## 2 Basic Concepts of DQ0-Based Models
### Modeling General Linear Networks  
This section presents a relatively simple method to develop \(dq0\) models based on prior knowledge of the system dynamics in the \(abc\) reference frame [23]. Let \(x\) represent the state vector of the system in the \(abc\) reference frame, and use the compact notation \(x_{abc}=[x_{a,1},x_{b,1},x_{c,1},\ldots,x_{a,m},x_{b,m},x_{c,m}]^{\mathsf{T}}\). Define the \(abc\) model  
\[\frac{\mathrm{d}}{\mathrm{d}t}x_{abc}=Ax_{abc}+Bu, \tag{29}\]  
and the composite \(dq0\) transformation and its inverse as  
\[\Lambda_{\theta}= \begin{bmatrix}T_{\theta}&&0\\ &\ddots&\\ 0&&T_{\theta}\end{bmatrix} ,\quad\Lambda_{\theta}^{-1}= \begin{bmatrix}T_{ \theta}^{-1}&&0\\ &\ddots&\\ 0&&T_{\theta}^{-1}\end{bmatrix} , \tag{30}\]  
such that \(x_{dq0}=\Lambda_{\theta}x_{abc}\), \(x_{abc}=\Lambda_{\theta}^{-1}x_{dq0}\). Substitute these definitions into (29) to get  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\Lambda_{\theta}^{-1}\right)x_{dq0}+\Lambda _{\theta}^{-1}\frac{\mathrm{d}}{\mathrm{d}t}x_{dq0}=A\Lambda_{\theta}^{-1}x_{ dq0}+Bu. \tag{31}\]  
Derivative of the inverse \(dq0\) transformation is given by \(\frac{\mathrm{d}}{\mathrm{d}t}(\Lambda_{\theta}^{-1})=-\Lambda_{\theta}^{-1}W_{c}\), where  
\[W_{c}= \begin{bmatrix}\mathcal{W}&&0\\ &\ddots&\\ 0&&\mathcal{W}\end{bmatrix} ,\quad\mathcal{W}= \begin{bmatrix}0&\frac{\mathrm{d} }{\mathrm{d}t}\theta&0\\ -\frac{\mathrm{d}}{\mathrm{d}t}\theta&0&0\\ 0&0&0\end{bmatrix} . \tag{32}\]  
Substituting this expression into (31) results in  
\[\frac{\mathrm{d}}{\mathrm{d}t}x_{dq0}=\left(\Lambda_{\theta}A\Lambda_{\theta} ^{-1}+W_{c}\right)x_{dq0}+\Lambda_{\theta}Bu. \tag{33}\]  
This equation describes the dynamics of the system based on \(dq0\) quantities. Note that in general this model depends on \(\theta(t)\), and is not time-invariant. However, for the special case of symmetrically configured networks, it is typically true that \(\Lambda_{\theta}A=A\Lambda_{\theta}\), and therefore Equation (33) takes the form \[\frac{\mathrm{d}}{\mathrm{d}t}x_{dq0}=\left(A+W_{c}\right)x_{dq0}+\Lambda_{\theta}Bu, \tag{34}\]  
which often results in a linear and time-invariant model. As an example, consider the circuit in Figure 5.  
The network dynamics in the \(abc\) reference frame is  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} =-\frac{R}{L} \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} +\frac{1}{L} \begin{bmatrix}v_{a,1}-v_{a,2}\\ v_{b,1}-v_{b,2}\\ v_{c,1}-v_{c,2}\end{bmatrix} , \tag{35}\]  
In this example \(\Lambda_{\theta}=T_{\theta}\). As a result, the dynamic model in the \(dq0\) reference frame is  
\[\frac{\mathrm{d}}{\mathrm{d}t}i_{dq0}=\left(A+\mathcal{W}\right)i_{dq0}+ \Lambda_{\theta}Bu, \tag{36}\]  
which yields  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} =\left(A+\mathcal{W}\right) \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +T_{\theta}B \begin{bmatrix}v_{a,1}-v_{a,2}\\ v_{b,1}-v_{b,2}\\ v_{c,1}-v_{c,2}\end{bmatrix} , \tag{37}\]  
or  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} =\left(-\frac{R}{L} \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix} + \begin{bmatrix}0&\frac{\mathrm{d}}{\mathrm{d}t}\theta&0\\ -\frac{\mathrm{d}}{\mathrm{d}t}\theta&0&0\\ 0&0&0\end{bmatrix} \right) \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +\frac{1}{L}T_{\theta} \begin{bmatrix}v_{a,1}-v_{a,2}\\ v_{b,1}-v_{b,2}\\ v_{c,1}-v_{c,2}\end{bmatrix} . \tag{38}\]  
This last equation can be written as  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} = \begin{bmatrix}-\frac{R}{L}&\frac{\mathrm{d}}{\mathrm{d}t} \theta&0\\ -\frac{\mathrm{d}}{\mathrm{d}t}\theta&-\frac{R}{L}&0\\ 0&0&-\frac{R}{L}\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +\frac{1}{L} \begin{bmatrix}v_{d,1}-v_{d,2}\\ v_{q,1}-v_{q,2}\\ v_{0,1}-v_{0,2}\end{bmatrix}