\sqrt{2} \operatorname{Im}\{V(t)\}. \tag{49}\]  
Using this result, in a quasi-static system, the powers, amplitude and phase relates to \(dq\) quantities by  
\[P(t) =\operatorname{Re}\left\{V(t)I^{*}(t)\right\}=\frac{1}{2}\left(v _{d}(t)i_{d}(t)+v_{q}(t)i_{q}(t)\right), \tag{50}\] \[Q(t) =\operatorname{Im}\left\{V(t)I^{*}(t)\right\}=\frac{1}{2}\left(v _{q}(t)i_{d}(t)-v_{d}(t)i_{q}(t)\right),\] \[|V(t)|^{2} =\operatorname{Re}\left\{V(t)\right\}^{2}+\operatorname{Im}\left\{ V(t)\right\}^{2}=\frac{1}{2}\left(v_{d}^{2}(t)+v_{q}^{2}(t)\right),\] \[\angle V(t) =\operatorname{atan2}(v_{q},v_{d}).\]  
Assume now a symmetrically configured linear network with \(N\) buses. The network is described by the admittance matrix \(Y(s)\). In a quasi-static system, the network equations are given by  
\[I=Y(j\omega_{s})V. \tag{51}\]  
This can be rewritten as \[ \begin{split}\operatorname{Re}\{I\}&=\operatorname{Re}\{Y(j \omega_{s})\}\operatorname{Re}\{V\}-\operatorname{Im}\{Y(j\omega_{s})\} \operatorname{Im}\{V\},\\ \operatorname{Im}\{I\}&=\operatorname{Re}\{Y(j \omega_{s})\}\operatorname{Im}\{V\}+\operatorname{Im}\{Y(j\omega_{s})\} \operatorname{Re}\{V\},\end{split}  \tag{52}\]  
or equivalently  
\[ \begin{bmatrix}\operatorname{Re}\{I\}\\ \operatorname{Im}\{I\}\end{bmatrix} = \begin{bmatrix}\operatorname{Re}\{Y(j \omega_{s})\}&-\operatorname{Im}\{Y(j\omega_{s})\}\\ \operatorname{Im}\{Y(j\omega_{s})\}&\operatorname{Re}\{Y(j\omega_{s})\} \end{bmatrix}\begin{bmatrix}\operatorname{Re}\{V\}\\ \operatorname{Im}\{V\}\end{bmatrix} . \tag{53}\]  
This expression applies only to a quasi-static network.